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OUTLINE SCHAUM'S OF THEORY AND PROBLEMS OF COLLEGE CHEMISTRY Eighth Edition JEROME L. ROSENBERG, Ph.D. Professor of Biological Sciences, Emeritus University of Pittsburgh LAWRENCE M. EPSTEIN, Ph.D. Associate Professor of Chemistry, Emeritus University of Pittsburgh SCHAUM'S OUTLINE SERIES McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogotci Caracas London Madrid Mexico City Milan .Montreal New Dehli San Juan Singapore Sydney Tokyo Toronto Lisbon JEROME L. ROSENBERG did his graduate work at Columbia University in physical chemistry, receiving his M.A. in 1944 and his Ph.D. in 1948. His research activities at several institutions resulted in many papers in photosynthesis, general photochemistry, and molecular electronic structure. Formerly Professor of Chemistry, Dean of the Faculty of Arts and Sciences, and Vice Provost at the University of Pittsburgh, he is now Research Integrity Officer and Professor Emeritus of Biological Sciences there. Dr. Rosenberg has been associated with the preparation of Schaum's Outline of College Chemistry since the third edition, published in 1949. LAWRENCE M. EPSTEIN started his career as a chemical engineer, then earned his M.S. in 1952 and Ph.D. in 1955 from Polytechnic University in the field of physical chemistry. He did research in the fields of radiation chemistry and in Mossbauer effect spectroscopy at Westinghouse Research Laboratories, and then served as Associate Professor and supervisor of the General Chemistry program at the University of Pittsburgh until he retired in 1986. Schaum's Outline of Theory and Problems of COLLEGE CHEMISTRY Copyright 0 1997, 1990, 1980 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publishers. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 1 0 9 I S B N 0-03-053709-3 Sponsoring Editor: Arthur Biderman Production Supervisor: Pamela Pelton Editing Supervisor: Maureen B. Walker Library of Congress Cataloging-in-Publication Data Rosenberg, Jerome Laib, date Schaum's outline of theory and problems of college chemistry/ Jerome L. Rosenberg, Lawrence M. Epstein.-8th ed. p. cm.-(Schaum's outline series) Includes index. ISBN 0-07-053709-7 (pbk.) 1. Chemistry-Problems, exercises, etc. 2. Chemistryautlines, syllabi, etc. I. Epstein, Lawrence M. 11. Title. QD4 1.R67 1997 96-46610 540'.7&dc2 1 CIP McGraw-Hill of TheMcGrawHiU Companies A Division v This book is designed to help the student of college chemistry by summarizing the chemical principles of each topic and relating the solution of quantitative problems to those fundamentals. Although the book is not intended to replace a textbook, its solved problems, with complete and detailed solutions, do cover most of the subject matter of a first course in college chemistry. The student is referred to one of the many standard General Chemistry textbooks for such matters as full treatment of nomenclature, descriptive chemistry of the elements, and more extensive exposition and illustration of principles. Both the solved and the supplementary problems are arranged to allow a progression in difficulty within each topic. Several important features had been introduced into the sixth edition, notably the kinetic theory of gases, a more formal treatment of thermochemistry, a modern treatment of atomic properties and chemical bonding, and a chapter on chemical kinetics. In the seventh edition the early chapters were revised to conform more closely to the methods used in current textbooks to introduce calculational skills to the beginning student. Some changes in notation were made, and the usage of SI units was expanded. An attempt was made to increase the variety of stoichiometry problems, especially in the chapters on gases and solutions, while eliminating some of the very complex problems that arise in gaseous and aqueous equilibria. In the treatment of chemical bonding the subject of molecular orbitals was deemphasized in favor of VSEPR theory. A new chapter on Organic Chemistry and Biochemistry was added, conforming to the trend in current texts. In this eighth edition we now carefully conform to the language and style of the currently most-used textbooks, for example, using the term "molar mass" broadly, and eliminating "molecular weight" and the like. At least 1 5 O / O of the problems in each chapter are new, and some old ones were dropped, so that the problems better reflect the practical situations of the laboratory, industry, and the environment. The use of SI units has been expanded further, but liter and atmosphere are retained where appropriate. JEROMEL. ROSENBERG LAWRENCE M. EPSTEIN This page intentionally left blank Contents Chapter 1 QUANTITIES AND UNITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction. Systems of measurement. International system of units. Temperature. Other temperature scales. Use and misuse of units. Factor-label method. Estimation of numerical answers. 1 Chapter Chapter 2 ATOMIC AND MOLECULAR MASS; MOLAR MASS. . . . . . . . . . . . . Atoms. Nuclei. Relative atomic masses. Mole. Symbols, formulas, molar masses. 17 27 3 Empirical formula from composition. Composition from formula. Nonstoichiometric factors. Nuclidic molecular masses and chemical formulas. FORMULAS AND COMPOSITION CALCULATIONS . . . . . . . . . . . . Chapter 4 CALCULATIONS FROM CHEMICAL EQUATIONS. . . . . . . . . . . . . . Introduction. Molecular relations from equations. Mass relations from equations. Limiting reactant. Types of chemical reactions. 45 Chapter 5 MEASUREMENT OF GASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas volumes. Pressure. Standard atmospheric pressure. Pressure gages, Standard conditions (S.T.P.). Gas laws. Boyle's law. Charles' law. GayLussac's law. Combined gas law. Density of an ideal gas. Dalton's law of partial pressures. Collecting gases over a liquid. Deviations from ideal behavior. 61 Chapter 6 THE IDEAL GAS LAW AND KINETIC THElORY . . . . . . . . . . . . . . . . Avogadro's hypothesis. Molar volume. Ideal gas law. Gas volume relations from equations. Gas stoichiometry involving mass. Basic assumptions of the kinetic theory of gases. Predictions of the kinetic theory. 77 Chapter 7 Heat. Heat capacity. Calorimetry. Energy and enthalpy. Enthalpy changes for various processes. Rules of thermochemistry. Scope of thermochemical calculations. THERMOCHEMISTRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Chapter 8 ATOMIC STRUCTURE AND THE PERIODIC LAW. . . . . . . . . . . . . . 107 Absorption and emission of light. Interaction of light with matter. Particles and waves. The Pauli principle and the periodic law. Electron configurations. Atomic radii. Ionization energies. Electron affinity. Magnetic properties. Chapter 9 CHEMICAL BONDING AND MOLECULAR STRUCTURE. . . . . . . . . 123 Ionic compounds. Covalence. Valence-bond representation. Molecular-orbital representation. 7r bonding and multicenter T bonds. Shapes of molecules. VSEPR bond angles. Coordination compounds. Isomerism. Bonding in metals. Chapter 10 Chapter 11 Crystals. Crystal forces. Ionic radii. Forces in liquids. SOLIDS AND LIQUIDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OXIDATION-REDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 175 Oxidation-reduction reactions. Oxidation state. Balancing oxidation-reduction equations. Ionic notation for equations. V vi CONTENTS Chapter 12 CONCENTRATIONS OF SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . Solute and solvent. Concentrations expressed in physical units. Concentrations expressed in chemical units. Comparison of the concentration scales. Summary of concentration units. Dilution problems. 189 Chapter 13 Chapter REACTIONS INVOLVING STANDARD SOLUTIONS. . . . . . . . . . . . . 205 Advantages of volumetric standard solutions. Solution stoichiometry. 14 Introduction. Vapor pressure. Freezing-point lowering, A Tr. Boiling-point elevation, AT,. Osmotic pressure. Deviations from laws of dilute solutions. Solutions of gases in liquids. Law of distribution. PROPERTIES OF SOLUTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Chapter 15 ORGANIC CHEMISTRY AND BIOCHEMISTRY. . . . . . . . . . . . . . . . . 227 Introduction. Nomenclature. reactions. Biochemistry. Isomerism. Functional groups. Properties and Chapter 16 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM . . . . . . . . . . 243 The first law. The second law. The third law. Standard states and reference tables. Chemical equilibrium. The equilibrium constant. Le Chatelier's principle. Chapter 17 ACIDS AND BASES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acids and bases. Ionization of water. Hydrolysis. tors. Weak polyprotic (polybasic) acids. Titration. Buffer solutions and indica- 265 Chapter 18 COMPLEX IONS; PRECIPITATES . . . . . . . . . . . . . . . . . . . . . . . . . . . Coordination complexes. precipitation. Solubility product. Applications of solubility product to 299 Chapter 19 ELECTROCHEMISTRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical units. Faraday's laws of electrolysis. Galvanic cells. Standard half-cell potentials. Combination of couples. Free energy, nonstandard potentials, and the direction of oxidation-reduction reactions. Use of Eo'-values. 315 Chapter 20 RATES OF REACTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate constants and the order of reactions. reactions. Energy of activation. Mechanism of 337 Chapter 21 Appendix A Appendix B Appendix C Appendix D NUCLEAR PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental particles. Binding energies. Nuclear equations. Radiochemistry. 353 365 368 371 377 EXPONENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SIGNIFICANT FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LOGARITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FOUR-PLACE COMMON LOGARITHMS . . . . . . . . . . . . . . . . . . . . . INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 TABLE OF ATOMIC MASSES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 NUCLIDIC MASSES OF SELECTED RADIONUCLIDES . . . . . . . . . . . . . . . . . . . . . . 386 PERIODIC CHART OF THE ELEMENTS . . . . . . . . . . . . . . . . . . . . . . . .Inside front cover Chapter 1 Quantities and Units INTRODUCTION Most of the measurements and calculations in chemistry and physics are concerned with different kinds of quantities, e.g., length, velocity, volume, mass, energy. Every measurement includes both a number and a unit. The unit simultaneously identifies the kind of dimension and the magnitude of the reference quantities used as a basis for comparison. Many units are commonly used for the dimension of length, e.g., inch, yard, mile, centimeter, kilometer. The number obviously indicates how many of the reference units are contained in the quantity being measured. Thus the statement that the length of a room is 20 feet means that the length of the room is 20 times the: length of the foot, which in this case is the unit of length chosen for comparison. Although 20 feet has the dimension of length, 20 is a pure number and is dimensionless, being the ratio of two lengths, that of the room and that of the reference foot. It will be assumed that readers are familiar with the use of exponents, particularly powers-of-ten notation, and with the rules for significant figures. If not, Appendices A and B should be studied in conjunction with Chapter 1. SYSTEMS OF MEASUREMENT Dimensional calculations are greatly simplified if the unit for each kind of measure is expressed in terms of the units of selected reference dimensions. The three independent reference dimensions for mechanics are length, mass, and time. As examples of relating other quantities to the reference dimensions, the unit of speed is defined as unit length per unit time, the: unit of volume is the cube of the unit of length, etc. Other reference dimensions, such as those used to express electrical and thermal phenomena, will be introduced later. There are several systems of units still in use in the English-speaking nations, so that one must occasionally make calculations to convert values from one system to another, e.g., inches to centimeters, or pounds to kilograms. INTERNATIONAL SYSTEM OF UNITS Considerable progress is being made in the acceptance of a common international system of reference units within the world scientific community. This system, known as SI from the French name, Systeme International d'Unites, has been adopted by many international bodies, including the International Union of Pure and Applied Chemistry. In SI, the reference units for length, mass, and time are the meter, kilogram, and second, with the symbols m, kg, and s, respectively. To express quantities much larger or smaller than the standard units, use may be made of multiples or submultiples of these units, defined by applying as multipliers of these units certain recommended powers of ten, listed in Table 1-1. The multiplier abbreviation ILSto precede the symbol of the base unit without any space or punctuation. Thus, picosecond (10-l2 s) is ps, and kilometer (103 m) is km. Since for historical reasons the SI reference unit for mass, kilogram, already has a prefix, multiples for mass should be derived by applying the multiplier to the unit gram rather than to the kilogram. Thus 10-9 kg is a microgram (10-6 g), abbreviated pg. Compound units can be derived by applying algebraic operations to the simple units. EXAMPLE 1 The unit for volume in SI is the cubic meter (m3), since: Volume = length x length x length := m x m x m = m3 1 2 QUANTITIES AND UNITS [CHAP. 1 Table 1-1. Multiples and Submultiples for Units Prefix deci centi millli micro nano pico femto atto Abbreviation d C Multiplier Prefix deka hecto kilo mega gigs tera Abbreviation da h k M G T P E Multiplier 10 1o2 1o3 1o6 m P n P f a peta exa 109 10l2 1015 1olS The unit for speed is the unit for length (distance) divided by the unit for time, or the meter per second (m/s), since distance m Speed = ____ = time s The unit for density is the unit for mass divided by the unit for volume, or the kilogram per cubic meter (kg/m3), since mass - kg Density = - volume m3 Table 1-2. Some SI and Non-SI Units Physical Quantity Length Unit name angstrom inch square meter (SI) cubic meter (SI) liter cubic centimeter atomic mass unit (dalton) pound kilogram per cubic meter (SI) gram per milliliter, or gram per cubic centimeter Force Pressure newton (SI) pascal (SI) bar atmosphere torr (millimeter of mercury) Unit Symbol Definition 10-I' m 2.54 x 10-2 m exactly A Area Volume I I in m2 I dm3, 10-3 m3 Mass I Density U lb 1.66054 x 10-27 kg 0.453 592 37 kg g / m L or g/cm3 N Pa bar a tm torr (mm Hg) kg . m/s2 ~~~~~ N/m2 105 Pa 101 325 Pa atm/760 or 133.32 Pa CHAP. 11 QUANTITIES AND UNITS 3 Symbols for compound units may be expressed in any of the following formats. 1. Multiple of units. Example: kilogram second. (a) Dot kg - s (b) Spacing kg s (not used in this book) 2. Division of units. Example: meter per second. m (a) Division sign - or m/s S (b) Negative power m * s-l (or ms-') Note that the term per in a word definition is equivalent to divide by in the mathematical notation. Note also that symbols are not followed by a period, except at the end of a sentence. In recognition of long-standing traditions, the various international commissions have acknowledged that a few non-SI units will remain in use in certain fields of science and in certain countries during a transitional period. Table 1-2 lists some units, both SI and non-SI, which will be used in this book, involving the quantities length, mass, and time. Other units will be introduced in subsequent chapters. TEMPERATURE Temperature may be defined as that property of a body which determines the flow of heat. Two bodies are at the same temperature if there is no transfer of heat when they are placed together. Temperature is an independent dimension which cannot be defined in terms of mass, length, and time. The SI unit of temperature is the kelvin, and 1 kelvin (K) is defined as 1/273.16 times the triple point temperature. The triple point is the temperature at which water coexists in equilibrium with ice at the pressure exerted by water vapor only. The triple point is 0.01 K above the normal freezing point of water, at which water and ice coexist in equilibrium with air at standard atmospheric pressure. The SI unit of temperature is so defined that OK is the absolute zero of temperature; the SI or Kelvin scale is often called the absolute temperature scale. Although absolute zero is never actually attainable, it has been approached to within 10-~ K. OTHER TEMPERATURE SCALES On the commonly used Celsius scale (also called the centigrade scale), a temperature difference of one degree is 1 K (exactly). The normal boiling point of water is' 100 "C,the normal freezing point 0 "C, and absolute zero -273.15 "C. A temperature difference of one degree on the Fahrenheit scale is 5/9 K (exactly). The boiling point and freezing point of water, and absolute zero are 212 O F , 32 "E;, and -459.67 respectively. The relationships among these three scales, illustrated in Fig. 1-1, are described by the following linear equations, in which temperature on the SI scale is designated by T , and on the other scales by t. O F , t T -=--273.15 "C K or t= t ,=:($-32) or 5 t t = - (7- 32) "C 9 .F 4 QUANTITIES AND UNITS [CHAP. 1 Boiling point of water Freezing point of water 373.15 K 273.15 K ,100 "C ~ 212 O 32 O F F 0C Absolute zero of temperature OK Kelvin - ~ -273.1 5 "C - l i -459.67 O F Celsius Fahrenheit Fig. 1-1 Here the ratios T / K , t / "C, t/ are the dimensionless numerical measures of the temperature on the Kelvin, Celsius, and Fahrenheit scales, respectively. Read the upper left equation as follows: The temperature in degrees Celsius equals the temperature in kelvins minus 273.15. O F USE AND MISUSE OF UNITS The units (e.g., cm, kg, g/mL, ft/s) must be regarded as a necessary part of the complete specification of a physical quantity. It is as foolish to separate the number of a measure from its unit as it is to separate a laboratory reagent bottle from its label. When physical quantities are subjected to mathematical operations, the units must be carried along with the numbers and must undergo the same operations as the numbers. Quantities cannot be added or subtracted directly unless they have not only the same dimensions but also the same units. EXAMPLE 2 It is obvious that we cannot add 5 hours (time) to 20 miles/hour (speed) since time and speed have different physical significance. If we are to add 2 lb (mass) and 4 kg (mass), we must first convert lb to kg or kg to lb. Quantities of various types, however, can be combined in multiplication or division, in which the units as well as the numbers obey the algebraic laws of squaring, cancellation, etc. Thus: 1. 2. 3. 4. 6L+2L=8L ( 5 cm)(2 cm2)= 10 cm3 (3 ft3)(200 lb/ft3)=600 lb (2 s)(3 m/s2)=6 m/s 15 g 5. -=5cm3 3 g/cm3 FACTOR-LABEL METHOD In solving problems, one can be guided consciously by the units to the proper way of combining the given values. Such techniques are referred to in textbooks as the factor-label method, the unitfactor method, or dimensional analysis. In essence one goes from a given unit to the desired unit by multiplying by a fraction called a unit-factor in which the numerator and the denominator must represent the same quantity. EXAMPLE 3 Convert 5.00 inches to centimeters. The appropriate unit-factor is 2.54 cm/l in, and 5.00 in x 2.54 cm/l in = 12.7 cm. Note that 1 in and its equal, 2.54 cm, clearly represent the same quantity. Also note that the reciprocal of a unit-factor must also be a unit-factor. If one used the reciprocal by mistake in the above calculation, however, the answer would have been 1.97 in2/cm, and these ridiculous units would reveal that a mistake had been made. CHAP. 1 1 QUANTITIES AND UNITS 5 The factor-label method can be extended to the use of unit-factors (or conversionfactors)in which numerator and denominator are equivalent but of different dimensions. EXAMPLE 4 What is the weight in grams of seven nails, which weigh 0.765 kg per gross? 7 nails x 1 gross nails 0.765 kg 1000 g = 37.2 X x1 gross nails 1 kg 144 nails The fraction in the middle above is a unit-factor of mixed dimensions. Also, in this case the numerator and denominator are not universally equivalent, but are associated measures for this particular batch of nails. Many similar examples are encountered in the chemical calculations shown in, subsequent chapters. ESTIMATION OF NUMERICAL ANSWERS If one's calculator is working correctly and is accurately used, the answer will be correct. But if not, will an incorrect answer be recognized? A very important skill is to determine first, by visual inspection, an approximate answer. Especially important is the correct order of magnitude, represented by the location of the decimal point or the power of 10, which may go astray even though the digits are correct. EXAMPLE 5 Consider the multiplication: 122g x 0.05 1 8 = 6.32 g. Visual inspection shows that 0.051 8 is a little more than &, and of 122 is a little more than 6. Hence the answer should be a little more than 6g, which it is. If the answer were given incorrectly as 63.2g or 0.632g, visual inspection or mental checking of the result would indicate that the decimal point had been misplaced. i i EXAMPLE 6 Calculate the power required to raise a 639 kg mass 20.74m in 2.120 minutes. The solution is: 639 kg x 20.74m x 9.81 m . s - ~ = 1 02:2J/s = 1 022 watts 2.120 min x 60 s/min Check the numerical answer by estimation. This example involves concepts and units unfamiliar to you, so that you can't easily judge whether the result "makes sense" or not. Write each term in exponential notation, using one significant figure. Then mentally combine the powers of ten and the multipliers separately to estimate the result, as shown below. NUM = 6 x io2 x 2 x 10' x 1 x 10' = 12 x 104 DEN = 2 x 6 x 10' = 12 x 10' NUM/DEN = 1000 estimated, compared to 1022 calculated Solved Problems UNITS BASED ON MASS OR LENGTH 1.1. The following examples illustrate conversions among various units of length, volume, or mass. 1 inch = 2.54 cm = 0.0254 m = 25.4 mm = 2.54 x 107 nm 1 foot = 12 in = 12 in x 2.54 cm/in = 30.48 cm = 0.304 8 m = 304.8 mm 1 liter = 1 dm3 = 10-3 m3 1 mile=5280 f t = 1.609 x 105cm= 1.609 x 103 m = 1.609 km= 1.609 x 106 mm 1 pound = 0.453 6 kg = 453.6 g = 4.536 x 105mg 1 metric ton = 1 000 kg = 106 g 6 QUANTITIES AND UNITS [CHAP. 1 1.2. Convert 3.50 yards to (a) millimeters, (b) meters. The bridge between the English units of length and the SI (metric system) is the unit factor (see Table 1-2) 1 in/2.54 cm. 36in 2.54cm 10mm 3.50 yd x -x - - 3.20 x 103mm x = 1 in lcm 1yd Note that the use of three successive conversion factors was necessary. The units yd, in, and cm canceled out leaving the desired unit, mm. (b) lm = 3.20 x 103 mm x - 3.20 m 103 mm 1.3. Convert (a) 14.0 cm and (b) 7.00 m to inches. 14.0 cm = (14.0 cm) - = 5.51 in (2.:i:m) or 14.0 cm = 14.0 cm = 5.51 in 2.54 cm/in 1 in will frequently appear in the form 1 in/2.54 cm, especially in com2.54 cm puter displays, because it allows the entire calculation to appear on one line of type, i.e., (14.0 cm)(l in/2.54 cm) = 5.51 in. Factors such as ~ (b) 7.00 m = (7.00 m)( 100 cm/l m)(1 in/2.54 cm) = 276 in 1.4. How many square inches are in one square meter? 1 m = (1 m)(100 cm/l m)(l in/2.54 cm) = 39.37 in 1 m2 = (1 m)* = (39.37 in)2 = 1550 in2 alternatively, 1 m2 = (1 m)2(100 cm/l m)'( 1 in/2.54 cm)2 = [( 100)2/(2.54)2] = 1 550 in2 in2 Note that since a conversion factor is equal to 1, it may be squared (or raised to any power) without changing its value. 1.5. (a) How many cubic centimeters are in one cubic meter? (b) How many liters are in one cubic meter? ( c ) How many cubic centimeters are in one liter? 1 L = 1 dm3 = (1 dm)3 (10)3: ~ = 103 cm3 1.6. Find the capacity in liters of a tank 0.6 m long, 10 cm wide, and 50 mm deep. Convert to decimeters, since 1 L = 1 dm3. CHAP. 11 QUANTITIES AND UNITS 7 Volume = (0.6 m) ( 10 cm) (50 mm) = (0.6 m) 10 dm ( x ) (gi)mm) (&) x cm) (10 x (50 = (6 dm)( 1 dm)(0.5 dm) = 3 dm3 = 3 L 1.7. Determine the mass of 66 Ib of sulfur in (a) kilograms and (h) grams. (c) Find the mass of 3.4 kg of copper in pounds. (4 66 lb = (66 lb)(0.453 6 kg/lb) = 30 kg or 66 lb = (66 lb)(l kg/2.2 lb) = 30 kg 66 lb=(66 1b)(453.6 g/lb)=30000 g, or 3 . 0 ~ 1 g ~ 0 3.4 kg= (3.4 kg)(2.2 lb/kg,)= 7.5 lb COMPOUND UNITS 1.8. Fatty acids spread spontaneously on water to form a rnonomolecular film. A benzene solution containing 0.10 mm3 of stearic acid is dropped into a tray full of water. The acid is insoluble in water but spreads on the surface to form a continuous film of area 400 cm2 after all of the benzene has evaporated. What is the average film thickness in (a) nanometers, (b) angstroms? 1 mm3 = ( 1 0 - ~ m ) ~ 1 0 - ~ = m3 1 cm2 = ( 1 0 - ~m)2 = 1 0 - ~ m2 volume - (0.10 mm3)( 1 o - ~ /mm3) m3 Film thickness= ___ ___ = 2.5 x lO-'m = 2.5 nm area (400 cm2)(10-4 m2,/cm2) (b) = 2.5 x 10-9 m x 1O1'A/m = 25 A 1.9. A pressure of one atmosphere is equal to 101.3 kPa. Express this pressure in pounds force (lbf) per square inch. The pound force (lbf) is equal to 4.448 N 1 atm = 101.3 kPa = (101.3 x 103 N/m2) - 2*54 (4iEN) 7 ( m, 2= 14.69 Ibf/in2 Notice that the conversion factor between m and in is squared to give the conversion factor between m2 and in2. 1.10. An Olympic-class sprinter can run 100 meters in about 10.0 seconds. Express this speed in (a) miles per hour, and (b) kilometers per hour. Part (b) should be solved first, for it involves only metric units. 60 rnin 60 s 100 m 1 km Xx -x - = 36.0 km/h 10.0s 1OOOm 1 min Ih (a) Using the information in problem 1.1: 36.0 km/h x 1 mi/ 1.609 km := 22.4 mi/h 1.11. New York City's 7.9 million people in 1978 had a daily per capita consumption of 656 liters of water. How many metric tons (103 kg) of sodium fluoride (45% fluorine by weight) would be required per year to give this water a tooth-strengthening dose of 1 part (by weight) fluorine per million parts water? The density of water is 1.000 g/cm3, or 1.000 kg/L. 8 QUANTITIES AND UNITS [CHAP. 1 Mass of water, in tons, required per year = (7.9 x 1 0 persons) (6,5,6,s:?i:i) ~ = 1.89 ( 1 L water) ( 1000 kg 7 ( ) 365 days 1 kg water 1 metric ton ) 109 metric tons water Y r Note that all units cancel out except metric tons water/yr, which appears in the result. Mass of sodium fluoride, in tons, required per year = (1.89 = 4.2 109 103 tons (metric) water Y r 1 ton sodium fluoride 0.45 ton fluorine tons (metric) sodium fluoride Y r 1.12. In a measurement of air pollution, air was drawn through a filter at the rate of 26.2 liters per minute for 48.0 hours. The filter gained 0.0241 grams in mass because of entrapped solid particles. Express the concentration of solid contaminants in the air in units of micrograms per cubic meter. [(0.0241 g)(106 pg/l g)/(48.0 h)(60 min/h)] x ( l min/26.2 L) x (1 L / l dm3) x (10 dm/l m)3 = 319 pg/m3 The answer was estimated as follows: NUM = 2 x I O - ~ x io6 x 1 x I x 1 x 103 = 2 x 107 DEN = 5 x 10' x 6 x 10' x 3 x 10' = 9 x 104 NUM/DEN = 222, or approximately 200 1.13. Calculate the density, in g/cm3, of a body that weighs 420 g (i.e., has a mass of 420 g) and has a volume of 52 cm3. mass 420 g Density = ____ = 8.1 g/cm3 volume 52cm3 ~ 1.14. Express the density of the above body in the standard SI unit, kg/m3. 1.15. What volume will 300 g of mercury occupy? Density of mercury is 13.6 g/cm3. mass - 300 g Volume= - 22.1 cm3 density 13.6 g/cm3 - 1.16. The density of cast iron is 7200 kg/m3. Calculate its density in pounds per cubic foot. Density = (7 200 5)(o.4i:: ( kg) 0.3048 m ft ) = 449 lb/ft3 The two conversion factors were taken from Problem 1.1. CHAP. I] QUANTITIES AND UNITS 9 1.17. A casting of an alloy in the form of a disk weighed 50.0 g,. The disk was 0.250 inch thick and had a circular cross section of diameter 1.380 in. What is the density of the alloy, in g/cm3? Volume= ($),= ( .-( 1 380 in)'(0.250 in) . -)( ) r 2.54 cm 3 = 6.13 cm3 mass Density of the alloy = - - 50*0 = 8.15 g/cm3 volume 6.13 cm3 1.18. The density of zinc is 455 lb/ft3. Find the mass in grams of 9.00 cm3 of zinc. First express the density in g/cm3 (455 $) (-) 1 ft fz.) = 7.29 g cm3 (9.00 cm3)(7.29g/cm3) =: 65.6 g 1.19. Battery acid has a density of 1.285 g/cm3 and containls 38.0% by weight H2SO4. How many grams of pure H2SO4 are contained in a liter of battery acid? 1 cm3 of acid has a mass of I .285 g. Then 1 L of acid (1 000 cm3) has a mass of 1 285 g. Since 38.0% by weight (or by mass) of the acid is pure H2S04, the number of grams of H2SO4 in 1 L of battery acid is 0.380 x 1285 g = 488 g Formally, the above solution can be written as follows: 38 g H2SO4 100 acid = 488 Mass of H2S04 = ( I 285 g acid) ~- g H2S04 Here the conversion factor 38 g H2S04 100 g acid is taken to be equal to 1. Although the condition 38 g H2S04 = 100 g acid is not a universal truth in the same sense that 1 in always equals 2.54 cm, the condition is a rigid one of association of 38 g H2S04, with every 100 g acid for this particular acid preparation. Mathematically.,these two quantities may be considered to be equal for this problem, since one of the quantities implies the other. Liberal use will be made in subsequent chapters of conversion factors that are valid only for particular cases, in addition to the conversion factors that are universally valid. ( ) 1.20. (a) Calculate the mass of pure HNOl per cm3 of the concentrated acid which assays 69.8% by weight H N 0 3 and has a density of 1.42 g/cm3. (h) Calculate the mass of pure H N 0 3 in 60.0 cm3 of concentrated acid. (c) What volume of the concentrated acid contains 63.0 g of pure HN03? (a) 1 cm3 of acid has a mass of 1.42 g. Since 69.8% of the total mass of the acid is pure HN03, then the number of grams of H N 0 3 in 1 cm3 of acid is 0.698 x 1.42 g = 0.991 g (b) Mass of H N 0 3 in 60.0 cm3 of acid = (60.0 cm3)(0.991g/cm3) = 59.5 g H N 0 3 (c) 63.0 g H N 0 3 is contained in 63*0 = 63.6 crr? acid 0.991 g/cm3 10 QUANTITIES AND UNITS [CHAP. 1 TEMPERATURE 1.21. Ethyl alcohol (a) boils at 78.5 "C and (b) freezes at - 1 17 "C, at one atmosphere pressure. Convert these temperatures to the Fahrenheit scale. Use t = W(78.5) + 321 "F = (141 + 32) "F = 173"F (-211 + 3 2 ) " F = -179F t = [!(-117)+32]"F= 1.22. Mercury (a) boils at 675 "F and (b) solidifies at -38.0 OF, at one atmosphere pressure. Express these temperatures in degrees Celsius. Use (4 (b) t = $(675 - 32) "C = (643) "C = 357 "C t = (-38.0 5 3 - 32.0) "C = 3 (-70.0) "C = -38.9 "C 1.23. Change (a) 40 "C and (b) -5 "C to the Kelvin scale. T = (4,273) T = (40 K T = (-5 + 273) K = 313 K + 273) K = 268K 1.24. Convert (a) 220 K and (b) 498 K to the Celsius scale. Use t = (g-273) "C t = (220 - 273) "C = -53 "C t = (498 - 273) "C = 225 "C 1.25. During the course of an experiment, laboratory temperature rose 0.8"C. Express this rise in degrees Fahrenheit. Temperature intervals are converted differently than temperature readings. For intervals, it is seen from Fig. 1-1 that 100C = 180F Hence or 5C = 9 F 0.8 "C = (0.8 "C) - (E) = 1.4"F CHAP. 13 QUANTITIES AND UNITS 11 Supplementary Problems UNITS BASED ON MASS OR LENGTH 1.26. (a) Express 3.69 m in kilometers, in centimeters, and in millimeters. (b) Express 36.24 mm in centimeters and in meters. Ans. (a) 0.003 69 km, 369 cm, 3 690 mm; 1.27. (6) 3.624 cm, 0.036 24 m (c) centimeters in 4 ft 3 in. Determine the number of (a) millimeters in 10 in, (b) feet in 5 m, Ans. (a) 254 mm; (b) 16.4 ft; (c) 130 cm 1.28. Convert the molar volume, 22.4 liters, to cubic centimeters, to cubic meters, and to cubic feet. Ans. 22400 cm3, 0.0224 m3, 0.791 ft3 1.29. Express the weight (mass) of 32 g of oxygen in milligrams, in kilograms, and in pounds. Ans. 32000 mg, 0.032 kg, 0.070 5 lb 1.30. How many grams in 5.00 lb of copper sulfate? How many pounds in 4.00 kg of mercury? How many milligrams in 1 lb 2 oz of sugar? Ans. 2270 g, 8.82 lb, 510000 mg 1.31. Convert the weight (mass) of a 2 176 lb compact car to (a) kilograms, (b) metric tons, (c) U.S. tons (1 ton = 2 000 lb). Ans. (a) 987 kg; (b) 0.987 metric ton; (c) 1.088 ton (US.) 1.32. The color of light depends on its wavelength. The longest visible rays, at the red end of the visible spectrum, are 7.8 x 10-7 m in length. Express this length in micrometers, in nanometers, and in angstroms. Ans. 0.78 pm, 780 nm, 7 800 A 1.33. An average person should have no more than 60 grams of fat in his or her daily diet. A certain package of chocolate chip cookies is labeled "1 portion is 3 cookies" and also "fat: 6 grams per portion." How many cookies may one eat before exceeding 50% of one's recommended fat intake? Ans. 15 cookies 1.34. In a crystal of platinum, centers of individual atoms are 2.8 apart along the direction of closest packing. How many atoms would lie on a one-centimeter length of a line in this direction? A hs. x 107 3.5 1.35. The blue iridescence of butterfly wings is due to striations that are 0.15 pm apart, as measured by the electron microscope. What is thisodistance in centimeters? How does this spacing compare with the wavelength of blue light, about 4 500 A Ans. 1.5 x 10-5 cm, f wavelength of blue light 13. .6 An average man requires about 2.00 mg of riboflavin (vitamin B2) per day. How many pounds of cheese would a man have to eat per day if this were his only source of riboflavin and if the cheese contained 5.5 pg riboflavin per gram? Ans. 0.80 lb/day 12 QUANTITIES AND UNITS [CHAP. 1 1.37. When a sample of healthy human blood is diluted to 200 times its initial volume and microscopically examined in a layer 0.10 mm thick, an average of 30 red corpuscles are found in each 100 x 100 micrometer square. (a) How many red cells are in a cubic millimeter of blood? (b)The red blood cells have an average life of 1 month, and the adult blood volume is about 5 L. How many red cells are generated every second in the bone marrow of the adult? Ans. (a) 6 x 106 cells/mm3; (b) 1 x 107 cells/s 1.38. A porous catalyst for chemical reactions has an internal surface area of 800 m2 per cm3 of bulk material. Fifty percent of the bulk volume consists of the pores (holes), while the other 50 percent of the volume is made up of the solid substance. Assume that the pores are all cylindrical tubules of uniform diameter d and length I , and that the measured internal surface area is the total area of the curved surfaces of the tubules. What is the diameter of each pore? (Hint: Find the number of tubules per bulk cm3, n, in terms of 1 and d, by using the formula for the volume of a cylinder. V = ird21.Then apply the surface-area formula, S = rdl, to the cylindrical surfaces of n tubules.) Ans. 25 A 1.39. Suppose that a rubber tire loses from its surface, on the average, a layer one molecule thick each time it goes around on th! pavement. (By "molecule" you should infer one monomer unit.) Assume that the molecules average 7.50 A in thickness, and that the tire tread is 35.6 cm in radius and 19.0 cm wide. On a 483 km drive from Pittsburgh to Philadelphia (a) how much is the radius reduced (in mm), and (b)what volume of rubber (in cm3) is lost from each tire? Ans. (a) 0.162 mm; (b) 68.8 cm3 COMPOUND UNITS 1.40. The density of water is 1.OOO g/cm3 at 4 "C. Calculate the density of water in pounds per cubic foot at the same temperature. Ans. 62.4 lb/ft3 1.41. Refer to problem 1.39. If the tire tread has a density of 963 kg/m3, calculate the mass in grams lost by each tire on the trip. Ans. 66.3 g 1.42. The silica gel which is used to protect sealed overseas shipments from moisture seepage has a surface area of 6.0 x 105m2 per kilogram. What is this surface area in square feet per gram? Ans. 6.5 x 103 ft2/g 1.43. There is reason to think that the length of the day, determined from the earth's period of rotation, is increasing uniformly by about 0.001 s every century. What is this variation in parts per billion? A ~ S .3 x 1 0 - ~s per 109 s 1.44. The bromine content of average ocean water is 65 parts by weight per million. Assuming 100 percent recovery, how many cubic meters of ocean water must be processed to produce 0.61 kg of bromine? Assume that the density of seawater is 1.0 x 103 kg/m3. Ans. 9.4 m3 1.45. An important physical quantity has the value 8.314 joules or 0.08206 litereatmosphere. What is the conversion factor from liter atmospheres to joules? e Ans. 101.3 J/L atm CHAP. 11 QUANTITIES AND UNITS 13 1.46. Find the density of ethyl alcohol if 80.0 cm3 weighs 63.3 g. Ans. 0.791 g/cm3 1.47. Find the volume in liters of 40 kg of carbon tetrachloride, whose density is 1.60 g/cm3. Ans. 25 L 1.48. A type of plastic foam has a density of 17.7 kg/m3. Calculate the mass in pounds of an insulating slab 4.0 ft wide, 8.0 ft long, and 4.0 in thick. Ans. 11.8 Ib 1.49. Air weighs about 8 lb per 100 cubic feet. Find its density in (a) grams per cubic foot, (b) grams per liter, (c) kilograms per cubic meter. Ans. (a) 36 g/ft3; (b) 1.3 g/L; (c) 1.3 kg/m3 1.50. The estimates for the caloric content of food are: 9.0 Cal/g for fat, and 5.0 Cal/g for carbohydrates and protein. A certain breakfast muffin contains 14% by weight of fat, 64% carbohydrate, and 7% protein (the rest being water, which has no calories). Does it meet the criterion of 30% or less calories from fat, which is recommended for the U.S. population? Ans. Yes, 26% of the calories are from fat. 1.51. A wood block, 10 in x 6.0 in x 2.0 in, weights 3 lb 10 oz. What is the density of the wood in SI units? Ans. 840 kg/m3 1.52. An alloy was machined into a flat disk. 31.5 mm in diameter and 4.5 mm thick, with a hole 7.5 mm in diameter drilled through the center. The disk weighed 20.2 g. What is the density of the alloy in SI units? Ans. 6100 kg/m3 1.53. A glass vessel weighed 20.237 6 g when empty and 20.310 2 g when filled to an etched mark with water at 4 "C. The same vessel was then dried and filled to the same mark with a solution at 4 "C. The vessel was now found to weigh 20.3300 g. What is the density of the solution'? Ans. 1.273 g/cm3 1.54. A sample of lead shot weighing 321 g was added to a graduated cylinder partially filled with isopropyl alcohol (enough to immerse the lead completely). As a result the alcohol level rose 28.3 mL. What is the density of the lead in SI units? (The density of isopropyl alcohol is 0.785 g/cm3.) Ans. 1.13 x 104 kg/m3 1.55. A sample of concentrated sulfuric acid is 95.7% H2SO4 by weight and its density is 1.84 g/cm3. (a) How many grams of pure H2S04 are contained in one liter of the acid? (b) How many cubic centimeters of acid contain 100 g of pure H2S04? Ans. (a) 1760 g; (b) 56.8 cm3 1.56. A quick method of determining density utilizes Archimedes' principle, which states that the buoyant force on an immersed object is equal to the weight of the liquid displaced. A bar of magnesium metal attached to a balance by a fine thread weighed 31.13 g in air and 19.35 g when completely immersed in hexane (density 0.659 g/cm3). See Fig. 1-2. Calculate the density of this sample of magnesium in SI units. Ans. 1741 kg/m3 14 QUANTITIES AND UNITS [CHAP. 1 I A % Fig. 1-2 1.57. An electrolytic tin-plating process gives a coating 30 millionths of an inch thick. How many square meters can be coated with one kilogram of tin, density 7300 kg/m3? Ans. 180 m2 1.58. A piece of gold leaf (density 19.3 g/cm3) weighing 1.93 mg can be beaten further into a transparent film covering an area of 14.5 cm'. (a) What is the volume of 1.93 mg of gold? (b)What is the thickness of the transparent film, in angstroms? Ans. (a) 1.00 x 10-4 cm3; (b) 690 A 1.59. A piece of capillary tubing was calibrated in the following manner. A clean sample of the tubing weighed 3.247 g. A thread of mercury, drawn into the tube, occupied a length of 23.75 mm, as observed under a microscope. The weight of the tube with the mercury was 3.489 g. The density of mercury is 13.60 g/cm3. Assuming that the capillary bore is a uniform cylinder, find the diameter of the bore. Ans. 0.98 mm 1.60. The General Sherman tree, located in Sequoia NatiGrial Park, is believed to be the most massive of living things. If the overall density of the tree trunk is assumed to be 850 kg/m3, calculate the mass of the trunk by assuming that it may be approximated by two right conical frusta having lower and upper diameters of 11.2 and 5.6 m, and 5.6 and 3.3 m, respectively, and respective heights of 2.4 and 80.6 m. A frustum is a portion of a cone bounded by two planes, both perpendicular to the axis of the cone. The volume of a frustum is given by where h is the height and r1 and r2 are the radii of the circular ends of the frusta. Ans. 1.20 x 106 kg = 1 200 metric tons TEMPERATURE 1.61. (a) Convert 88 "F to "C; 16 "F to "C; 130 "F to "C. (b) Convert 35 "C to Ans. (U) 31 "C, -9"C, 54C; (b) 95"F, 36"F, -20F O F ; 2 "C to OF; -29 "C to O F . 1.62. Convert the following temperatures: -149.7 "C to Ans. O F ; -396.0 "F to "C; 1 555 "C to "F. -237.5 OF, -237.8 "C, 2 83 1 "F CHAP. 11 QUANTITIES AND UNITS 15 1.63. The temperature of dry ice (sublimation temperature at normal pressure) is - 109 OF. Is this higher or lower than the temperature of boiling ethane (a component of bottled gas), which is -88 "C? Ans. higher 1.64. Gabriel Fahrenheit in 1714 suggested for the zero point on his scale the lowest temperature then obtainable from a mixture of salts and ice, and for his 100" point he suggested the highest known normal animal temperature. Express these "extremes" in degrees Celsius. Ans. - 17.8 "C, 37.8 "C 1.65. Sodium metal has a very wide liquid range, melting at 98 "C and boiling at 892 "C. Express the liquid range in degrees Celsius, kelvins, and degrees Fahrenheit. Ans. 794 "C, 794 K, 1429 "F 1.66. Convert 298 K, 892 K, 163K, to degrees Celsius. Ans. 25 "C, 619 "C, -1 10 "C 1.67. Express 11 I ,298 K, in degrees Fahrenheit. S Ans. -440F. 77F 1.68. Convert 23 O F to degrees Celsius and kelvins. Ans. -5"C, 268 K 1.69. At what temperature have the Celsius and Fahrenheit readings the same numerical value? Ans. -40" 1.70. A water-stabilized electric arc was reported to have reached ii temperature of 25 600 OF. On the absolute scale, what is the ratio of this temperature to that of an oxyacetylene flame, 3 500 "C? Ans. 3.84 1.71. Construct a temperature scale on which the freezing and boiling points of water are 100" and 400", respectively, and the degree interval is a constant multiple of the Celsius degree interval. What is the absolute zero on this scale, and what is the melting point of sulfur, which is 444.6"C? Ans. -719 ", 1433.8 O This page intentionally left blank Chapter 2 Atomic and Molecular Mass; Molar Mass ATOMS In the form of the atomic theory proposed by John Dalton in 1805, all atoms of a given element were thought to be identical. Chemists in the following decades set themselves the task of finding the relative masses of the atoms of the different elements by precise quantitative chemical analysis. Over a hundred years after Dalton's proposal was made, investigations with radioactive substances showed that not all atoms of a given element are identical. Modern atomic mass tables must recognize this fact because one of the properties that distinguishes these different kinds of atoms for a given element is the atomic mass. An element can exist in several isotopic forms, and all atoms of the same isotope are identical. NUCLEI Every atom has a positively charged nucleus which contains over 99.9 percent of the total mass of the atom. Although the structure of the nucleus is not thoroughly understood, every nucleus may be described as being made up of two different kinds of particles. These two kinds of particles, called nucleons, are the proton and the neutron, and they have nearly the same mass. Of these two, only the proton is charged and the sign of its charge is positive. The total charge of any nucleus is equal to the number of protons times the charge of one proton. The magnitude of the protonic charge may indeed be considered the fundamental unit of charge for atomic and nuclear phenomena, since no smaller charge than this has been discovered in any free particle. The nuclear charge expressed in this unit is then equal to the number of protons in the nucleus. The atoms of all isotopes of any one element have the same number of protons. This number is called the atomic number, 2, and is a characteristic of the element. The nuclei of the different isotopes differ, however, in the number of neutrons and therefore in the total number of nucleons per nucleus. The total number of nucleons, A , is called the mass number, and the number of neutrons is therefore A - 2. Atoms of the different isotopic forms of an element are called nuclides and are distinguished by using the mass number as a superscript to the elementary symbol on the left. Thus 15N refers to the nitrogen of mass number 15. RELATIVE ATOMIC MASSES The masses of the individual atoms are very small. Even the heaviest atom has a mass less than 5 x 10-25 kg. It has been convenient, therefore, to define a special unit in which the masses of the atoms can be expressed without having to use exponents. This unit is called the atomic mass unit and is designated by the symbol U. (Biochemists sometimes refer to this unit as the dalton.) It is defined as exactly the mass of a 12C atom. Thus the mass of the 12C atom is exactly 12 U; the mass of the 23Na atom is 22.9898 U. Table 2-1 lists the masses of some nuclides to which reference will be made in this chapter and in Chapter 2 1. Most chemical reactions do not discriminate significantly among the various isotopes. For example, the percentages of iron atoms which are 54Fe,56Fe,57Fe,and 58Feare 5.8,91.8, 2.1, and 0.3, respectively, in all iron ores, meteorites, and iron compounds prepared synthetically. For chemical purposes it is of interest to know the average mass of an iron atom in this natural isotopic mixture. These average masses are also tabulated in terms of the unit U and are designated by A,(E), where E is the symbol for the particular element. The term atomic mass will be understood to mean average atomic mass, and nuclidic mass will be used when referring to one particular isotope of an element. A , values, which are listed at the end of this book, form the basis for practically all chemical weight calculations. A , values used to be 17 18 ATOMIC AND MOLECULAR MASS; MOLAR MASS [CHAP. 2 Table 2-1. Some Nuclidic Masses 16.999 13 U 17.999 16 18.00094 18.00571 27.976 93 28.976 49 29.973 77 3 1.97207 3 5 ~ ~ `H 2H 3H 4He 6He `Li 7 ~ i 7Be 1.007 83 U 2.014 10 3.016 05 4.002 60 6.018 89 6.015 12 7.01600 7.01693 12.00000 U 13.00335 14.00324 16.014 70 14.00307 15.000 11 16.006 10 15.99491 3 7 ~ 1 36Ar 38Ar 40 Ar 87Rb 34.968 85 U 36.965 90 35.967 55 37.962 73 39.962 38 86.909 19 determined by precise chemical analysis, but nearly all modern values are the weighted average of the nuclidic masses measured by mass spectroscopy. MOLE An ordinary chemical experiment involves the reaction of enormous numbers of atoms or molecules. It has been convenient to define a new term, the mole, as denoting a collection of a large fixed number of fundamental chemical entities, comparable to the quantity that might be involved in an actual experiment. In fact, the mole is recognized in SI as the unit for one of the dimensionally independent quantities, the amount of substance. The abbreviation of the unit is rnol. A mole of atoms of any element is defined as that amount of substance containing the same number of atoms as there are C atoms in exactly 12 g of pure 12C.This number is called Avogadro's constant, NA; its value may be related to the value of the U, listed in Table 1-2, as follows. Mass of 1 mol of 12Catoms = NA x (mass of one 12Catom) 12 g/mol= NA x 12 U = 6.022 I x 10~~/m01 All units in the expression for N A canceled except for mol, which remained in the denominator, and thus appears as mol-'. Let us now consider a mole of atoms of some other element, of atomic mass A,. The average mass of an atom of this element is A,u and the mass of a mole of such atoms is NA x A,u, or simply A,g/mol. In other words, the mass in grams of a mole of atoms of an element is equal to the atomic mass, and A , may be considered to have the units g/mol. Thus a "mole of gold" is 197.0 g of gold. As will be seen below, the mole may be used to refer to NA elementary entities of types other than atoms, such as formula units, molecules, and charged particles. SYMBOLS, FORMULAS, MOLAR MASSES The symbol for an element is used to designate that element as distinct from other elements. In a formula the symbol designates an atom of the element. Molecular substances consist of independent molecules in each of which two or more atoms are tightly bound together. The formula for a molecular substance consists of the symbols for the atoms in one molecule. For instance, the formula for carbon dioxide is C02. Note the use of the subscript, 2, to show that each molecule contains two oxygen atoms CHAP. 21 ATOMIC AND MOLECULAR MASS; MOLAR MASS 19 in addition to one carbon atom. The molecular mass of CO2 is the sum, in U, of the atomic mass of carbon plus twice the atomic mass of oxygen. The molar mass (of CO2 is (as explained above) the mass in grams numerically equal to the molecular mass in U. The molecular mass in this case is 12.0 + 2(16.0) = 44.0 U and the molar mass is 44.0 g. Thus a "mole of carbon dioxide' is 44.0 g of carbon dioxide, and the formula CO2 can also be used to refer to 1 mole of carbon dioxide. Many common substances are ionic in nature. The atoms, in the form of charged particles called ions, are arranged in a vast regular spatial array. In such cases the formula indicates the relative number of atoms of each element present. Fe304refers to a compound in which 3 atoms of iron are present for every 4 atoms of oxygen. If the subscripts are the smallest possible integral numbers that express the relative numbers of atoms, the formula is called an empiricalformula. Fe304is an example; it is a descriptive notation for this particular oxide of iron. Fe,304 also used in some contexts to mean a particular amount of the compound, the is amount that contains 3 moles of iron atoms and 4 moles of oxygen atoms. This amount of compound contains N A simplest formula units (where the formula unit contains 3 iron atoms and 4 oxygen atoms) and is, therefore, called a mole of Fe304. The molar mass of Fe304 is numerically the same as the mass in U of aformula unit, in this case 3(55.85) + 4(16.00) = 231.55 U. Thus a "mole of Fe30$' is 231.55 g of Fe304. The term "atomic weight" had been widely used rather than "atomic mass," and "molecular weight" rather than "molar mass." (Many writers used "molecular weight" for "molar mass" even for ionic substances.) Because "weight" is a force rather than a mass, such usage is discouraged. The beginning student, however, must be aware of the old terms because they dominate the chemical literature. The term "molar mass" is a particularly welcome change because of its universal applicability, referring to Avogadro's number of molecules, ions, formula units, or individual atoms (e.g., the molar mass of gold is 197.0 g/mol; the molar mass of hydroxide ion, OH-, is 17.0 glmol). Solved Problems ATOMIC MASS 2.1. It has been found by mass spectrometric analysis that in nature the relative abundances of the various isotopic atoms of silicon are 92.23% 28Si, 4.67% 29Si, and 3.10% 30Si. Calculate the atomic mass of silicon from this information and from the nuclidic masses. The atomic mass is the average mass of the three nuclides, each weighted according to its own relative abundance. The nuclidic masses are given in Table 2- 1. A , = (0.922 3)(27.977) + (0.046 7)(28.976) + (0.031 0)(29.974) = 25.803 + 1.353 + 0.929 = 28.085 2.2. Naturally occurring carbon consists of two isotopes, I2C and 13C. What are the percentage abundances of the two isotopes in a sample of carbon whose atomic mass is 12.011 12? Let x == O h abundance of 13C; then 100 - x is A , = 12.011 12 = = O h (12.00000)(100 - X) 100 + (13.003 3 5 ) ~ - of 12C. = 12.00000 + (13'00335 - 12'ooooo)x 12.00000 + (0.010033 5). 100 20 ATOMIC AND MOLECULAR MASS; MOLAR MASS [CHAP. 2 Thus and X = (12.011 12 - 12.00000 - 0.011 12 = 1.1080,~ 13c 0.010033 5 0.010033 5 100 - x = 98.892% 12C 2.3. Before 1961, a physical atomic mass scale was used whose basis was an assignment of the value 16.00000 to l 6 0 . What would have been the physical atomic mass of 12C on the old scale? The ratio of the masses of any two nuclides must be independent of the establishment of the reference point. Of 1 6 0 oldscale A , of l60 present scale 12.00000 15.99491 or ( A , of 12C)oldscale (16.0000) = (:::;E;) ~ = 12.00382 2.4. A 1.527 6 g sample of CdC12 was converted to metallic cadmium and cadmium-free products by an electrolytic process. The weight of the metallic cadmium was 0.9367 g. If the atomic mass of chlorine is taken as 35.453, what must be the atomic mass of Cd from this experiment? Throughout this book we will specify the amount of a substance in terms of the chemist's unit, the mole. We will use the symbol n ( X ) to refer to the number of moles of substance or chemical constituent X. Since in most laboratory work mass is determined by weighing, the word "weight" (as in the second sentence of this problem) is commonly used where "mass" would be more precise. Unless it leads to an ambiguity, we will continue to follow common usage and not bother to distinguish between "mass" and "weight." In this case, we calculate first the number of moles of C1 atoms in the weighed sample. Weight of CdC1, = 1.5276 g Weight of Cd in CdCl, = 0.9367 g Weight of C1 in CdC1, n(C1) = 0.5909 g x ~ = 0.5909 g 1 mol = 0.016667 mol 35.453 g From the formula CdC12 we see that the number of moles of Cd is exactly half the number of moles of c1. n(Cd) = fn(C1) = i(0.016667) = 0.008 333 mol The atomic mass is the mass per mole. A,(Cd) = 0*9367 = 112.41 g/mol 0.008 333 mol 2.5. In a chemical determination of the atomic mass of vanadium, 2.8934 g of pure V0Cl3 was allowed to undergo a set of reactions as a result of which all the chlorine contained in this compound was converted to AgCl. The weight of the AgCl was 7.180 1 g. Assuming the atomic 1 masses of Ag and C are 107.868 and 35.453, what is the experimental value for the atomic mass of vanadium? CHAP. 21 ATOMIC AND MOLECULAR MASS; MOLAR MASS 21 This problem is similar to Problem 2.4, except that n(C1) must be obtained by way of n(AgC1). The three C1 atoms of VOC13 are converted to 3 formula units of AgCl, the molar mass of which is 143.321 (the sum of 107.868 and 35.453). n(AgC1) = 7.180 1 g x From the formula AgCl, n(C1) = n(AgC1) = 0.050098 mol Cl Also, from the formula VOC13, 1 mol ==0.050098 mol 143.321 g n(V)+ fn(C1) = f (0.050 098) = 0.016 699 mol V To find the weight of vanadium in the weighed sample of VOC13, we must subtract the weights of chlorine and oxygen contained. If we designate the mass of any substance or chemical constituent X by m(X), then m(X)= n ( X ) x &(X) where &(X) is the molar mass of X. Note that if X is a single atom, then &(X) is A , ( X ) . m(C1) = n(C1) x A,(Cl) = (0.050098 mo1)(35.543 g/mol) = 1.776 1 g C1 Noting from the formula V0Cl3, that n ( 0 ) = n(V), m(0)= n ( 0 ) x A,(O) = (0.016699 mo1)(15..999 g/mol) = 0.2672 g 0 By difference, m ( V ) = m(VOC13) - m(0)- m(C1) = (2.8934 - 0.267 2 - 1.776 1) g = 0.850 1 g Then Note that this result differs slightly from the accepted value. The difference can be ascribed to experimental error in this determination. MOLAR MASSES 2.6. Determine the molar mass of (a) potassium hexachloroiridate(IV), K21rC16and (b) molar mass of trifluorosilane, SiHF3. Potassium hexachloroiridate(1V) does not exist as discrete molecules represented by the empirical formula, but trifluorosilane does. The term "molar mass" in either case refers to the mass of NA formula units, which in grams is numerically equal to the sum of all A,, which appear in the formula (for each element, multiplying its A , by the number of atoms of that element in the formula). (U) 2 K = 2(39.098) = 78.20 1 Ir =: l(192.22) = 192.22 (b) 1 Si = l(28.086) = 28.086 1 H = l(1.008) = 1.008 6 C1 = 6(35.453) = 212.72 Molar mass = 483.14 3 F = 3(18.9984) = 56.005 Molar mass =: 86.089 Note that the atomic masses are not all known to the same number of significant figures or to the same number of decimal places in U units. In general, the rules for significant figures discussed in Appendix B apply. The value of A,(Ir) is known to only 0.01 U. Note that in order to express 6 times the atomic mass of C1 to 0.01 U, it was necessary to use the atomic mass to 0.001 U. Similarly, an extra figure was used in the atomic mass for fluorine to give the maximum significance to the last digit in the sum column. 22 ATOMIC AND MOLECULAR MASS; MOLAR MASS [CHAP. 2 27 .. How many (a) grams of H2S, (b) moles of H and of S, (c) grams of H and of S, ( d ) molecules of H2S, (e) atoms of H and of s, are contained in 0.400 mol HZS? Atomic mass of H is 1.008; of S, 32.066. Molecular mass of H2S is 2(1.008) + 32.066 = 34.08. Note that it was not necessary to express the molecular mass to 0.001 U, even though the atomic masses are known to this significance. Since the limiting factor in this problem is n(H2S), known to one part in 400, the value 34.08 (expressed to one part in over 3000) for the molecular mass is more than adequate. (4 Number of grams of compound = (number of moles) x (mass of 1 mole) Number of grams of H2S = (0.400 mo1)(34.08 g/mol) = 13.63 g H2S (b) One mole of H2S contains 2 moles of H and 1 mole of S. Then 0.400 mol H2S contains (0.400 rnol H2S) 2 rnolH2s) = 0.800 mol H mol H and 0.400 mol S. ( (4 Number of grams of element = (number of moles) x (mass of 1 mole) Number of grams of H = (0.800 mol)(1.008 g/mol) = 0.806 g H Number of grams of S = (0.400 mo1)(32.066 g/mol) = 12.83 g S (4 Number of molecules = (number of moles) x (number of molecules in 1 mole) = (0.400 mol)(6.02 x 1023molecules/mol) = 2.41 x 1023molecules (e) Number of atoms of element = (number of moles) x (number of atoms per mole) Number of atoms of H = (0.800 mo1)(6.02 x 1023atoms/mol) = 4.82 x 1023atoms H Number of atoms of S = (0.400 mol)(6.02 x 1023atoms/mol) = 2.41 x 1023atoms S 2.8. How many moles of atoms are contained in (a) 10.02 g calcium, (b) 92.91 g phosphorus? (c) How many moles of molecular phosphorus are contained in 92.91 g phosphorus if the formula of the molecule is Pq? ( d ) How many atoms are contained in 92.91 g phosphorus? (e) How many molecules are contained in 92.91 g phosphorus? Atomic masses of Ca and P are 40.08 and 30.974. Hence 1 mol Ca = 40.08 g Ca 1 mol P = 30.974 g P n(Ca) = n(P) = mass of Ca 10.02 g = 0.250 mol Ca atoms atomic mass of Ca 40.08 g/mol mass of P atomic mass of P 92.91 g - 30.974 g/mol = 3.000 mol P atoms (c) Molar mass of P4 is (4)(30.974) = 123.90. Then 92.91 g mass of P4 = 0.7500 rnol P4 molecules n(p4) = molar mass of P4 123.90 g/mol (4 (4 Number of atoms of P = (3.000 mol)(6.022 x 1023atoms/mol) = 1.807 x 1024atoms P Number of molecules of P4 = (0.7500 mo1)(6.022 x 1023molecules/mol) = 4.5 17 x 1023molecules P4 CHAP. 21 ATOMIC AND MOLECULAR MASS; MOLAR MASS 23 2.9. How many moles are represented by Ca(NO&? (a) 6.35 g of CO2, (b) 9.11 g of SO2, ( c ) 15.02 g of Refer to the table at the end of the book or to the periodic table inside the back cover for the appropriate atomic masses. From these, Molar mass of CO2 = 12.01 + 2(16.00) = 44.01 g/mol Molar mass of Si02 = 28.09 + 2(16.00) = 60.09 g/mol Molar mass of Ca(N03), = 40.08 + 2[14.01 + 3(16.00)] = 164.10 g/mol (a) Amount of CO2 = 6.35 g x (1 mo1/44.01 g) = 0.1443 mol (302 (b) Amount of Si02 = 9.1 1 g x (1 mo1/60.09 g) = 0.1516 mol Si02 (c) Amount of Ca(N03), = 15.02 g x (1 mo1/164.10 g) = 0.0915 mol Ca(N03), The result (a) is a measure of the number of CO2 molecules (CO2 is normally a gas in which the CO2 molecules are separated from each other and have individual physical identities). Si02 on the other hand is a complicated crystalline solid (quartz), in which each silicon is surrounded by more than two oxygens and each oxygen by more than one silicon, so that there is no physically distinct cluster of one silicon with two oxygens. The result (b) represents a count of the number of SiO2 formula units. Ca(N03)2 is an ionic crystal, the given sample containing 0.09 15 moles of calcium ions and twice that number of moles of nitrate ions. MULTIPLE PROPORTIONS 2.10. Three common gaseous compounds of nitrogen and oxygen of different elementary composition are known, (A) laughing gas containing 63.65% nitrogen, (B) a colorless gas containing 46.68% nitrogen, and (C) a brown gas containing 30.45% nitrogen. Show how these data illustrate the law o multiple proportions. f According to the law of multiple proportions, the relative amounts of an element combining with some fixed amount of a second element in a series of compounds are the ratios of small whole numbers. On the basis of 100 g of each compound, we tabulate below the mass of N, the mass of 0 (obtained by difference), and the mass of N per gram of 0. Compound A g of N g of 0 (g of N)/(g of 0) Compound B 46.68 53.32 0.875 5 Compound C 30.45 69.55 0.437 8 63.65 36.35 1.7510 The relative amounts of nitrogen in the three cases are not affected if all three amounts are divided by the smallest of them. The relative amounts are indeed the ratios of small whole numbers, 4, 2, and 1, within the precision of the analyses. The law of multiple proportions was an important contribution to the credibility of Dalton's atomic theory. It was discovered before relative atomic masses were well known (note that A , values were not involved in the calculation above). However, it follows logically from the premises that all atoms of the same element have the same mass (which is unchangeable) and that compounds contain elements in the relative proportions of simple whole numbers. 24 ATOMIC AND MOLECULAR MASS; MOLAR MASS [CHAP. 2 Supplementary Problems ATOMIC MASS 2.11. Naturally occurring argon consists of three isotopes, the atoms of which occur in the following abundances: 0.34% 36Ar,0.07% 38Ar, and 99.59% 40Ar. Calculate the atomic mass of argon from these data and from data in Table 2-1. Ans. 39.948 2.12. Naturally occurring boron consists of 80.22% I'B (nuclidic mass = 11.009) and 19.78% another isotope. To account for the atomic mass, 10.810, what must be the nuclidic mass of the other isotope? Ans. 10.01. 2.13. 35C1and 37C1 the only naturally occurring chlorine isotopes. What percentage distribution accounts for are the atomic mass, 35.4527? Ans. 24.23% 37Cl 2.14. To account for nitrogen's atomic mass of 14.00674, what must be the ratio of "N to 14N atoms in natural nitrogen? Ignore the small amount of 16N. Ans. 0.00369 2.15. At one time there was a chemical atomic mass scale based on the assignment of the value 16.0000 to naturally occurring oxygen. What would have been the atomic mass, on such a table, of silver, if current information had been available? The atomic masses of oxygen and silver on the present table are 15.9994 and 107.868 2. Ans. 107.872 2.16. The nuclidic mass of 90Sr had been determined on the old physical scale (l6O= 16.0000) as 89.936. Recompute this to the present atomic mass scale, on which l 6 0 is 15.9949. Ans. 89.907 2.17. In a chemical atomic mass determination, the tin content of 3.769 2 g of SnC14 was found to be 1.7170 g. If the atomic mass of chlorine is taken as 35.453, what is the value for the atomic mass of tin determined from this experiment? Ans. 118.65 2.18. A 12.5843-g sample of ZrBr4 was dissolved and, after several chemical steps, all of the combined bromine was precipitated as AgBr. The silver content of the AgBr was found to be 13.2160 g. Assume the atomic masses of silver and bromine to be 107.868 and 79.904. What value was obtained for the atomic mass of Zr from this experiment? Ans. 91.23 2.19. The atomic mass of sulfur was determined by decomposing 6.2984 g of Na2C03 with sulfuric acid and weighing the resultant Na2S04 formed. The weight was found to be 8.438 0 g. In this reaction, all the sodium in the starting material (Na2C03)appears in the product (Na2S04).Calculate the atomic mass of sulfur from this experiment. Ans. 32.017 CHAP. 21 ATOMIC AND MOLECULAR MASS; MOLAR MASS 25 MOLAR MASS 2.20. (d) SS, (e) ca3(po4)2, Determine the molecular mass (or formula unit mass) to 0.01 (f Fe4[Fe(CN)613- U for (a) LiOH, (b) H2SO4, (c) 02, Ans. (a) 23.95, (b) 98.08, (c) 32.00, ( d) 256.53, (e) 310.18, ( f ) 859.28 2.21. How many grams of each of the constituent elements are contained in one mole of (a) CH4, (b) Fe303, (c) Ca3P2? How many atoms of each element are contained in the same amount of compound? Ans. (a) 12.01 g C, 4.032 g H; 6.02 x 1023atoms C, 2.41 x 1024atoms H (b) 1 11.69 g Fe, 48.00 g 0; 1.204 x 1024atoms Fe, 1.81 x 1024atoms 0 ( c ) 120.23 g Ca, 61.95 g P; 1.81 x 1024atoms Ca, 1.204 x 1024atoms P 2.22. Calculate the number of grams in a mole of each of the following common substances: (a) calcite, CaCO,; (b) quartz, Si02; (c) cane sugar, C12H22011; (d) gypsum, CaS04.2H20; (e) white lead, Pb(OH)2. 2YbC03. Ans. (a) 100.09 g; (b) 60.09 g; (c) 342.3 g; ( d ) 172.2 g; (e) 775.7 g 2.23. What is the average mass in kilograms of (a) a helium atom, ( b ) a fluorine atom, (c) a neptunium atom? Ans. (a) 6.65 x 10-27 kg; (b) 3.15 x 10-26 kg; (c) 3.94 x 10-25 kg 2.24. What is the mass of one molecule of (a) CH30H, (b) C60H122, (c) C I2~~ H2000 O1ooo? Ans. (a) 5.32 x 10-26 kg; (b) 1.40 x 10-24 kg; ( c ) 5.38 x 10-23 kg 2.25. How many moles of atoms are contained in (a) 32.7 g Zn, (b) 7.09 g C1, (c) 95.4 g Cu, ( d ) 4.31 g Fe, (e) 0.378 g S? Ans. (a) 0.500 mol; (b)0.200 mol; (c) 1.50 mol; ( d ) 0.077 2 mol; (e) 0.01 1 8 mol 2.26. How many moles are represented by (a) 24.5 g H2SO4, (b) 4.00 g 02? Ans. (a) 0.250 mol; (b) 0.125 mol 2.27. (a) How many moles of Cd and of N are contained in 132.4 g of Cd(N03)2-4H20? (b) How many molecules of water of hydration are in this same amount? Ans. (a) 0.429 mol Cd, 0.858 mol N, (6) 1.033 x 1024molecules H 2 0 2.28. How many moles of Fe and of S are contained in (a) 1 mol of FeS2 (pyrite), (b) 1 kg of FeS2? (c) How many kilograms of S are contained in exactly 1 kg of FeS2? Ans. (a) 1 mol Fe, 2 rnol S; (b) 8.33 mol Fe, 16.7 mol S; (c) 0.535 kg S 2.29. A certain public water supply contained 0.10 ppb (part per billion) of chloroform, CHC13. How many molecules of CHC13 would be contained in a 0.05-mL drop of this water? Ans. 2.5 x 10" MULTIPLE PROPORTIONS 2.30. Verify the law of multiple proportions for an element, X, which forms oxides having percentages of X equal to 77.4%, 63.2%, 69.6%, and 72.0/0. If the compound with 77.4% X is XO, what is X, and what are the other compounds? Ans. The relative amounts of X combining with a fixed amount of oxygen are 2, 1, , and The relative ; amounts of oxygen combining with a fixed amount of X are, 1, 2, $, and Since A , ( X ) = 54.8, X is Mn, and the other compounds have the empirical formulas Mn02, Mn2C+, and Mn304. :. 1. This page intentionally left blank Chapter 3 Formulas and Composition Calculations EMPIRICAL FORMULA FROM COMPOSITION As defined in Chapter 2, an empirical formula expresses the relative numbers of atoms of the different elements in a compound using the smallest integers possible. These integers may be found by converting analytical weight composition data to the amount of each element, expressed in moles of atoms, contained in some fixed weight of the compound. Consider a compound that analyzes 17.09% magnesium, 37.93% aluminum, and 44.98% oxygen. (Unless stated to the contrary, percentage is a weight percentage, i.e., number of grams of the element per 100 g of the compound.) Table 3-1 gives a systematic scheme for handling the data. Table 3-1 (2) (3) (4) (5) Element, E Mg A1 0 Mass of E per Fixed Amount of Compound (in this case, 100 g), m(E) 17.09 g 37.93 g 44.98 g Amount of E in Moles of Atoms, Atomic Mass of E, A m 24.31 g/mol 26.98 g/mol 16.00 g/mol 0.703 mol 1.406 mol 2.812 mol n(E Smallest n ( E ) 1.oo 2.00 4.00 The numbers in column (4) are the numbers of moles of atoms of the component elements in the fixed weight of compound, 100 g, chosen as the basis. Any set of numbers obtained by multiplying or dividing each of the numbers in column (4) by the same factor will be in the same ratio to each other as the numbers in (4). The set in column ( 5 ) is such a set, obtained by dividing each n(E) value in (4) by the smallest entry in (4), 0.703. Column ( 5 ) shows that the relative numbers of moles of atoms, and therefore of atoms themselves, of Mg, Al, and 0 in this compound are 1 :2: 4. Therefore the empirical formula is MgA1204. COMPOSITION FROM FORMULA The existence of a formula for a compound implies that fixed relationships exist between the weights of any two elements in the compound or between the weight of any element and the weight of the compound as a whole. These relationships can best be seen by writing the formula in a vertical form, as illustrated in Table 3-2 for the compound A1203. The sum of the entries in column (4) for the elements equals the molar mass of the compound. The entries in column ( 5 ) represent the fractionaz content of the various elements in the compound. These numbers are really dimensionless (g/g) and are the same in any unit of mass. Thus, 1 gram (pound, ton, etc.) of A1203contains 0.529 gram (pound, ton, etc.) of A1 and 0.471 gram (pound, ton, etc.) of 0. It is obvious that the sum of the constituent fractions of any compound must equal 1.000. The percentage of aluminum in A1203is the number of parts by weight of A1 in 100 parts by weight of A1203.It follows that the percentage is expressed by a number 100 times as great as the fraction. Thus, the percentages of aluminum and oxygen are 52.9% and 47.1 %, respectively. The sum of the constituent percentages of any compound must equal 100.0%. 27 28 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 Table 3-2 (3) (atomic mass of element) A m (4) (5) m(E) per Mole of Compound = n(E) x ALE) I m(E) per g of Compound 2 mol 3 mol 1 mol 27.0 g/mol 16.0 g/mol 54.0 g 48.0 g 102.0 g A1203= 0.529 g Al/g A1203 54*0 ~ 102.0 g A1203 Check: 48'0 = 0.471 g O/g A1203 1 Molar mass= 102.0 I 1.000 Sometimes the composition of a substance with respect to a particular element is expressed in terms of a simple compound that can be prepared from that element. For example, the aluminum content of a glass may be expressed as 1.3% A1203. This means that if all the aluminum contained in 100 g of the glass were converted to A1203, the weight of the A1203 would be 1.3 g. This convention is not meant to imply that the aluminum in the glass is in the chemical form A1203. In many cases oxide notations are the result of historical errors in the assignment of chemical structures to complex substances. Whatever the origin, it is a straightforward procedure to convert the data in such a form to direct elementary composition, or vice versa, by the use of a quantitative factor such as that found in column (5) of Table 3-2. The ratio 54.0 g AI 102.0 g A1203 or 102.0 g A1203 54.0 g A1 is called a quantitative factor and may be used as a special conversion factor in numerical problems. (See the solution to Problem 1.19.) NONSTOICHIOMETRIC FACTORS The calculation of quantities based on formulas and equations (Chapter 4) is called stoichiometry. Use of conversion factors is common even when the relative proportions are not fixed by a chemical formula. Consider a silver alloy used for jewelry containing 86% silver. Factors based on this composition, such as 100 g alloy 86 g Ag may be used as conversion factors in all problems involving alloys of this particular composition, and are termed nonstoichiometric factors. or 0.86 g Ag 1 g alloy NUCLIDIC MOLECULAR MASSES AND CHEMICAL FORMULAS The molecular mass of a compound, computed from the atomic masses of the constituent elements, is the average mass (in U) of a molecule, weighted among the various isotopic forms of the different component elements. A nuclidic molecular mass may be defined for a molecule made up of particular nuclides by adding nuclidic atomic masses in the same way that the usual molecular mass is computed from the atomic masses. A mass spectrometer is an instrument that separates particles of different isotopic composition and measures their individual relative masses. If a nuclidic mass of an unknown compound is known with great precision from mass spectrometry, the exact molecular formula can often be deduced directly from this information without resort to a quantitative chemical composition analysis. CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 29 EXAMPLE 1 Consider the three gases CO, N2, and C2H4. Since I2C, l60, 14N, and 'H dominate over all other isotopes, the mass spectrometer will reveal the presence of a particle of approximate mass 28 in all three cases. If the measurements are made with great precision, the three gases can easily be distinguished on the basis of their nuclidic masses, which are calculated below. 12~160 12.0000 15.9949 ____27.9949 U 2(12.0000) = 24.000 0 4( 1.007 83) = 4.03 1 3 28.031 3 14N2 2( 14.00307) = 28.006 1 U 12Cz1H4 U EXAMPLE 2 Find the formula of an organic compound whose dominant nuclidic species was found to have a precise molecular mass of 44.025. It is known that no elements other than C, H, 0, and N are present. The number of carbon atoms in the molecule, n(C), must be at least 1, otherwise the compound would not be organic. n(C) cannot be greater than 3, because 4 carbon atoms would contribute 48 to the total mass number of the molecule, 44. Similar constraints limit the number of 0 or N atoms per molecule. The possible combinations of C, 0, and N consistent with the limiting mass are listed in column (1) of Table 3-3. Column (2) lists the mass numbers of the (C, 0, N) skeletons. Column (3) lists the number of hydrogen atoms needed to bring the mass number of the molecule to 44. Column (,4) lists the maximum number of H atoms consistent with rules for molecular structure as discussed in Chapters 9 and 15. One such rule is that n(H, max) is equal to twice the number of carbon atoms plus the number of nitrogen atoms plus 2. Column (5) lists the allowed formulas consistent with the total mass number and with all the assumptions and rules. Note that all skeletons for which the number in column (3) (the mass shortage to be made up by hydrogen) exceeds the number in column (4) (the amount of hydrogen allowable for the skeleton by the rules of valence) are rejected. Column (6) tabulates the nuclidic molecular masses for the allowed formulas, computed from the nuclidic masses in Table 2-1. When the computed molecular masses are compared with the experimental value, 44.025, it is seen that C2OH4 is the only allowable formula that fits the data within the claimed precision; therefore this must be the formula of the substance. Table 3-3 (1) (2) (C, 0, N) Skeleton C c 2 Mass Number of Skeleton (3) 44 minus Mass Number of Skeleton (4) (5) n(H, max) Molecular Formula (6) Nuclidic Molecular Mass I I 12 24 36 28 44 I I 32 20 I 1 4 6 I 1 C3H8 1 I c 3 44.063 CO I I 4 I I 1 43.990 38 CON 42 6 2 1 7 5 C2NH6 CONH2 44.050 44.014 30 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 (Not all the allowable molecular formulas in column (5) represents stable chemical substances. Nevertheless, they might possibly show up in a mass spectrometer reading, which details not only stable compounds but also ions and decomposition fragments.) Solved Problems CALCULATION OF FORMULAS 3.1. Derive the empirical formula of a hydrocarbon that on analysis gave the following percentage composition: C = 85.63%, H = 14.37%. The tabular solution, based on 100 g of compound, is as follows: I 1 C H 85.63 g 14.37 g 1 12.011 g/mol 1.008 g/mol 7.129 mol 14.26 rnol 1 .ooo where E = element; m(E) =mass of element per 100 g of compound; A,(E) = atomic mass of element; n(E) = amount of element per 100 g of compound, expressed in moles of atoms. The procedure of dividing n(E) by n(C) is equivalent to finding the number of atoms of each element for every atom of carbon. The ratio of H to C atoms is 2: 1. Hence, the empirical formula is CH2. (The formulas C2H4,C3H6,C4H8,etc., imply the same percentage composition as does CH2, but for the empirical formula the smallest possible integers are chosen.) There is no stable substance with the formula CH2. It is necessary to determine the molar mass to determine the molecular formula. If this hydrocarbon were a gas or an easily volatilized liquid, its molar mass could be estimated from the density of the gas, as shown in Chapter 5. Supposing such a determination yields a molar mass of about 55 g/mol, what is the molecular formula? The formula unit mass of CH2 is 14. Since 55 is very close to 4 x 14 (but far from 3 x 14 or 5 x 14), the compound must be C4H8. 32 .. A compound gave on analysis the following percentage composition: K = 26.57%, CR = 35.36%, 0 = 38.07%. Derive the empirical formula of the compound. The standard tabular solution, as applied to 100 g of compound, follows. (1) (2) m(E) 26.57 g 35.36 g 38.07 g (3) Ar(E) 39.10 g/mol 52.00 g/mol 16.00 g/mol (4) (5) E K Cr 0 n(E) E--- m(E) A,(E) n(E) 0.680 0 rnol 1.ooo 1 .ooo 3.499 (6) 4E) 0.6800 mol 2 2 7 2 0.6800 mol 0.6800 mol 2.379 rnol In contrast to previous examples, the numbers in column (5) are not all integers. The ratio of the numbers of atoms of two elements in a compound must be the ratio of small whole numbers, in order to satisfy one of the postulates of Dalton's atomic theory. Allowing for experimental and calculational uncertainty, we see that the entry of oxygen in column (9,3.499, is, to within the allowed error, 3.500 or $, indeed the ratio of small whole numbers. By rounding off in this way and multiplying each of the entries in column (5) by 2, we arrive at the set of smallest integers that correctly represent the relative numbers of atoms in the compound, as tabulated in column (6). The formula is thus K2Cr207. CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 31 3.3. A 15.00-g sample of a hydrated salt, Na2S04-xH20, was found to contain 7.05 g of water. Determine the empirical formula of the salt. Hydrates are compounds containing water molecules loosely bound to the other components. H 2 0 may usually be removed intact by heating such compounds and ma:y then be replaced by wetting. The Na2S04 and H 2 0 groups may thus be considered as the units of which the compound is made, and their formula unit masses are used in place of atomic masses. It is clearly more convenient to base the tabular solution in this case on 15.00 g of compound (which contains 15.00 - 7.05 = 7.95 g Na2S04). X Na2S04 H20 m(X> 7.95 g 7.05 g 142.1 g/mol 18.02 g/mol 0.0559 mol 01.391 mol 6.99 The mole ratio of H20 to Na2S04 is, to within the allowed error, 7 to 1, and the empirical formula is Na2S04. 7H10. 3.4. A 2.500-g sample of uranium was heated in the air. The resulting oxide weighed 2.949 g. Determine the empirical formula of the oxide. 2.949 g of the oxide contains 2.500 g U and 0.449 g 0 (obtained by subtracting the initial mass from the final mass). Calculation shows that 2.949 g of the oxide contains 0.02806 mol oxygen atoms and 0.010 50 mol uranium atoms, or 2.672 mol oxygen atoms per mol uranium atoms. The smallest multiplying integer that will give whole numbers is 3. n ( 0 ) - 2.672 m o l 0 - 3(2.672 mol 0) - 8.02 mol 0 .(U) 1 .OOO mol U - 3( 1.OOO mol 1 - 3.00 mol U 3 The empirical formula is U30g. Emphasis must be placed on the importance of carrying out the computations to as many significant figures as the analytical precision requires. If numbers in the ratio 2.67:l had been multiplied by 2 to give 5.34:2 and these numbers had been rounded off to 5:2, the wroing formula would have been obtained. This would have been unjustified because it would have assumed an error of 34 parts in 500 in the analysis of oxygen. The weight of oxygen, 0.449 g, indicates a possible error of only a few parts in 500. When the multiplying factor 3 was used, the rounding off was from 8.02 to 8.00, the assumption being made that the analysis of oxygen may have been in error by 2 parts in 800; this amount of error is more reasonable. 3.5. A 1.367-g sample of an organic compound was combusted in a stream of dry oxygen to yield 3.002 g CO2 and 1.640 g H20. If the original compound contained only C , H, and 0, what is its empirical formula? It is necessary to use quantitative factors for CO;?and H 2 0 to find how much C and H are present in the combustion products and, thus, in the original sample. 1 molC ( 4 ~ ~ ~ (3.002 g CO2) = 0.819 g C ~ ~ ~ - - ) 2 mol H nz(H) = (1 mol H 2 0 ) (1*640g H 2 0 ) = [2(1*oo8 91 18.02 g H20 (1.640 g H 2 0 ) = 0.183 5 g H The amount of oxygen in the original sample cannot be obtained from the weight of combustion products, since the CO2 and H20 contain oxygen that came partly from the combined oxygen in the compound and partly from the oxygen stream used in the combustion process. The oxygen content of the sample can be obtained, however, by difference. m(0)= m(compound) - m(C) - n(H) = 1.367 - 0.819 - 0.184 = 0.364 g 32 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 The problem can now be solved by the usual procedures. The numbers of the moles of the elementary atoms in 1.367 g of compound are found to be C, 0.068 2; H, 0.182 0; 0, 0.022 8. These numbers are in the ratio 3:8:1, and the empirical formula is C3H80. COMPOSITION PROBLEMS 3.6. A strip of electrolytically pure copper weighing 3.178 g is strongly heated in a stream of oxygen until it is all converted to 3.978 g of the black oxide. What is the percentage composition of this oxide? Total weight of black oxide = 3.978 g Weight of copper in oxide = 3.178 g Weight of oxygen in oxide = 0.800 g Fraction of copper = Fraction of oxygen = weight of copper in oxide - 3.178 g - - 0.799 = 79.9% = total weight of oxide 3.978 g weight of oxygen in oxide - 0.800 g -= 0.201 = 20.1% total weight of oxide 3.978 g Check: 100.0% 37 .. (a) Determine the percentages of iron in FeC03, Fe2O3, and Fe304. (b) How many kilograms of iron could be obtained from 2.000 kg of Fe203? (a) Molar mass of FeC03 is 115.86; of Fe203, 159.69; of Fe304,23 1.54. Consider 1 mol of each compound. Fraction of Fe in FeCO Fraction of Fe in Fe203 m(1 mol Fe) - 55.847 g - 0.4820 = - m(1 mol FeC03) - 115.86 g = 1 48.20% m(2 mol Fe) - 2(55*847g, = 0.6994 = 69.94% m(1 mol Fe203) - 159.69 g m(3 mol Fe) - 3(55*847 = 0.723 6 g) m(1 mol Fe2O4) - 231.54 g =1 1 Fraction of Fe in Fe304= 72.36% (b) From (a), the weight of Fe in 2.000 kg Fe203 is 0.699 4 x 2.000 kg = 1.399 kg Fe. 3.8. Given the formula K2CO3, determine the percentage composition of potassium carbonate. One mole of K2CO3 contains 2 moles K = 2(39.098 3) = 78.197 g K 47.998 g 0 I moIeC = l(12.011) = 12.011 g C 3 moles 0 = 3( 15.9994) = Molar mass of K2CO3 = 138.206 g 78 197 = Fraction of K in K2CO3 = - 0.565 8 = 56.58% 138.208 12.011 = Fraction of C in K2C03= - 0.0869 = 8.69% 138.206 47.998 Fraction of 0 in K2CO3 = - 0.347 3 = 34.73% = 138.206 Check: 100.0% CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 33 3.9. (a) Calculate the percentage of CaO in CaC03. (6) How many pounds of CaO can be obtained from 1 ton of limestone that is 97.0% CaC03? One ton is 2000 lb. ( a ) The quantitative factor can be written by considering the conservation of Ca atoms. 1 rnol CaC03 contains 1 mol Ca, which is the same amount of Ca as is contained in 1 rnol CaO. Therefore, Fraction of CaO in CaC03 = molar mass Ca.0 - ___ __ = 0.560 = 56.0% molar mass CaC03 100.1 (4 Weight of CaC03 in 1 ton limestone = 0.970(2000 lb) = 1 940 lb CaC03 Weight of CaO = (fraction of CaO in CaC03)(weight of CaC03) = 0.560(1 940 lb) = 1090 lb CaO in 1 ton limestone 3.10. How much 58.0% sulfuric acid solution is needed to provide 150 g of H2S04? Let x := mass of sulfuric acid solution x = 259 1 s o h ; Using the concept of an inverse conversion factor avoids the use of algebra: (I50 g H2S04) (58.0 100 g soln g H2SO4 = 259 g soln 3.11. How much calcium is in the amount of Ca(N03)2that contains 20.0 g of nitrogen? It is not necessary to find the weight of Ca(N03)2 containing 20.0 g of nitrogen. The relationship between two component elements of a compound may be found directly from the formula. Weight of Ca = (20.0 g N) [~p40081gLgc~)] = 28.6 g Ca 3.12. (a) How much H2S04could be produced from 500 kg of sulfur? (b) How many kilograms of Glauber's salt, Na2S04.10H20, could be obtained from 1 000 kg H2S04? (a) The formula H2SO4 indicates that 1 rnol S (32.07 g S) will give 1 rnol H2SO4 (98.08 g H2SO4). We assume that the sulfur is utilized completely. Then, since the ratio of any two constituents is independent of the mass units, Weight of H2SO4 = (500 kg S) 98'08 kg H2So 32.07 kg$) (b) ( 1 529 kg H2S04 1 rnol H2SO4 (98.08 g H2S04) will give 1 rnol Na2S04.10H20 (322.2 g Na2S04.10H20), since each substance contains one sulfate (SO4) group per formula unit. Then Weight of Na2S04.10H20 (1 .OOO kg H2S04) = ( 322.2 kg Na2S04- 10H20 98.08 kg H2S04 = 3.285 kg N a 2 S 0 4 .10H20 3.13. How many tons of Ca3(P0& must be treated with carbon and sand in an electric furnace to make 1 ton of phosphorus? Assume complete conversion of thle phosphorus. 34 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 The formula Ca3(P04)2 indicates that 2 mol P (2 x 30.974 g=61.95 g P) is contained in 1 mol Ca3(P0& (310.2 g Ca3(P04)2).Then, changing grams to tons in the weight ratio, we obtain Weight of Ca3(P04)2 = (1 ton p) (310*2 tons Ca3(P04)2 61.95 tons P = 5.01 tons Ca3(P04)2 3.14. A 5.82-g silver coin is dissolved in nitric acid. When sodium chloride is added to the solution, all the silver is precipitated as AgCl. The AgCl precipitate weighs 7.20 g. Determine the percentage of silver in the coin. Fraction of Ag in AgCl = molar mass Ag molar mass AgCl 107.9 - - 0.753 = - 143.3 Mass of Ag in 7.20 g AgCl = (0.753)(7.20 g) = 5.42 g Ag Hence the 5.82-g coin contains 5.42 g Ag. Fraction of Ag in coin = - 0.93 1 = 93.1 % = 5 42 g 5.82 g Ag 3.15. A sample of impure sulfide ore contains 42.34% Zn. Find the percentage of pure ZnS in the sample. The formula ZnS shows that 1 mole of ZnS contains 1 mole Zn; hence a suitable conversion factor is 1 molar mass ZnS - 97.46 g ZnS 65.39 g Zn 1 molar mass Zn Consider 100.0 g of sample; it contains 42.34 g Zn. Then 97.46 g ZnS (42.34 g Zn) 65.39 Zn ( ) = 63.11 g ZnS in 100 g of sample, or 63.1 1 YOpure ZnS 3.16. Fertilizers are compounds usually used to supply potassium, nitrogen, and phosphorus to soil. If a bag of fertilizer consists of almost pure K N 0 3 (potassium nitrate), what percentages of those three important elements should appear on the label? Consider one mole of pure KN03, which contains: 1 mole of K = 1 mole of N = 39.10 g 14.01 g 48.00 g 101.11 g 3 moles of 0 = 3(16.00) = Molar mass of K N 0 3 = Percent of K=(39.10 g K/IOl.ll g compound x 100/~ =38.67% Percent of N=(14.01 g N/101.11 g compound x 100% = 13.86% (The percent of P of course is zero.) 3.17. (a)A Pennsylvania bituminous coal is analyzed as follows: Exactly 2.500 g is weighed into a fused silica crucible. After drying for 1 hour at 110 "C, the moisture-free residue weighs 2.415 g. The crucible, next, is covered with a vented lid and strongly heated until no volatile matter remains. The residual coke button weighs 1.528 g. The crucible is then heated without the cover until all specks of carbon have disappeared, and the final ash weighs 0.245 g. What is the proximate CHAP. 31 FORMULAS A N D COMPOSITION CALCULATIONS 35 analysis of this coal, i.e., the percents of moisture, volatile combustible matter (VCM), fixed carbon (FC), and ash? Moisture = 2.500 g VCM = 2.41 5 g FC 1.528 g - 2.41.5 g - = 0.085 g 1.528 g = 0.887 g = 1.283 g - 0.2458 Ash = 0.245 g Total Fraction of moisture 0.085 g 2.500 g 2.500 g coal = ___ = 0.034 = 3.4% Similarly, the other percentages are calculated to be 35.5% VCM, 51.3% FC, 9.8% ash. (b)On the "dry basis" a sample of coal analyzes as follows: VCM, 21.06%; fixed carbon, 71.80%; ash, 7.14%. If the moisture present in the coal is 2.49%, what is the analysis on the "wet basis"? Consider as a basis 100.0 g of wet coal. Of this, 100.0 g VCM, FC, and ash then are VCM FC Ash - 2.49 g = 9.75 g is dry coal. The weights of (0.2106)(97.5) = 20.5 g or 20.5%' of the lOOg of wet coal (0.7180)(97.5) = 70.0 g or 70.0o/Clof the 1OOg of wet coal (0.071 4)(97.5) = 7.0 g or 7.0% of the lOOg of wet coal The sum of the above percentages plus the 2.5% moisture comes to 100%. 3.18. A certain fertilizer, "A", contains 38.7% K, 13.9% N, and no P. Another, "B", contains 12.2% N, 26.9% P, and no K. What will be the percentages lof K, N, and P in a fertilizer made by blending equal weights of A and B? We may choose to mix 100 g of each, containing: I K 38.7 g 0 I I P 0 - 1 I I I From A From E3 Total nt 1 1 I I I I 13-9 g 12.2 g 26.1 g I I I I 26.9 g 26.9 g 38.7 g I I Percent = (rn/200 g ) x 100% = 19.4% K, 13.1% N , 13.5% P The manufacturer wishes to market a mixture of A and B in which the two elements K and P are present in equal proportions. What proportions of A and B should be used? Consider 100 g of mixture; let x = g of A; then 100 - x -= g of B. Setting % K = % P in the mixture, 0.387.~ 0.269(100 - x ) . Solving this we find: x = 41.0 g of A, and (100 - x) = 59.0 g of B. = 3.19. When the Bayer process is used for recovering aluminunn from siliceous ores, some aluminum is always lost because of` the formation of an unworkab1,e "mud" having the following average formula: 3 N a 2 0 .3Al2O3- 5Si02. 5H20. Since aluminum and sodium ions are always in excess in the solution from which this precipitate is formed, the precipitation of the silicon in the "mud" is complete. A certain ore contained 13% (by weight) kaolin (A1203- 2Si02 2H20) and 87% gibbsite (A1203-3H20). What percent of the total aluminurn in this ore is recoverable in the Bayer process? a 36 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 Consider 100 g ore, which contains 13 g kaolin and 87 g gibbsite. Weight of A1 in 13 g kaolin = 13 g kaolin x 2 mol A1 1 mol kaolin = 54.0 13 ~ - = 2 . 7 g A l 258 Weight of A1 in 87 g gibbsite = 87 g gibbsite x Total weight of A1 in 100 g ore = 2.7 g 2 rnol A1 54.0 = 87 x -= 30.1 8 A1 156 1 rnol gibbsite = 32.8 g + 30.1 g Kaolin has equal numbers of A1 and Si atoms, and 13 g kaolin contains 2.7 g Al. The mud takes 6 A1 atoms for 5 Si atoms or 6 AI atoms lost for every 5 Si atoms in the kaolin. Hence the precipitation of all the Si from 13 g kaolin involves the loss of (2.7 g ) = 3.2 g AI. 4 Fraction of AI recoverable = recoverable A1 - (32.8 - 3.2) g = 0.90 = 90% total AI 32.8 g 3.20. A clay was partially dried and then contained 50% silica and 7% water. The original clay contained 12% water. What is the percentage of silica in the original sample? Assuming that the only substance lost in the drying process was water, the original and dried clays have the compositions 1 I Original Dried YO Water 12 7 1 ?40 Silica 1 I % Other I I I 88-x 43 50 The ratio of silica to the other dry constituents must be the same in both clays; hence x Solving, x = 47; 88-x-43 __ 50 i.e., there is 47% silica in the original clay. 3.21. Bronze is an alloy of copper and tin. A 0.655 4 g sample of a certain bronze was dissolved in nitric acid and the tin removed. After appropriate treatment of the solution, titration with sodium thiosulfate revealed that it contained 8.35 1 millimoles of copper. Calculate the percentages of copper and tin in this bronze. %Cu = 100% x [8.351 x IOp3 rnol Cu x 63.55 g Cu/mol Cu]/O.6554 g sample = 80.97% %Sn = 100.00% - 80.97% = 19.03% 3 2 . A nugget of gold and quartz weighs 100 g and has a density of 6.4 g/cm3. The densities of gold .2 and quartz are 19.3 and 2.65 g/cm3, respectively. Determine the weight of gold in the nugget. Let x = grams of gold in nugget; then 100 g - x = grams of quartz in nugget. Volume of nugget = (volume of gold in nugget) + (volume of quartz in nugget) 100 g - X + 6.4 g/cm3 - 19.3 g/cm3 from which x = 68 g gold. 1oog-x 2.65 g/cm3 CHAP. 3 1 FORMULAS AND COMPOSITION CALCULATIONS 37 MOLAR MASSES 3.23. A purified cytochrome protein isolated from a bacterial preparation was found to contain 0.376% iron. What can be deduced about the molar mass of the protein. 55.8 The protein must contain at least one atom of iron per molecule. If it contains only one, of weight U = 55.8 daltons, then the molar mass A is given by 0.003 7 6 A = 55.8 daltons A = 14800 daltons If the protein molecule contained x atoms of Fe, the molar mass would be 14800x daltons. The method given here is useful for determining the minimum molar mass of a macromolecular substance when an analysis can be done for one of the minor components. Frequently the approximate molar mass can be determined by a physical method such as osmotic pressure or sedimentation rate, thus fixing the value of the integer x. 3 2 . A purified pepsin isolated from a bovine preparation was subject to an amino acid analysis of its .4 hydrolytic product. The amino acid present in smallest amount was lysine, C6HI4N2o2,and the amount of lysine recovered was found to be 0.43 g per 100 g protein. What is the minimum molar mass of the protein? Proteins do not contain free amino acids, but they do contain chemically linked forms of amino acids, which on degradative hydrolysis can be reconverted to the free amino acid form. The molar mass of free lysine is 146, and we let A be the minimum molar mass of the protein. As in Problem 3.23, the protein molecule must be at least heavy enough to contain one lysine residue. No. of moles lysine = No. of moles protein 1 mol 1 mol 0.43 g x - 100 g x = 146 g "K g from which = 34 000. 3 2 . Unsaturated polyesters, made from maleic acid (C4H404) and ethylene glycol (C2H602),are .5 widely used (with styrene and fiberglass) to make reinforce:d resin structures. A 5.00 gram portion of a batch of polyester was dissolved and treated with 0.00420 moles of sodium hydroxide (NaOH), an amount just sufficient to neutralize all of the acid "end-groups" present. Since there are two end-groups in each molecule, what is the average molar mass of the polyester? 2 x moles of polyester = moles of NaOH Moles of polyester = 0.00420/2 : 0.002 10 moles = A = molar mass = 5.00 g/0.002 10 mol = 2 380 g/mol. FORMULA FROM PRECISE NUCLIDIC MOLECULAR MASS 3 2 . An organic compound was prepared containing at least one and no more than two sulfur atoms .6 per molecule. The compound had no nitrogen, but oxygen could have been present. The massspectrometrically determined molecular mass of the predominant nuclidic species was 110.020. (a) What are the allowable molecular formulas consistent with the mass number 110 and with the facts about the elementary composition? (b) What is the molecular formula of the compound? (a) The nonhydrogen skeleton would be made up of the elements C, 0, and S. The number of possible skeletons can be reduced by the following considerations: (i) The maximum number of carbon atoms is 6, since the mass number of 7 carbons plus 1 sulfur would be 116, in excess of the given value. (ii) The maximum number of hydrogen atoms is 2n(C) + 2. = 14. (iii) The (C, 0, S) skeleton must 38 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 therefore contribute between 96 and 110, inclusive, to the mass number. A rather short list, Table 3-4, will now suffice. (1) (C, 0, S) Skeleton (2) Mass Number of Skeleton 108 (3) (4) n(H, max) 4 4 6 6 8 8 10 (5) 110 - (2) 2 2 6 6 10 10 Molecular Formula C04SH2 C02S2H2 C203SH6 C20S2H6 (6) Nuclidic Molecular Mass co4s c 22 o s 109.967 109.949 110.004 108 104 104 100 100 96 108 104 C2O3s c o2 2 s 109.986 C302s c3os2 C40S C50S 14 2 6 12 14 CSOSH;! C6SH6 109.983 110.019 c6s (b) Of the six formulas consistent with the known mass number, only C6SH6 is consistent with the precise molecular mass. Supplementary Problems CALCULATION OF FORMULAS 3.27. What are the empirical formulas for the chlorides of vanadium containing 58.0%, 67.8%, and 73.6% chlorine? Ans. vc12, vc13, v 1 c4 3.28. A compound contains 21.6% sodium, 33.3% chlorine, 45.1TOoxygen. Derive its empirical formula. Take N = 23.0, C1= 35.5, 0 = 16.0. Ans. NaC103 32. .9 When 1.010 g of zinc vapor is burned in air, 1.257 g of the oxide is produced. What is the empirical formula of the oxide? Ans. ZnO 3.30. A compound has the following percentage composition: H = 2.24%, C = 26.69%, 0 = 71.07%. Its molar mass is 90. Derive its molecular formula. Ans. H2C204 CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 39 3.31. Determine the simplest formula of a compound that has the following composition: Cr = 26.52%, S = 24.52%, 0 = 48.96%. Ans. Cr2S3OI2 Cr2(S04)3 or 3.32. A 3.245-g sample of a titanium chloride was reduced with sodium to metallic titanium. After the resultant sodium chloride was washed out, the residual metal was dried, and weighed 0.819 g. What is the empirical formula of this titanium chloride'? Ans. TiC14 3.33. Calculate the formula of the compound, which exists as a hydrate, and contains 44.6% ytterbium and 27.5% chlorine. Ans. YbCln.6 H 2 0 3.34. An organic compound was found on analysis to contain 47.37% carbon and 10.59% hydrogen. The balance was presumed to be oxygen. What is the empirical formula of the compound? Ans. C3HsO2 3.35. Derive the empirical formulas of the minerals that have the following compositions: (a) ZnS04 = 56.14%, H20=43.86%; (6) MgO=27.16%, SiO2=6O.70%, H 2 0 = 12.14%; ( c ) N a = 12.10%, AI= 14.19%, Si =22.14%, 0=42.09%, H@=9.48%. Ans. (a) Z n S 0 4 . 7H20; (b) 2Mg0.3si02.2H20; ( c ) Na2A12Si3010. 2H20 3.36. A borane (a compound containing only boron and hydrogen) arialyzed 88.45% boron. What is its empirical formula? 3.37. An experimental catalyst used in the polymerization of butadiene is 23.3% CO, 25.3% MO, and 51.4% C1. What is its empirical formula? View Full Document

O ~ M O ~ C I ~ 1 3.38. A 1.500-g sample of a compound containing only C, H, and 0 was burned completely. The only combustion products were 1.738 g CO2 and 0.71 1 g H20. What is the empirical formula of the compound? Ans. C2H403 3.39. Elementary analysis showed that an organic compound contained C, H, N, and 0 as its only elementary constituents. A 1.279-g sample was burned completely, as a result of which 1.60 g of CO2 and 0.77 g of H20 were obtained. A separately weighed 1.625-g sample contained 0.216 g nitrogen. What is the empirical formula of the compound? Ans. C3H703N 3.40. Plaster of Paris is an unusual hydrate of calcium sulfate (CaS04) which contains 6.20% H20. What is its formula? Ans. 2CaS04. HzO, or CaS04.fHzO 3.41. A hydrocarbon containing 92.3% C and 7.74% H was found (by measuring its gas density) to have a molar mass of approximately 79. What is the molecular formula? Ans. C6H6 40 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 COMPOSITION 3.42. A fusible alloy is made by melting together 10.6 Ib bismuth, 6.4 lb lead, and 3.0 lb tin. (a) What is the percentage composition of the alloy? (b) How much of each metal is required to make 70.0 g of alloy? (c) What weight of alloy can be made from 4.2 Ib of tin? Ans. (a) 53% Bi, 32% Pb, 15% Sn; (b) 37.1 g Bi, 22.4 g Pb, 10.5 g Sn; (c) 28 lb alloy 3.43. Calculate the percentage of copper in each of the following minerals: cuprite, Cu20; copper pyrites, CuFeS2; malachite, CuCO3 . Cu(OH)2. How many kilograms of cuprite will give 500 kg of copper? Ans. 88.82%, 34.63%, 57.48%; 563 kg 3.44. What is the nitrogen content (fertilizer rating) of NH4N03? Of (NH4)2S04? of NH3? A ~ s . 35.0% N, 21.2% N, 82.3% N 3.45. Determine the percentage composition of Ca2P207. (a) silver chromate, Ag2Cr04; (b) calcium pyrophosphate, Ans. (a) 65.03% Ag, 15.67% Cr, 19.29% 0; (b) 31.54% Ca, 24.38% P, 44.08% 0 3.46. Find the percentage of arsenic in a polymer having the empirical formula C2H8AsB. Ans. 63.6% As 3.47. The specifications for a transistor material called for one boron atom in 10" silicon atoms. What would be the boron content of 1 kg of such material? Ans. 4 x 10-" kg B 3.48. The purest form of carbon is prepared by decomposing pure sugar, Cl2H22OI1(driving off the contained H20). What is the maximum number of grams of carbon that could be obtained from 500 g of sugar? Ans. 211 g c 3.49. The so-called "vinyl" plastic is a polymer of vinyl chloride, CH2CHC1. (a) What is the percentage of chlorine in this plastic, and (b) how does it compare to the percentage of chlorine in common table salt, NaCl? Ans. (a) 56.7%, (b) common table salt is 60.7% chlorine 3.50. What weight of CuO will be required to furnish 200 kg copper? Ans. 250 kg CuO 3.51. Ordinary salt, NaC1, can be electrolyzed in the molten state to produce sodium and chlorine. Electrolysis of an aqueous solution produces sodium hydroxide (NaOH), hydrogen, and chlorine. The latter two products may be combined to form hydrogen chloride (HCl). How many pounds of metallic sodium and of liquid chlorine can be obtained from 1 ton of salt? Alternatively, how many pounds of NaOH and how many pounds of hydrogen chloride? Ans. 787 lb Na, 1213 lb liquid C12, 1370 lb NaOH, 1248 lb HC1 3.52. Compute the amount of zinc in a metric ton of ore containing 60.0% zincite, ZnO. Ans. 482 kg Zn 3.53. How much phosphorus is contained in 5.00 g of the compound CaC03. 3Ca3(P04)2? How much PZOs? Ans. 0.902 g P, 2.07 g P2O.j CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 41 3.54. A 10.00-g sample of crude ore contains 2.80 g of HgS. What is the percentage of mercury in the ore? A ~ s . 24.1% Hg 3.55. A procedure for analyzing the oxalic acid content of a solution involves the formation of the insoluble complex M0403(C204)3. 12H20. (a) How many grams of this complex would form per gram of oxalic acid, H2C204, 1 mol of the complex results from the reaction with 3 mol of oxalic acid? (b) How many if grams of molybdenum are contained in the complex formed by reaction with 1 g of oxalic acid? Ans. (a) 3.38 g complex; (b) 1.42 g MO 3.56. An agricultural insecticide contains 18% As. Express this as the percentage As205 Ans. 28% As205. 3.57. Express the potassium content of a fertilizer in percent potassium if it is labeled 6.8% K 2 0 . A ~ s . 5.6% K 3.58. A typical analysis of a Pyrex glass showed 12.9% B 2 0 3 , 2.20/0 A1203, 3.8% Na20, 0.4% K20, and the balance SiO?. Assume that the oxide percentages add up to 100%. What is the ratio of silicon to boron atoms in the glass? Ans. 3.6 3.59. A piece of plumber's solder weighing 3.00 g was dissolved in dilute nitric acid, then treated with dilute H2S04. This precipitated the lead as PbS04, which after washing and drying weighed 2.93 g. The solution was then neutralized to precipitate stannic acid, which was decomposed by heating, yielding 1.27 g Sn02. What is the analysis of the solder as YOPb and YOSn. Ans. 66.7% Pb, 33.3% Sn 3.60. Determine the weight of sulfur required to make 1 metric ton of H2SO4. Ans. 327 kg S 3.61. A sample of` impure cuprite C u 2 0 contains 66.6% copper. What is the percentage of pure Cu20 in the sample? A ~ s . 75.00/;1C U ~ O 3.62. A cold cream sample weighing 8.41 g lost 5.83 g of moisture on heating to 110 "C. The residue on extracting with water and drying lost 1.27 g of water-soluble glycerol. The balance was oil. Calculate the composition of this cream. Ans. 69.3%) moisture, 15.1YOglycerol, 15.6% oil 3.63. A household cement gave the following analytical data: A 28.5-g sample, on dilution with acetone, yielded a residue of 4.6 g of aluminum powder. The filtrate, on evaporation of the acetone and solvent, yielded 3.2 g of plasticized nitrocellulose, which contained 0.8 g of benzene-soluble plasticizer. Determine the composition of this cement. Ans. 16.2% Al, 72.6% solvent, 2.8% plasticizer, 8.4% nitrocellulose 3.64. A coal contains 2.4% water. After drying, the moisture-free residue contains 71 .O% carbon. Determine the percentage of carbon on the "wet basis." Ans. 69.3% C 42 FORMULAS AND COMPOSITION CALCULATIONS [CHAP. 3 3.65. A certain breakfast preparation contains 0.637% salt (NaC1). Express this as milligrams of sodium per 60.0 gram serving. Ans. 150 mg sodium 3.66. A liter flask contains a mixture of two liquids (A and B) of specific gravity 1.4. (Specific gravity means the density relative to that of water.) The specific gravity of liquid A is 0.8 and of liquid B, 1.8. What volume of each must have been put into the flask? Assume no change of volume on mixing. Ans. 400 mL of A, 600 mL of B 3.67. When a zinc sulfide ore (ZnS) is roasted, all the sulfur is released to the atmosphere as SO2. If a maximum of 0.060 mg of SO2 is permitted per cubic meter of air, (a) how many cubic meters of air will be needed for safe disposal of the effluent from the roasting of 1.00 metric ton of zinc sulfide, and (b) how large an area would be covered by such a volume of air if it were 1.00 km high? Ans. (a) 1.10 x 10'' m3; (b) 1.10 x 107 m2 (about 4.2 square miles!) 3.68. A taconite ore consisted of 35.0% Fe304 and the balance siliceous impurities. How many tons of the ore must be processed in order to recover a ton of metallic iron (a) if there is 100% recovery, (b) if there is only 75% recovery? Ans. (a) 3.94 tons; (b) 5.25 tons 3.69. A typical formulation for a cationic asphalt emulsion calls for 0.5% tallow amine emulsifier and 70% asphalt, the rest consisting of water and water-soluble ingredients. How much asphalt can be emulsified per pound of the emulsifier? Ans. 140 lb 3.70. Uranium hexafluoride, UF6, is used in the gaseous diffusion process for separating uranium isotopes. How many kilograms of elementary uranium can be converted to UF6 per kilogram of combined fluorine? Ans. 2.09 kg MOLAR MASSES 3.71 One of the earliest methods for determining the molar mass of proteins was based on chemical analysis. A hemoglobin preparation was found to contain 0.335% iron. (a) If the hemoglobin molecule contains 1 atom of iron, what is its molar mass? (b) If it contains 4 atoms of iron, what is its molar mass? Ans. (a) 16700; (b) 66700 3.72. A polymeric substance, tetrafluoroethylene, can be represented by the formula (C2F4)x, where x is a large number. The material was prepared by polymerizing C2F4 in the presence of a sulfur-bearing catalyst that served as a nucleus upon which the polymer grew. The final product was found to contain 0.012% S. What is the value of x if each polymeric molecule contains (a) 1 sulfur atom, (b) 2 sulfur atoms? In either case, assume that the catalyst contributes a negligible amount to the total mass of the polymer. Ans. (a) 2 700; (b) 5 300 3.73. A peroxidase enzyme isolated from human red blood cells was found to contain 0.29% selenium. What is the minimum molecular mass of the enzyme? Ans. 27000 daltons (27000 U) 3.74. A sample of polystyrene prepared by heating styrene with tribromobenzoyl peroxide in the absence of air has the formula Br3C6H3(C8H&. The number n varies with the conditions of preparation. One sample of polystyrene prepared in this manner was found to contain 10.46% bromine. What is the value of n? Ans. 19 CHAP. 31 FORMULAS AND COMPOSITION CALCULATIONS 43 FORMULA FROM PRECISE NUCLIDIC MOLECULAR MASS In all the following problems the nuclidic molecular mass reported is that of the species containing the most prevalent nuclide of each of its elements. 3 7 . An alkaloid was extracted from the seed of a plant and purified. The molecule was known to contain 1 atom .5 of nitrogen, no more than 4 atoms of oxygen, and no other elements besides C and H. The massspectrometrically determined nuclidic molecular mass was found to be 297.138. (a) How many molecular formulas are consistent with mass number 297 and with the other known facts except the precise molecular weight? (b) What is the probable molecular formula? An$. (U) 17; (b) C1803NH19 3 7 . An organic ester was decomposed inside a mass spectrometer. An ionic decomposition product had the .6 nuclidic molecular mass 11 7.090. What is the molecular formula of this product, if it is known in advance that the only possible constituent elements are C, 0, and H, and that no more than 4 oxygen atoms are present in the molecule? Ans. C6O2HI3 3 7 . An intermediate in the synthesis of a naturally occurring alkaloid had a mass-spectrometrically determined .7 nuclidic molecular mass of 205.147. The compound is known to' have no more than 1 nitrogen atom and no more than 2 oxygen atoms per molecule. (a) What is the most probable molecular formula of the compound? (b) What must the precision of the measurement be to exclude the next to most probable formula? Ans. (a) C l 3 0 N H I 9 (nuclidic molecular mass is 205.147). (b) The next closest molecular mass is 205.159 for C140H2]. The range of uncertainty in the experimental value should not exceed half the difference between 205.147 and 205.159, i.e., it should be less than 0.006, or about 1 part in 35000. This page intentionally left blank Chapter 4 Calculations from Chemical Equations INTRODUCTION The balanced equation is the basis for all calculations of the quantities of substances involved in chemical reactions. In the course of working out these numerical problems, the alert student will not only develop skill in calculation, but will acquire much knowledge of descriptive and practical chemistry. If two or more substances interact (or if one substance decomposes) to form a new substance or substances, we say that a chemical change has occurred. The new substances are termed the products of the reaction, and the former the reactants. A chemical reaction is a statement of such an event in which the formulas of the reactants are on the left, followed by a right-pointed arrow and then the formulas of the products. In a balanced equation, coefficients specify the number of molecules (or formula units) of each species involved. The coefficients must satisfy Dalton's requirement that no atoms can be created or destroyed in a chemical reaction. MOLECULAR RELATIONS FROM EQUATIONS The relative numbers of reacting and resulting molecules are indicated by the coefficients of the formulas representing these molecules. For example, the combustion of ammonia in oxygen is described by the balanced chemical equation 4NH3 + 302 ---f 2N2 + 6H20 (4 molecules) (3 molecules) (2 molecules) (6 molecules) in which the algebraic coefficients 4, 3, 2, and 6 indicate that 4 molecules of NH3 react with 3 molecules of O2 to form 2 molecules of N2 and 6 molecules of H20. The balanced equation does not necessarily mean that if 4 molecules of NH3 are mixed with 3 molecules of 0 2 the reaction as indicated will go to completion. Some reactions between chemical substances occur almost instantaneously upon mixing, some occur completely after sufficient time has elapsed, and some reactions go to only a partial extent even after an infinite time. The common interpretation of the balanced equation for all categories is as follows: If a large number of NH3 and 0 2 molecules are mixed, a certain number of N2 and H 2 0 molecules will be formed. At a given instant it is not necessary that either the NH3 or 0 2 is all consumed, but whatever reaction does occur takes place in the molecular ratio prescribed by the equation. In the above reaction, the atoms in seven indicated molecules (4NH3, 302) rearrange to form eight molecules (2N2, 6H20); there is no algebraic rule governing these numbers of molecules, but the number of atoms on each side of the equation must balance for each element, since the reaction obeys the laws of conservation of matter and of nontransmutability of the elements. Thus the equation is balanced and checked by counting the atoms of each kind (4N, 12H, 60), not the molecules. The number of atoms of any element occurring in a given substance is found by multiplying the subscript of that element in the formula by the coefficient of the formula. Thus 4NH3 represents 12 atoms of H, because there are 3 atoms of H in each of 4 molecules of NH3. In some more complex formulas, several subscripts must be multiplied together before multiplying by the coefficient of the entire formula. Thus 3(NH4)2S04 would represent 24 atoms of H, because each of the 3 formula units of (NH4)2S04contains 2 NH4 groups, each of which in turn contains 4H atoms. MASS RELATIONS FROM EQUATIONS Because 1 mol of any substance is N A molecules (Chapter 2), the relative numbers of moles undergoing reaction ure the same as the relative numbers of molecules. In terms of the molar masses, NH3 = 17, 45 46 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 O2= 32, N2 = 28, and H20 = 18, the above combustion equation, 4NH3 + 3 0 2 --+ 2N2 + 6 H 2 0 (4 mol= 68 g) (3 rnol = 96g) (2 mol = 56g) (6 mol = 108 g) to shows that 4 mol of NH3 (4 x 17 g NH3) reacts with 3 rnol of 0 2 (3 x 32 g 02) form 2 mol of N2 (2 x 28 g N2) and 6 mol of H 2 0 (6 x 18 g H20). More generally, the equation shows that the masses of NH3, 02, and H 2 0 consumed or formed in the reaction-expressed in any mass unit whatever-are N2, in the ratio 68 :96 : 56 : 108 (or 17 :24 : 14 : 27). In all cases, the law of conservation of mass requires that the sum of the reactant masses (68 96) be equal to the sum of the resultant product masses (56 108). The importance of the above mass relations may be summarized as follows. + + 1. Mass relations are as exacting as the law of conservation of mass. 2. Mass relations do not require any knowledge about the variable conditions; for example, whether the H20 is water or steam. 3. Mass relations do not require any knowledge of the true molecular formulas. In the above example, the masses or the number of atoms would be unchanged if the oxygen were assumed to be ozone, 203, instead of 302. In either case, the equation would be balanced with 6 oxygen atoms on each side. Similarly, if the water molecules were polymerized, mass relations would be the same whether the equation contained 6H20, 3H402, or 2H602. This principle is very important in cases where the true molecular formulas are not known. Mass relations are valid for the many equations involving molecules that may dissociate (S8, P4, H6F6,N2O4, 12, etc.) or those that associate to form complex polymers, such as the many industrially important derivatives of formaldehyde, starch, cellulose, nylon, synthetic rubbers, silicones, etc., regardless of whether empirical or molecular formulas are used. LIMITING REACTANT Ordinarily one is given the mass of one designated reactant or product, and has the task of calculating the mass of one (or more) of the other substances involved. It will always be assumed that a sufficient (or excess) amount of all the undesignated reactants are present. If the masses of two designated reactants are given, you have the task of determining which one is in excess. You may calculate the yield of some product, using each designated reactant mass in turn. The reactant that yields the smaller result is the limiting reactant and the other is the excess reactant. Alternatively, you may convert the given masses to moles and compare the molar ratio to that demanded by the balanced equation. (See Problems 4.7 and 4.8.) TYPES OF CHEMICAL REACTIONS Skill in balancing equations will increase rapidly with practice, especially as one learns to recognize common types of reactions. Also recognition of the reaction type often helps in the prediction of the products of the reaction, if they are not stated. A few examples of the more predictable types are given below. 1. Combustion reactions: Oxygen in excess (usually from the air) combines with organic compounds (carbon, hydrogen, and oxygen) to produce carbon dioxide (CO,) and water (H20) as the sole products. C2H60 + 302 --j 2C02 + 3H20 2. Replacement (displacement) reactions: A more active element replaces a less active one in a compound. CHAP. 41 CALCULATIONS FROM CHEMICAL EQUATIONS 47 + CuS04 2Mg + TiC14 Zn AgN03 + NaCl --+ -+ + ZnS04 + Cu 2MgC12+ Ti Double displacement reactions (metathesis): Particularly common for ionic reactions in solution; atoms or groups of atoms will "exchange partners" if an insoluble salt results. BaC12 + Na2S04 N aN 0 3 + AgCl (insoluble) 2NaCl+ BaS04 (insoluble) + Acid-base reactions (neutralization): An acid, which contributes H, and a base, which contributes OH, undergo metathesis to produce water (HOH, or H20)i and a salt. H2SO4 + Mg(OH), HCl + NaOH --+ NaCl --+ + HOH MgS04 -t2HOH Two very simple reaction types, combination and decomposition, unfortunately afford no simple ways to predict the products. Combination reactions: Elements or compounds simply combine into one product. 2s02 + 0 2 P4 + 6C12 + 4PCI3 or --+ 2s03 P4 + 1 OCl2 + 4PC15 depending upon the ratio of reactants and the conditions of temperature and pressure. Decomposition: A single reactant is transformed by heat or electricity into two or more products. 2Hg0 depending on the temperature and oxygen pressure. (heat) 2H20 2Hg+02 (elec) 2H2+C)2 4Hg0 or (heat) 2Hg,0+02 Solved Problems 4.1. Balance the following skeleton equations: (a) FeS2 + 0 2 --+ Fe203 SO2 + (b) C7H602 +0 2 --+ CO2 + H20 (a) There are no fixed rules for balancing equations. Often ai trial-and-error procedure must be used. It is commonly helpful to start with the most complex formula. Fe203 has two different elements and a greater total number of atoms than any of the other substances, so we might start with it. We note that oxygen atoms occur in pairs in the molecules 0 2 and SO2 but not in the formula unit Fe203. If we write the equation with symbols representing the integral coefficients, wFeS2 + x02 -+yFe203 + zS02 then the total number of oxygen atoms on the left, 2x, is even for any integral value of x. The total number on the right, 3y + 22, can be even or odd, depending on whether y is even or odd. We conclude from the required equality of 2x with 3y + 22 that y must be even. We can now try the smallest even number, 2, and proceed from there. wFeS2 + x02 --+ 2Fe203+ zSO2 equal 4. To balance iron atoms, U' must 4FeS2 + x02 -+ 2Fe203+ z S 0 2 4FeS2 + x02 2Fe203+ 8S02 = To balance sulfur, z must equal 8. ---f Finally, to balance oxygen, 2x = 6 + 16, or x 1 1. 48 CALCULATIONS FROM CHEMICAL EQUATIONS 4FeS2 + 1 102 2Fe203 8S02 -+ [CHAP. 4 + Note that the coefficient of the simplest substance, elementary oxygen in this case, was evaluated last. This is the usual consequence of beginning the balancing procedure with the most complex substance. The most complex substance in this equation is C7H602. may assume 1 molecule of this substance We (b) and immediately write the coefficients for CO2 and H20 that will lead to balance of C and H, respectively. C7H602 + x02 ---t 7co2 + 3H20 The balance of oxygen atoms is saved for last because an adjustment of x would not interfere with the balance of any other element. An arithmetic balance now demands that x equal y, giving an equation C7H602 + 902 -+ 7co2 + 3H20 that, while balanced, violates the rule of integral coefficients. The correct ratio is preserved and the fraction eliminated by multiplying each coefficient by 2. 2C7H6O2 + 1502 4 14CO2 + 6 H 2 0 This is the correct form. This can be recognized as a combustion reaction. The balancing procedure is the same for countless other combustion reactions. 4.2. Complete and balance the following equations. [Note: Barium phosphate, Ba3(P04)2,is very insoluble; tin is a more active metal than silver.] (a) Ba(NOB), (a) + Na3P04 -+ (6) Sn + AgN03 -+ ( c ) HC2H302 + Ba(OH), -+ Write the products of this metathesis reaction by exchanging partners to get the skeleton equation. Note that the left-hand partner in each compound is a single metallic element but the right-hand partner is a group of atoms. Chapter 9 addresses the question of how to write the proper formulas of the product compounds. Ba(N03)2+ Na3P04-+ NaN03 + Ba3(P04)2 Consider 1 formula unit of the most complex compound Ba3(P04)2and insert the coefficients 3 and 2 in the reactants to balance the Ba atoms and the PO4 groups. Then the coefficient 6 for NaN03 simultaneously balances both the Na atoms and the NO3 groups. The final result is 3Ba(N03)2 2Na3P04 + + 6NaN03 + Ba3(P04)2 (b) Write the products of this replacement reaction by exchanging Ag and Sn to get the skeleton equation Sn + AgN03 -+ Ag + Sn(N03), One unit of the substance with the most complex formula Sn(N03)2demands 2 of the AgN03, which in turn requires 2Ag among the products. Sn + 2AgN03 -+2Ag + Sn(N03), (c) Recognizing that this is an acid-base reaction you can write it in the balanced form at once by supplying enough acid molecules to furnish the same number of H atoms as OH groups, which in turn is the number of water molecules. (The formula for acetic acid, HC2H302, was written in this way to show that only one H atom in the molecule has an acid function.) 2HC2H302 Ba(OH), + + Ba(C2H302), 2 H 2 0 + 4.3. Caustic soda, NaOH, is often prepared commercially by the reaction of Na2C03with slaked lime, Ca(OH)2. How many grams of NaOH can be obtained by treating 1 kg of Na2C03 with Ca(OH),? CHAP. 4 1 CALCULATIONS FROM CHEMICAL EQUATIONS 49 First write the balanced equation for the reaction. Na2C03 ( I mol= 106.0g) + Ca(OH2) + 2NaOH [2 mol= 2(40.0)=80.0g] + CaC03 The pertinent mass ratio, 106/80, is all that is needed to solve the problem. We show four conceptual methods, all equivalent arithmetically. First Method. 106.0 g Na2C03 gives 80.0 g NaOH hence and 1 g Na2C03 gives - 1 000 g Na2C03 gives NaoH 80.0 I 000 x -- = 755 g NaOH 106..0 80.0 106.0 Mole Method. As in Chapter 2, the symbol n ( X ) will be used to refer to the number of moles of a substance whose formula is X, and m(X) will denote the mass of substance X. Consider 1000 g Na2C03. n(Na2C03)= 1000 g = 9.434 mol Na2C03 106.0 g/mol From the coefficients in the balanced equation, n(Na0H) = 2n(Na2C03) = 2(9.434) = 18.87 mol NaOH. m(Na0H) = (18.87 mol NaOH)(40.0 g NaOH/mol NaOH) = 755 g NaOH Proportion Method. Let x = number of grams of NaOH obtained from 1000 g Na2C03.It is known that 106.0 g Na2C03 gives 80.0 g NaOH; then, by proportion. 106.0 g Na2C03- 1 000 g Na2C03 80.0 g NaOH X s= (1 000 g Na2C03) 80.0 g NaOH = 755 g NaOH Note: It should be evident that 1000 pounds Na2C03 will g,ive 755 pounds NaOH, and that 1000 tons Na2C03will give 755 tons NaOH. Factor-Label Method. The amount of NaOH is obtained by multiplying the 1000 g Na2C03 by successive conversion factors until it has the desired units g NaOH. Amt. of NaOH = ( I 000 g Na2C03) ( 1 mol Na2C03 106.0 g Na2C03 )( mO1NaoH (40'0 g NaoH) 1 mo'l Na2CO3 1 mol NaOH ) = 755 NaOH The functions of the above conversion factors are: convert starting information from grams to moles; using the coefficients in the balanced equation convert to moles of desired substance; convert from moles to grams of desired substance. Be sure to label each factor completely with its units. Then if canceling of units does not result in the correct units for the answer you will know that there has been an error in your reasoning. 4.4. The equation for the preparation of phosphorus in an electric furnace is 2Ca3(PO,). + 6Si02 + 1 OC -+ 6CaSi03 + 1 OCO + P4 Determine (a) the number of moles of phosphorus formed for each mole of Ca3(PO4)2 used, (b)the number of grams of phosphorus formed for each mole of Ca3(P0& used, (c) the number of grams of phosphorus formed for each gram of Ca3(PC)4)2 used, ( d ) the number of pounds of phosphorus formed for each pound of Ca3(P04)2used, (e) the number of tons of phosphorus 50 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 formed for each ton of Ca3(P04)2used, for each mole Ca3(P04)2 used. (a) (b) (c) (d) (e) ( . f ) the number of moles each of Si02 and C required From the equation, 1 mol P4 is obtained for 2 mol Ca3(P04)2used, or, f mol P4 per mole Ca3(P04)2. Molar mass of P4 is 124. Then mol P4 = f x 124 = 62 g P4. = 0.20 g P4. One mol of Ca3(P04)2(319 g) yields rnol P4 (62 g). Then 1 g Ca3(P04)2gives 0.20 Ib; the relative amounts are the same regardless of units. 0.20 ton; as above. (f) From the coefficients in the balanced equation, I mol Ca3(PO4)2 required 3 mol SiO2 and 5 rnol C. -1. 1 4.5. Hydrochloric acid was once prepared commercially by heating NaCl with concentrated H2SO4. How much sulfuric acid containing 90.0% H2SO4 by weight is needed for the production of 1000 kg of concentrated hydrochloric acid containing 42.0% HC1 by weight? (1) (2) Amount of pure HC1 in 1 000 kg of 42.0% acid is (0.420)(1 000 kg) = 420 kg. Determine the amount of H2SO4 required to produce 420 kg HCl (H2S04= 98.1, HCl = 36.46). 2NaCl-t From the equation 72.92 g HCl requires 98.1 g H2S04 - H2S04 ( I mol=98.1&) --f Na2S04+ 2HC1 12 rnol = 2(36.46)= 72.92 g] 1 g HCI requires 1 kg HCI and (3) requires 98.1 72.92 gH2S04 98.1 -kg H2S04 72.92 420 kg HC1 requires (420) (79289: ~ kg) = 565 kg H2S04 Finally, determine the amount of sulfuric acid solution containing 90.0% H2S04that contains 565 kg of pure H2SO4. Since 0.900 kg of pure H2S04 makes 1 kg of 90.0% solution, then 565 kg H2SO4 x 1 kg soln = 628 kg soln 0.900 kg H2S04 Factor- Label Method. This method has the advantage of allowing one to write down the solution in one step. Carefully note the function of each factor, and check to confirm that all units cancel except the desired one. 42.0 kg HCl Amt. of 90.0% H2S04= (1 000 kg 42.0% HCI) (100 kg 42.0% HCI) 1000 g 1 mol HC1 (k) I g (36.46 g HCl) ( = 628 1 rnol H2SO4) (98.1 g H2SO4) (100 g 90.0% H2S04 2 mol HCl 1 mol H2S04 90.0 g H2SO4 kg 90.0% H2S04 4.6. Before the public became serious about pollution, it was common to improve the performance of gasoline by the addition of lead compounds. A particular 100-octane aviation gasoline used 1.00 cm3 of tetraethyl lead, (C2H5)4Pb, of density 1.66 g/cm3, per liter of product. This compound is made as follows: 4C2H5C1+ 4NaPb --+ (C2H5),Pb + 4NaCl+ 3Pb CHAP. 41 CALCULATIONS FROM CHEMICAL EQUATIONS 51 How many grams of ethyl chloride, C2HSC1, are needed to make enough tetraethyl lead for 1.00 L of gasoline? The mass of 1.00 cm3 (C2H5)4Pb is (1.00 crn"(l.66 g/cm3)= 1.66 g; this is the amount needed per liter. In terms of moles, 166g = 0.005 14: mol Number of moles of (C2H5)4Pb needed = 323 g/mol The chemical equation shows that 1 mol (C2H5)4Pb requires 4 niol ClH5C1. Hence 4(0.005 14) = 0.020 6 mol C2H5CI is needed. m(C2H5CI)= 0.0206 rnol x 64.5 g/mol = 1.33 g C2H5Cl Factor-Label Method. Amt. of C2HsCl = ( I .OO L gasoline) I .OO cm3 tet l e s ) 1 .OO L gasoline ( 1.66 ) g (1 mol tet lead 1 .OO cm3 323 g tet lead 4 mol C2H5CI) (64.5 g C2HSCl) 1 rnol C 2 H 5 a ( 1 mol tet lead 4.7. A solution containing 2.00 g of Hg(N03)2 was added to1 a solution containing 2.00 g of Na2S. Calculate the mass of HgS that was formed according to the reaction Hg(N03), + Na2S --+ HgS + 2NaN03 The special feature of this problem is that the starting quantities of two reactants are specified. It is necessary first to determine which, if any, substance is in excess. The simplest approach is to calculate the number of moles of each. n(Na2S) = 2`oo = 2.56 x 10--2mol Na2S 78.00 g/mol The equation indicates that equimolar quantities of reactants are required. Hence the given amount of Hg(N03)2 requires only 6.16 x 10-3 mol of NalS. The given amount of Na2S is far in excess of this. The Hg(N03)2is the limiting reagent in this case and determines the quantity of product. Amt. of HgS = [6.16 x 10-3mol Hg(N03),] [1 1 rnol HgS (232.6 g HgS) = 1.43 g HgS rnol Hg(P403)2 1 rnol HgS ] The amount of excess Na2S is 2.56 x 10-2 mol minus the 6.16 x 10-3 mol consumed, or 1.94 x 10-* mol, which amounts to 1.94 x 10-2 mol x 78.0 g/mol = 1.51 g. The amount of NaN03 is (6.16 x lOP3 mol Hg(N03),) (1 In summary, the original 4.00 g of reagents has been transformed to 1.43 g of HgS product, 1.05 g of byproduct NaN03, with 1.51 g of excess reagent Na2S remaining. (Note that the sum of the final masses is 3.99 g, within round-off error of the original 4.00 g.) zP:$z)) --+ 85.0 g NaN03 = 1.05 g NaN03 ( i m o l NaN03 4.8. How many grams of Ca3(P04)2 can be made according tlo the reaction 3CaC12+ 2K3P04 Ca3(P04)2 6KC1 t 52 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 by mixing a solution containing 5.00 g of CaClz with another containing 8.00 g of K3P04? n(CaC12) = 5.00 = 4.50 x 10-* rnol 11 1.O g/mol (4.50 x 10-2 rnol CaC12) n(K3P04) = 8.00 g 212.3 g/mol = 3.77 x lop2 mol 2 rnol K3P04 = 3.00 x IO-* mol K3P04 3 mol CaC12 required for the given CaC12. Thus the K3P04 is in excess and CaC12 is the limiting reagent. Amt. of Ca3(P04)2= (4.50 x lop' rnol CaC12) [ 1 mol Ca3 (po4)2] 310.2 g Ca3(P04)2 3 moI CaC12 1 rnol Ca3(P04), [ 1 If the calculation were begun with K3P04 instead of CaCI2: 3 mol CaC12 = 5.66 mol CaC12 (3.77 x 1 0 - ~ mol K ~ P O ~ ) 2 mol K3P04 required for the given K3P04. Since this is greater than the given amount of CaC12, again it is concluded that CaC12 is the limiting reagent. Completing the mass balance will provide a check on the work. Excess K 3 P 0 4 = [(3.77 - 3.00) x 10-2 moll (212.3 g/mol) Amt. of KCl = (4.50 x 10-2 mol CaCI2) ( 6 mol KCI ) (74.5 g KCI) 3 mol CaCI2 1 mol KCI = 1.63 g = 6.71 g KCl The total of the product, by-product, and excess reagent is 4.65 + 6.71 + 1.63 = 12.99 g, in agreement with the 8.00 + 5.00 = 13.00 g of reactants. 4.9. In one process for waterproofing, a fabric is exposed to (CH3)2SiC12vapor. The vapor reacts with hydroxyl groups on the surface of the fabric or with traces of water to form the waterproofing film [(CH3)2SiO],, by the reaction n(CH3)2SiC12 2nOH- + --$ 2nC1- + nH20 + [(CH,),SiO], where n stands for a large integer. The waterproofing film is deposited on the fabric layer upon layer. Each layer is 6 thick [the thickness of the (CH&SiO group]. How much (CH&SiC12 is needed to waterproof one side of a piece of fabric, 1 m by 2 m, with a film 300 layers thick? The density of the film is 1.0 g/cm3. A Mass of film = (volume of film)(density of film) = = (area of film)(thickness of film)(density of film) (100 cm x 200 cm)(300 x 6 A x 10-* cm/A)( 1.0 g/cm3) = 0.36 g 1 rnol [(CH3)2SiO], Amt. of (CH3),SiCI2 = (0.36 g [(CH,),SiOIn}{ 74n [(CH3hsi01, ! = 0.63 g (CH3),SiC12 Note that the unknown integer n canceled in the abovefactor-lahel calculation along with the units (except for grams). CHAP. 4 1 CALCULATIONS FROM CHEMICAL EQUATIONS 53 4.10. What is the % free SO3 in an oleum (considered as a solution of SO3 in H2SO4) that is labeled "109% H2S04"? Such a designation refers to the total weight of pure H2SO4, 109 g, that would be present after adding sufficient water to 100 g of the oleum to convert it to pure H2SO4. Nine g H20 will combine with all the free SO3 in 100 g of the oleum to give a total of 109 g H2SO4. The equation H 2 0 + SO3 + H2S04 indicates that I mol H 2 0 (18 g) combines with 1 mole SO3 (80 g). Then Thus 100 g of the oleum contains 40 g SO3, or the % free SO3 in the oleum is 40%. 4.11. KC104 may be made by the following series of reactions: C12 + 2KOH -+ KCl + KClO + H20 4KClO3 -+ 3KClO4 + KCI 3KClO -+ 2KC1+ KC103 How much Cl2 is needed to prepare 100 g KC104 by the above sequence? The mole method and the factor-label method are the simplest routes to the solution of this problem. In neither case is it necessary to find the masses of the intermediate products. Mole Method. n (KC10) ~1(C12: I n(KC103) = fn(KC10) = fn(C12) ~ ( K c ~ ~ ~ ( K c I(i)(f)n(c12)= in(c1,) =o ) = o~) n(KC104) = 100 g KC104 - = 0.721 5 mol KC104 138.6 g KC104/mol KC104 n(C12)= 4(0.721 5) = 2.886 mol C12 m(C12) = (2.886 mol Cl2)(7O.9g Cl2/rno1 C12) = 205 g C12 Factor-Label Method. Amt. of C12 = (100 g KC104)(:3?fgt?:j 1 mol KClO ( 1 mol C12 ) (70.9 g Cl2) 1 mol C12 4 mol KC103) (3 mol KC1O4 ( 3 mol KClO) 1 mol KC103 4.12. A 1.204 8-g impure sample of Na2C03 is dissolved and allowed to react with a solution of CaC12. The resulting CaC03, after precipitation, filtration, and drying, was found to weigh 1.0262 g. Assuming that the impurities do not contribute to the weight of the precipitate, calculate the percentage purity of the Na2C03. The equation for the reaction is as follows: The amount of CaC03 should first be found. n(CaC03) = 100.09 g CaC03/mol 0.010 253 mol Na2C03+ CaC12 -+ CaCO:! + 2NaCl From the coefficients in the balanced equation, 54 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 n(Na2C03)= n(CaC0,) = 0.010253 mol Now calculate the mass of pure Na2C03 in the sample. m(Na2C0,) = (0.010253 mol)( 105.99 g Na2C03/mol) = 1.0867 g Na2C03 The percentage purity is obtained by dividing the mass of Na2C03 by the mass of the sample, and multiplying by 100. 1.0867 g YOpurity = (100%) = 90.20% 1.204 8 g 4.13. A mixture of NaCl and KC1 weighed 5.4892 g. The sample was dissolved in water and reacted with an excess of silver nitrate in solution. The resulting AgCl weighed 12.705 2 g. What was the percentage of NaCl in the mixture? The two parallel reactions are NaCl + AgNO, -+ AgCl + N a N 0 3 KCl + AgNO, -+AgCl + KN03 Here the conservation of C1 atoms requires that the number of moles of AgCl formed equal the sum of the numbers of moles of NaCl and KCI. n(AgC1) = 12.7052 g AgCl = 0.088 649 mol = n(NaC1) + n(KC1) 143.321 g AgCl/mol = mass X Let x = mass of NaCl and y of KCI. Then = 0.088 649 mol Y 58.443 g/mol+ 74.551 g/mol (1) A second equation for the unknown masses is provided by the data: x+y=5.4892g Eliminating y between ( I ) and (2), and solving for x, we obtain x = m(NaC1) = 4.0624 g. Then 4.0624 g (100%) = 74.01 YONaCl 5.489 2 g (4 YONaCl = ~ Supplementary Problems BALANCING EQUATIONS Balance the following equations. 4.14. C2H4(OH)2 4.15. Li + H 2 0 -+ LiOH Ba(OH), +0 2 --+ CO2 + H20 + H2 4.16. 4.17. 4.18. 4.19. 4.20. 4.21. + AICl3 Al(OH), + BaC12 K2C8H404 + H20 KHC8H404 + KOH C2HC13+ CaC12+ H 2 0 C2H2C14+ Ca(OH), (NH4)2Cr207 N2 + Cr203 + H 2 0 Zn3Sb2+ H 2 0 Zn(OH), + SbH, --f Sn + SnC14 -+ SnCl, + -+ + -+ CHAP. 41 CALCULATIONS FROM CHEMICAL, EQUATIONS 55 4.22. 4.23. 4.24. 4.25. 4.26. 4.27. 4.28. 4.29. 4.30. 4.31. HC104 + P4010--+ H3P04 + C1207 C6H5CI + siCl4 + Na -+ (C6Hs)qSi + NaCl Sb2S3 + HCI -+ H3SbC16 + H2S NI3 IBr + NH3 -3 + NH4Br SF4 + H20 + SO2 + HF + Na2C03+ C + N2 Fe(CO), + NaOH NaCN + CO K4Fe(CN), + H2SO4 + H20 -+ K2S04 + FeS04 + (NH4),S04 --+ + CO Na2Fe(CO), + Na2C03+ H20 + . H3P04+ (NC'H4)2M004 H N 0 3 -+ ( N H 4 ) 3 P 0 412M003+ NH4N03+ H 2 0 Identify each of the following types of reactions, write the products, and balance the equation. ( a ) HCI Mg(OH)2 + (h) PbC12 -tK2SO4 ( c ) CH3C-120H +02(excess) -+ ( d ) NaOH + H2C6H606 -+ ( e ) F e + A g N 0 3 -+ Partial uizswers ( a ) acid-base (neutralization), products i3re HzO and MgC12; (b) double displacement (metathesis), products are KCI and PbSO4; (c) combustion, products are CO2 and H20; ( d ) acid-base (neutralization), products are Na2C6H606 and H20; ( e ) replacement (displacement), products are Fe(N03)2 and Ag. + - MASS RELATIONS 4.32. Consider the combustion of amyl alcohol, C5H,,OH. 2CSHilOH + 1502 -+ 10C02 + 12H20 ( a ) How many moles of 0 2 are needed for the combustion of 1 11101 of amyl alcohol? (6) How many moles of H 2 0 are formed for each mole of O2 consumed? (c) How rnany grams of CO2 are produced for each mole of amyl alcohol burned? ( d ) How many grams of CO2 are produced for each gram of amyl alcohol burned? (e) How many tons of CO2 are produced for each ton of amyl alcohol burned? Ans. 4.33. ( a ) 7.5 mol 0,; (6) 0.80 mol H 2 0 ; (c) 220 g CO,; ( d ) 2.49 g CO,; 3 (e) 2.49 tons CO2 A portable hydrogen generator utilizes the reaction CaH2 H2 can be produced by a 50 g cartridge of CaH2? A H S . 4.8 g H2 + 2H20 Ca(OH), + 2H2. How many grams of 4.34. Iodine can be made by the reaction 2NaI03 + 5NaHS03 -+ 3NaHS04 + 2Na2S04+ H 2 0 + 12.To produce each kilogram of iodine, how much NaI03 and how much NaHS03 must be used? Ans. 1.56 kg NaI03, 2.05 kg NaHS03 4NaOH + 2XeF2 3 2Xe + O2 + 4NaF + 2 H 2 0 4.35. The noble gas compound XeF2 can be safely destroyed by treatment with NaOH: Calculate the mass of oxygen resulting from the reaction above of 85.0 g of Xe02 with excess NaOH solution. ART. 8.03 g of 0 2 56 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 4.36. How much iron(II1) oxide can be produced from 6.76 g of FeC13 .6H20 by the following reactions? FeC13 - 6 H 2 0 + 3NH3 -+ Fe(OH), 2Fe(OH), Ans. 2.00 g of Fe203 -, Fe203 + 3H20 + 3NH4C1 4.37. (a) How many pounds of ZnO will be formed when 1 lb of zinc blende, ZnS, is strongly heated in air? 2211s + 302 ---f 2Zn0 + 2So2 (b) How many tons of ZnO will be formed from 1 ton ZnS? (c) How many kilograms of ZnO will be formed from 1 kg ZnS? Ans. (a) 0.835 lb; (b) 0.835 ton; (c) 0.835 kg 4.38. In a rocket motor fueled with butane, C4H10,how many kilograms of liquid oxygen should be provided with each kilogram of butane to provide for complete combustion? 2C4H10 + 1302 -+ 8C02 + 10H20 Ans. 4.39. 3.58 kg Chloropicrin, CC13N02, can be made cheaply for use as an insecticide by a process which utilizes the reaction CH3N02 Ans. + 3C12 -+ CC13N02 + 3HCl How much nitromethane, CH3N02,is needed to form 500 g of chloropicrin? 186 g 4.40. Ethyl alcohol (C2H50H) is made by the fermentation of glucose (C6H1206),as indicated by the equation C6H1206 2C2H50H+ 2C02 --+ How many metric tons of alcohol can be made from 2.00 metric tons of glucose? Ans. 4.41. 1.02 metric tons , How many kilograms of H2S04can be prepared from 1 kg of cuprite, C U ~ Sif each atom of S in Cu2S is converted into 1 molecule of H2SO4? Ans. 0.616 kg 4.42. (a) How much bismuth nitrate, Bi(N03)3 - 5H20, would be formed from a solution of 10.4 g of bismuth in nitric acid? Bi + 4HNO3 + 3H20 -+ Bi(N03)3 - 5 H 2 0 + NO (b) How much 30.0% nitric acid (containing 30.0% HN03 by mass) is required to react with this amount of bismuth? Ans. 4.43. (a) 24.1 g; (b) 41.8 g One of the reactions used in the petroleum industry for improving the octane rating of fuels is C7H14 + C7Hg + 3H2 The two hydrocarbons appearing in this equation are liquids; the hydrogen formed is a gas. What is the percentage reduction in liquid weight accompanying the completion of the above reaction? Ans. 6.2% CHAP. 41 CALCULATIONS FROM CHEMICAL, EQUATIONS 57 4.44. In the Mondprocess for purifying nickel, the volatile nickel carbonyl, Ni(C0)4, is produced by the following reaction: Ni + 4CO + Ni(CO), How much CO is used up in volatilizing each kilogram of nickel? Ans. 4.45. 1.91 kg CO When copper is heated with an excess of sulfur, Cu2S is formed. How many grams of Cu2S could be produced if 100 g of copper is heated with 50 g of sulfur? Ans. 125 g C U ~ S 4.46. The "thermite" process is of historical interest as a method of welding iron: 2A1+ Fe203 42Fe + A1203 Calculate the maximum amount of aluminum that can be mixed with 500 g of iron(II1) oxide to form a thermite charge that will yield pure iron, not alloyed with any excess aluminum. Ans. 169 g of aluminum 4.47. A mixture of 1 ton of CS2 and 2 tons of C12 is passed through a hot reaction tube, where the following reaction takes place: + CS2 + 3C12 -+ cc14 S2C12 (a) How much CC14 can be made by complete reaction of the limiting starting material? material is in excess, and how much of it remains unreacted? (b)Which starting Ans. (a) 1.45 tons CC14; (b) 0.28 ton CS2 4.48. A gram (dry weight) of green algae was able to absorb 4.7 x 10-3 mol CO2 per hour by photosynthesis. I f the fixed carbon atoms were all stored after photosynthesis as starch, (C6H1005)n,how long would it take for the algae to double their own weight? Neglect the increase in photosynthetic rate due to the increasing amount of living matter. Ans. 7.9 h 4.49. Carbon disulfide, CS2, can be made from by-product S02. The overall reaction is 5C + 2S02 + CS2 + 4420 How much CS2 can be produced from 450 kg of waste SO2 with excess coke, if the SO2 conversion is 82Y0? Ans. 219 kg 4.50. Silicate ores can be solubilized by fusion with sodium carbonate. A simplified equation for what occurs is: 2Na2C03+ Si02 --+ Na4Si04. 2C02 Calculate the minimum weight of Na2C03 required to dissolve a 0.500 g sample of an ore which analyzes 19.1YOsilica (Si02). A m . 0.337 g (Note: typically a large excess is used, e.g., about 3 g.) + 4.51. If The chemical formula of the chelating agent Versene is C2H4N2(C2H202Na)4. each mole of this compound could bind 1 mole of Ca2+, what would be the rating of pure Versene, expressed as mg CaC03 bound per gram of chelating agent? Here the Ca2+ is expressed in terms of the amount of CaC03 it could form. Ans. 264 mg CaC03 per g 58 CALCULATIONS FROM CHEMICAL EQUATIONS [CHAP. 4 4.52. When CaC2 is made in an electric furnace by the reaction CaO + 3C -+ CaC2 + CO the crude product is typically 85% CaC2 and 15% unreacted CaO. How much CaO is to be added to the furnace charge for each 50 tons (a) of CaC2 produced? (b) of crude product? Ans. 4.53. (a) 53 tons CaO; (b) 45 tons CaO The plastics industry uses large amounts of phthalic anhydride, C8H403,made by the controlled oxidation of naphthalene. 2CloHg + 902 -+ 2C8H4O3 + 4CO2 + 4H2O Since some of the naphthalene is oxidized to other products, only 70 percent of the maximum yield predicted by the above equation is actually obtained. How much phthalic anhydride would be produced in practice by the oxidation of 100 lb of CloHs? Ans. 81 lb 4.54. The empirical formula of a commercial ion-exchange resin is C8H7S03Na.The resin can be used to soften water according to the reaction Ca2++ 2CgH7S03Na --t (C8H7S03),Ca + 2Na' What would be the maximum uptake of Ca2+ by the resin, expressed in moles per gram of resin? Ans. 0.0024 mol Ca2+/gresin 4.55. The insecticide Chlordan is made by a two-step process: (hexachlorocyclopentadiene) C5H6 + (cyclopentadiene) + C1oH6C16 C5C16 C10H6C16 + CI2 -+ CloH6Cls (Chlordan) (a) Calculate how much Chlordan can be made from 500 g of C5C16. (b)What is the percentage of chlorine in Chlordan? Ans. (a) 751 g, (b) 69.2% C1 4.56. Commercial sodium "hydrosulfite" is 90% pure Na2S204.How much of the commercial product could be made by using 100 tons of zinc with a sufficient supply of the other reactants? The reactions are ZnS204+ Na2C03-+ ZnC03 + Na2S204 Ans. 296 tons Zn + 2S02 + ZnS204 4.57. Fluorocarbon polymers can be made by fluorinating polyethylene according to the reaction (CH2), + 4nCOF3 -+ (CF2), + 2nHF + 4nCOF2 2CoF3 where n is a large integer. The CoF3 can be regenerated by the reaction 2 2CoF2 + F -+ (a) If the H F formed in the first reaction cannot be reused, how many kilograms of fluorine are consumed per kilogram of fluorocarbon produced, (CF2),? (b) If the H F can be recovered and electrolyzed to hydrogen and fluorine, and if this fluorine is used for regenerating CoF3, what is the net consumption of fluorine per kilogram of fluorocarbon? Ans. (a) 1.52 kg; (b) 0.76 kg CHAP. 41 CALCULATIONS FROM CHEMICAL, EQUATIONS 59 4.58. A process designed to remove organic sulfur from coal prior to combustion involves the following reactions: X-S-Y + 2NaOH -+ X-0-Y CaC03 -+ CaO + CO2 + + Na2S+ H 2 0 Na2S + CO2 + H 2 0 Na2C03 Ca(OH), Na2C03i H2S CaC03 + 2NaOH CaO + H 2 0 -+ Ca(OH), + --+ In the processing of 100 metric tons of a coal having a 1.0% sulfur content, how much limestone (CaC03) must be decomposed to provide enough Ca(OH)2to regenerate the NaOH used in the original leaching step? Ans. 3.12 metric tons 4.59. Silver may be removed from solutions of its salts by reaction with metallic zinc according to the reaction Zn + 2Ag+ -+Zn2+ + 2Ag A 50-g piece of zinc was thrown into a 100-L vat containing 3.5 g Ag+/L. (a) Which reactant was completely consumed? (b) How much of the other substance remained unreacted? Ans. (a) zinc; (b) 1.9 g Ag+/L 4.60. The following reaction proceeds until the limiting substance is all consumed: 2A1+ 3Mn0 -+ A1203$- 3Mn A mixture containing 100 g A1 and 200 g MnO was heated to .initiate the reaction. Which initial substance remained in excess, and how much? Ans. Al, 49 g 4.61. A mixture of NaHC03 and Na2C03 weighed 1.023 5 g. The dissolved mixture was reacted with excess Ba(OH)2 to form 2.102 8 g BaC03, by the reactions Na2C03+ Ba(OH), NaHC03 Ans. 39.51% NaHC03 -+ + + Ba(OH), BaC03 + 2NaOH BaC03 NaOH + + H20 What was the percentage NaHC03 in the original mixture? 4.62. A mixture of NaCl and NaBr, weighing 3.5084 g, was dissolved and treated with enough AgN03 to precipitate all the chloride and bromide as AgCl and AgBr. The washed precipitate was treated with KCN to solubilize the silver and the resulting solution was electrolyzed. The equations are + AgN03 AgBr + NaN03 AgCl + 2KCN KAg(CN), + KCl AgBr + 2KCN -+ KAg(CN), + KBr 4KAg(CN)2 + 4KOH + 4Ag + 8KCN + 0 2 + 2H20 NaBr + NaCl + AgN03 --+ AgCl + NaN03 --+ After the final step was complete, the deposit of metallic silver weighed 5.502 8 g. What was the composition of the initial mixture? Ans. 65.23% NaC1, 34.77% NaBr This page intentionally left blank Chapter 5 Measurement of Gases GAS VOLUMES Gases, unlike liquids and solids, occupy volumes that depend very sensitively on pressure and temperature. Special attention must therefore be given to factors influencing the volume of gases. PRESSURE Pressure is defined as the force acting normally on a unit area of surface. Pressure = Pressure (in pascals) = force acting perpendicularly to an area area over which the: force is distributed force (in newtons) area (in square meters) Thus the pascal is defined by 1 Pa = 1 N/m2 = (1 kg . m/s2)/ni2 = 1 kg/m - s2. The pressure exerted by a column of fluid is Pressure = height x density of fluid x acceleration of gravity Pressure (in Pa) = height (in m) x density (in kg/m3) x 9.81 m/s2 STANDARD ATMOSPHERIC PRESSURE Air has weight and therefore exerts a pressure. The atmospheric pressure is due to the weight of the overlying air. A standard atmosphere (1 atm) is defined as exactly 101 325 Pa. It is approximately equal to the average pressure of the atmosphere at sea level. The standard atmosphere also is approximately equal (within several parts in 107) to the pressure exerted by a column of mercury 760 mm high, at 0 "C and at sea level. The torr is defined by: 760 torr= 1 atm. For problems in this book, the torr and the millimeter of mercury (mmHg) will be taken to be the same. The bar is defined by: 1 bar=exactly 105 Pa. One bar has recently replaced 1 atm as the standard pressure for reporting thermodynamic data. PRESSURE GAGES The pressure of a gas is frequently measured by attaching a manometer to the vessel. The manometer contains a liquid, usually mercury, the height of which is readable in mm Hg (i.e. torr) pressure units. A closed-tube manometer, Fig. 5-l(a),is originally filled all the way to the closed end, so that the difference in mercury levels is the absolute value of the pressure of the gas. An open-tube manometer shows the difference between the pressure in the gas and the barometric pressure. In Fig. 5-l(b) the gas pressure is less than barometric, and in Fig. 5-l(c) it is more. A variety of other gages are used to measure pressure. Those most commonly encountered are dial gages, which read relative pressure, that is the pressure in exce:ss of barometric, analogous to the open tube manometer in Fig. 5-l(c). Absolute gages (analogous to Fig. 5-l(a))are usually so marked on the dial. Most gages used by motorists to check their tires read the relative pressure. 61 62 MEASUREMENT OF GASES [CHAP. 5 (a) Closed tube manometer (b) Open tube manometer, `gas < `atmosphere (c) Open tube manometer, `gas > `atmosphere Fig. 5-1 STANDARD CONDITIONS (S.T.P.) Standard conditions (S.T.P.) denotes a temperature of 0C (273.15 K, rounded off for most problems in this book to 273 K) and normal atmospheric pressure (1 atm = 760 torr). As both the volume and density of any gas are affected by changes of temperature and pressure, it is customary to reduce all gas volumes to standard conditions for purposes of comparison. GAS LAWS At sufficiently low pressures and sufficiently high temperatures, all gases have been found to obey three simple laws. These laws relate the volume of a gas to the pressure and temperature. When a gas obeys these laws, it is said to behave as an ideal gas or perfect gas. These laws, which are described below, may be applied only to gases which do not undergo a change in chemical complexity when the temperature or pressure is varied. An example of a gas to which these laws do not apply is N02, which undergoes a dimerization to N204 at increasing pressures or decreasing temperatures. BOYLE'S LAW When the temperature is kept constant, the volume of a given mass of an ideal gas varies inversely with the pressure to which the gas is subjected. In mathematical terms, the product pressure x volume of a given amount of gas remains constant. Thus, in comparing the properties of a given amount of an ideal gas under two conditions, which we may call the initial andjinal states, we may write the following equation, applicable at constant temperature: A given subscript, 1 or 2, refers to a given state of the gas; 1 usually refers to an initial state and 2 to a final state. This law furnishes the most direct test of how well a real gas corresponds to ideal behavior . CHARLES' LAW At constant pressure, the volume of a given mass of gas varies directly with the absolute temperature. Then, at constant pressure, v - v 1 2 T 1 T2 where Tl and T2 denote the absolute temperatures of the gas at the two states being compared. CHAP. 51 MEASUREMENT OF GASES 63 GAY-LUSSAC'S LAW At constant volume, the pressure of a given mass of gas varies directly with the absolute temperature. Then, at constant volume, COMBINED GAS LAW Any two of the above three gas laws can be employed to derive a law which applies to all possible combinations of changes: = constant: T1 T2 for a fixed mass of gas. (See Problem 5.6.) Since many gas calculations are concerned with determining a final volume given an initial volume, the combined gas law is often written as PIV1 P -- -- v This expression indicates that the volume of a given mass of gas varies directly with the absolute temperature and inversely with the pressure. Notice that, while T1 and T2 must be in kelvins, any convenient absolute pressure unit may be used for PI and P2, and any convenient volume unit may be used for V1 and V2. DENSITY OF AN IDEAL GAS As the volume of a given mass of gas increases, the mass per unit volume (i.e., the density) decreases proportionately. Therefore the density ( d ) of a gas varies inversely with its volume. For an ideal gas, the combined gas law then gives DALTON'S LAW OF PARTIAL PRESSURES The partial pressure of a component of a gas mixture is the pressure which that component would exert if it alone occupied the entire volume. According to Dalton's law, the total pressure of a gaseous mixture is equal to the sum of the partial pressures of the components. Dalton's law is rigidly accurate only for ideal gases. At pressures of only a few atmospheres or below, gas mixtures may be regarded as ideal gases, and this law may be applied in calculations in this book. COLLECTING GASES OVER A LIQUID If a gas is collected over a volatile liquid, such as water, a correction is made for the amount of water vapor present with the gas. A gas collected over water is saturated with water vapor, which occupies the total gas volume and exerts a partial pressure. The partial pressure of the water vapor is a constant for each temperature and is independent of the nature or pressure of the confined gas. This definite value of the vapor pressure of water may be found tabulated as a function of temperature in handbooks or in 64 MEASUREMENT OF GASES [CHAP. 5 other reference books. If the total pressure (of the gas plus water vapor) is what is measured, the vapor pressure must be subtracted from the total pressure in order to obtain the partial pressure of the gas. Pressure of gas = (total pressure) - (vapor pressure of water) When a gas is collected over mercury, it is not necessary to make a correction for the vapor pressure of mercury, which is negligible at ordinary temperatures. DEVIATIONS FROM IDEAL BEHAVIOR The laws discussed above are strictly valid only for ideal gases. The very fact that all gases can be liquefied if they are compressed and cooled sufficiently is an indication that all gases become nonideal at high pressures and low temperatures. The ideal properties are observed at low pressures and high temperatures, conditions far removed from those of the liquid state. At pressures below a few atmospheres practically all gases are sufficiently dilute for the application of the ideal gas laws with a reliability of a few percent or better. Solved Problems PRESSURE DEFINED 5.1. Calculate the difference in pressure between the top and bottom of a vessel exactly 76 cm deep when filled at 25 "C with (a) water, (b) mercury. Density of mercury at 25 "C is 13.53 g/cm3; of water 0.997 g/cm3. Using the equivalent values in SI units for height and density, (4 or 7.43 kPa. Pressure = height x density x g = (0.76 m)(997 kg/m3)(9.81 m/s2) = 7.43 x 103 Pa Pressure = (0.76 m)(13 530 kg/m3)(9.81 m/s2) = 100.9 x 103 Pa or 100.9 kPa. 5.2. How high a column of air would be necessary to cause the barometer to read 76 cm of mercury, if the atmosphere were of uniform density 1.2 kg /m3? The density of mercury is 13.53 x 103kg /m3. Pressure of Hg = pressure of air height of Hg x density of Hg x g = height of air x density of air x g (0.76 m)(13 530 kg/m3) = h( 1.2 kg/m3) h = (0.76 m)(13 530 kg/m3) = 8.6 km 1.2 kg/m3) Actually the density of air decreases with increasing height so that the atmosphere must extend much beyond 8.6 km. Modern aircraft routinely fly higher than 8.6 km (28000 ft), CHAP. 51 MEASUREMENT O F GASES 65 5.3. GAS LAWS A mass of oxygen occupies 5.00 L under a pressure of 740 torr. Determine the volume of the same mass of gas at standard pressure, the temperature remaining constant. Figure 5-2 illustrates one way in which the change could be: carried out and observed. Standard pressure is 760 torr. Boyle's law gives p VI = p2V2 1 or PI 740 torr V2 = - Vl = (5.00 L) = 4.87 L P2 764) torr Observe that any convenient pressure and volume units can be chosen (in this case, torr and L), since only ratios of like quantities are involved. P , = 740 torr P , = 760 torr Fig. 5-2 5.4. A mass of neon occupies 200 cm3 at 100C. Find its volume at O"C, the pressure remaining constant. Figure 5-3 illustrates the change. Charles' law gives Absolute temperatures must be used in the gas laws. 66 MEASUREMENT OF GASES [CHAP. 5 Boiling water o o O ~ 100C Ice water 0 "C Fig. 5-3 5.5. A steel tank contains carbon dioxide at 27 "Cand a pressure of 12.0 atm. Determine the internal gas pressure when the tank and its contents are heated to 100C. See Figure 5-4. Gay-Lussac's law gives - _ -p2 P1 T 1 5.6. T2 or T2 P2 = -Pl Tl = (loo i273) (27 + 273) K (12.0 atm) = 14.9 atm Obtain the combined gas law from Boyle's law and Charles' law. Visualize the gas expanding in two stages as pictured below. T is constant P is constant P v, = P*v I j Tl v 2 whence Vi = T2 CHAP. 51 MEASUREMENT OF GASES 67 P, = 12.0atm 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 O O 0 0 0 0 0 0 0 0 0 0 0 Room temperature 27C 0 0 0 0 0 Boiling water 100 "C Fig. 5-4 In the left equation substitute the expression for Vi from the right equation. 5.7. Given 20.0 L of ammonia at 5 "C and 760 torr, determine its volume at 30 "C and 800 torr. See Figure 5-5. The combined gas law gives (5 "C= 278 K, 30 "C= 303 K): --1 p2v2 w - Tl 5.8. T2 or v2= v1 (5) (2) = (20.0 L ) 278 K ( E ) ( ) torr eL 800 = 20.7 One liter of gas measured at 1 .OO atm and -20 "C is allowed to warm to 40 "C. To how many atmospheres pressure must it then be subjected in order to reduce its volume to one-half liter? The combined gas law gives -'lV1 Tl -p2V2 T2 or 313 K P 2 = P I ( 2 ) ( 2 ) = ( l a t m ) ( e ) ( ~ ) =2.47atm 68 MEASUREMENT OF GASES [CHAP. 5 P, = 760 P, = 800 Fig. 5-5 5.9. A motorist filled his tires to a pressure of 30.0 pounds per square inch (psi) at a temperature of 27F. After a long drive, he found that the pressure had built up to 34.2 psi. Estimate the temperature (in OF) of the air in the tires. Assume that the tire gage reads relative pressure, that the volume of the tires remains constant, and that the ambient pressure is 1.00 atmosphere. In problem 1.9 it was calculated that 1 atm was equal to 14.7 psi. Hence the initial and final pressures are (30.0 14.7) and (34.2 14.7) psi, respectively. Since absolute zero is -460 O F (Fig. 1-1) the absolute temperature is (27 + 460) Fahrenheit degree intervals. Applying Gay-Lussac's law (for a constant volume change): + + 7'2 = 7 x P2/P1 = (27 + 460)(34.2 + 14.7)/(30.0 + 14.7) ' 1 = 533 Fahrenheit degree intervals; 533 - 460 = 73 O F This problem illustrates that any units of pressure or temperature may be used provided they are the same, but that they must be absolute values. Note: The absolute temperature scale using Fahrenheit degree intervals is used in some English-speaking countries and is called the Rankine scale, symbolized O R . It is not $1. 5.10. A certain container holds 6.00 g of CO;!at 150C and 100 kPa pressure. How many grams of CO2 will it hold at 30C and the same pressure? Let the volume of the container be V , and the volume of the 6.00 g of CO2 at 30C be V2. Then by Charles' law: V2/V, = (30 + 273)/(150 + 273) = 0.716 Since 6.00 g of CO2 occupies 0.716 Y , , to fill the container ( V I )will require 6.00 g/0.716 = 8.38 g of CO2 CHAP. 51 MEASUREMENT OF GASES 69 DENSITY OF A GAS 5.11. The density of helium is 0.178 6 kg/m3 at S.T.P. If a given mass of helium at S.T.P. is allowed to expand to 1.500 times its initial volume by changing the temperature and pressure, compute its resultant density. The density of a gas varies inversely with the volume Resultant density = (0.178 6 kg/m3) Note that 1 kg /m3 = 1 g/L. 5.12. The density of oxygen is 1.43 g/L at S.T.P. Determine the density of oxygen at 17 "C and 700 torr. The combined gas law shows that the density of an ideal gas varies inversely with the absolute temperature and directly with the pressure. PARTIAL PRESSURE 5.13. A mixture of gases at 760 torr contains 65.0% nitrogen, 15.0% oxygen, and 20.0% carbon dioxide by volume. What is the pressure of each gas in torr? A fundamental property of ideal gases is that each component of a gas mixture occupies the entire volume of the mixture. The percentage composition by volume refers to the volumes of the separate gases before mixing, each measured at the same pressure and temperature. Thus 65 volumes of nitrogen, 15 volumes of oxygen, and 20 volumes of carbon dioxide, each at 760 torr, are mixed to give 100 volumes of mixture at 760 torr. As the volume of each component is increased by mixing, its pressure must (Boyle's law) decrease proportionately. Partial pressure of N, = (760 torr)(%) = 494 torr Partial pressure of O2 = (760 torr)(G) = 114 torr Partial pressure of CO2 = (760 torr)(&) = 152 torr Check: Total pressure = sum of partial pressures = 760 torr 5.14. In a gaseous mixture at 20C the partial pressures of the components are: hydrogen, 200 torr; carbon dioxide, 150 torr; methane, 320 torr; ethylene, 105 torr. What is (a) the total pressure of the mixture, and (b) the volume percent of hydrogen? (4 (b) Total pressure of mixture = sum of partial pressures = 200 + 150 + 320 + 105 = 775 torr By applying Boyle's law, it can be shown that the fraction of the total pressure (when all gases occupy the same volume) is the same as the fraction of the total volume (when each gas is at the same pressure). Volume fraction of H2 = partial pressure of H2 - - 200 torr = 0.258 = 25.8% total pressure of mixture 775 torr ~ 5.15. A 200-mL flask contained oxygen at 200 torr, and a 300 niL flask contained nitrogen at 100 torr, as shown in Fig. 5-6. The two flasks were then interconnected so that each gas filled its combined 70 MEASUREMENT OF GASES [CHAP. 5 100 torr 200 torr PTOTAL PTOTAL Fig. 5-6 volumes. Assuming no change in temperature, what was the partial pressure of each gas in the final mixture and what was the total pressure? The final total volume was 500 mL. Oxygen Pfinal = Pinitial - = (200 torr) Vfinal) Nitrogen Pf = ( Pi );( Vinitial (Z) = = 80 torr = (100 torr) 300 (=) 60 torr Total pressure = (80 torr) + (60 torr) = 140 torr COLLECTING GASES OVER A LIQUID 5.16. Exactly 100 cm3 of oxygen is collected over water at 23 "C and 800 torr. Compute the standard volume of the dry oxygen. Vapor pressure of water at 23 "C is 21.1 torr. CHAP. 51 MEASUREMENT OF GASES 71 The gas collected is actually a mixture of oxygen and water vapor. The partial pressure of water vapor in the mixture at 23 "C is 21.1 torr. Hence Pressure of dry oxygen = (total pressure) - (vapor pressure of water) = (800 torr) - (21 tolrr) = 779 torr Thus, for the dry oxygen, V , = 100 cm', T , = 23 + 273 = 296 K, P = 779 torr. Converting to S.T.P., v, = V l ( 2 ) 5.17. (2) = (100 cm3) (29M() 273 K (G) cm3 = 94.5 779 torr In a basal metabolism measurement timed at exactly 6 min, a patient exhaled 52.5 L of air, measured over water at 20 "C.The vapor pressure of water at 20 "C is 17.5 torr. The barometric pressure was 750 torr. The exhaled air analyzed 16.75 volume YO oxygen, and the inhaled air 20.32 volume YOoxygen, both on a dry basis. Neglecting any solubility of the gases in water and any difference in the total volumes of inhaled and exhaled air, calculate the rate of oxygen consumption by the patient in cm3 (S.T.P.) per minute. 273 K (750 - 17.5) torr Volume of dry air at S.T.P. = (52.5 L) = 47.1 L (293 K) 760 torr Rate of oxygen consumption = - [- volume of oxygen (S.T.P.) consumed time in which this volume was consumed (0.203 2 - 0.167 5)(47.L L) = 0.280 L/min = 280 cm3/min 6 min 1 5.18. A quantity of gas is collected in a graduated tube over mercury. The volume of gas at 20C is 50.0 cm3, and the level of the mercury in the tube is 200 mm above the outside mercury level, as shown in Fig. 5-7. The barometer reads 750 torr. Find the volume at S.T.P. After a gas is collected over a liquid, the receiver is often adjusted so that the level of liquid inside and outside the receiver is the same. When this cannot be done conveniently, it is necessary to calculate the effect of the difference in levels. The vapor pressure of Hg at 20 "C is negligibly small, and the type of correction needed for collection above water, as in problems 5.16 and 5.17, is not necessary. Since the level of mercury inside the tube is 200 mm higher than outside, the pressure of the gas is 200 mmHg==200torr less than the atmospheric pressure of 750 torr. 273 K (750 - 200) torr = 33.7 cm3 Volume at S.T.P. = (50.0 cm3)(293 K) [-760 ____ torr 1 Supplementary Problems 5.19. The vapor pressure of water at 25C is 23.8 torr. Express this in (a) atmosphere, (b) kilopascals. Ans. (a) 0.031 3 atm; (b) 3.17 kPa 5.20. Camphor has been found to undergo a crystalline modification at a temperature of 148 "C and a pressure of 3.09 x 109 N/rn2. What is the transition pressure in atmospheres? Ans. 3.05 x 104 atm 72 MEASUREMENT O F GASES [CHAP. 5 t,,, = 20" c Pbarom 750 torr = /- V = 50.0 c ' m 200 m m I Fig. 5-7 5.21. A hydrogen-filled weather balloon occupies 48.0 ft3 at ground level where the pressure is 753 torr. Compute its volume at the top of the mountain where it is about to be launched, where the temperature is the same as on the ground, but the pressure is only 652 torr. (You may ignore any contribution to the pressure of the gas due to tension in the skin of the balloon.) Ans. 55.4 ft3 5.22. Ten liters of hydrogen under 1 atm pressure is contained in a cylinder which has a movable piston. The piston is moved in until the same mass of gas occupies 2 L at the same temperature. Find the pressure in the cylinder. Ans. 5 atm 5.23. Chlorine gas is evolved at the anode of a commercial electrolysis cell at the rate of 3.65 L/min, at a temperature of 647 "C. On its way to the intake pump it is cooled to 63 "C. Calculate its rate of intake to the pump assuming the pressure has remained constant. Ans. 1.33 L/min 5.24. A quantity of hydrogen is confined in a platinum chamber of constant volume. When the chamber is immersed in a bath of melting ice, the pressure of the gas is 1000 torr. (a) What is the Celsius temperature when the pressure manometer indicates an absolute pressure of 100 torr? (b) What pressure will be indicated when the chamber is brought to IOO'C? Ans. (a) -246C; (b) 1366 torr 5.25. Since helium gas is a precious resource, the contents of a small passenger balloon were pumped to storage during an overhaul. Normally the balloon contains 18 700 ft3 of helium at 31 "C and 1.00 atm (14.7 lbf/in2, CHAP. 51 MEASUREMENT OF GASES 73 or psi). How many steel cylinders will be required to store it, if each cylinder holds 2.50 ft3, storage temperature is 11 "C, and the cylinders are rated safe at 2 000 psi pressure? Ans. 52 cylinders (calculated 5 1.4 cylinders) 5.26. A mass of gas at 50C and 785 torr occupies 350 mL. What volume will the gas occupy at S.T.P.? Ans. 306 mL 5.27. The gas evolved in a reaction was collected in a vessel like that shown in Fig. 5-7. Its volume was 47.3 cm3 when first observed, with the barometric pressure 752 torr, room temperature 26 "C and the Hg level in the bulb 279 mm below that in the collection tube. The next day, with the room temperature 17 "C and the barometric pressure 729 torr, an assistant measured the volume again but first adjusted the bulb to equalize the levels. What volume reading is expected? Ans. 29.8 cm3 5.28. Exactly 500 cm3 of nitrogen is collected over water at 25C and 755 torr. The gas is saturated with water vapor. Compute the volume of the nitrogen in the dry condition at S.T.P. Vapor pressure of water at 25 "C is 23.8 torr. Ans. 441 cm3 5.29. A dry gas occupied 127 cm3 at S.T.P. If this same mass of gas were collected over water at 23 "C and a total gas pressure of 745 torr, what volume would it occupy? The vapor pressure of water at 23 "C is 21 torr. Ans. 145 cm3 5.30. A mass of gas occupies 0.825 L at -30 "C and 556 Pa. What is the pressure if the volume becomes 1 L and the temperature 20 "C? Ans. 553 Pa 5.31. A 57.3 L pressure vessel has a safety valve set to open at 875 kPa. A chemical reaction is expected to yield 472 L of gaseous product at S.T.P. Would it be prudent to store this in the vessel described if the ambient temperature can rise to 105 O F ? Ans. No; the final pressure could be 959 kPa 5.32. The density of a gas at S.T.P. provides an easy measurement of molar mass (see problem 6.1). For a normally liquid substance, however, it is necessary to measure the gas density at an elevated temperature and reduced pressure. Calculate the density in g/L (same as kg/nn3) at S.T.P. of a gas which has a density of 3.45 g/L at 98C and 638 torr. Ans. 5.59 g/L 5.33. A certain container holds 2.55 g of neon at S.T.P. What mass of neon will it hold at 100C and 10.0 atm? Ans. 18.7 g 5.34. At the top of a mountain the thermometer reads 10C and the barometer reads 700 mmHg. At the bottom of the mountain the temperature is 30 "C and the pressure is 760 mmHg. Compare the density of the air at the top with that at the bottom. Ans. 0.986 (top) to 1.000 (bottom) 5.35. A volume of 95 cm3 of nitrous oxide at 27C is collected over mercury in a graduated tube, the level of mercury inside the tube being 60 mm above the outside mercury level when the barometer reads 750 torr. (a) Compute the volume of the same mass of gas at S.T.P. (b) What volume would the same 74 MEASUREMENT OF GASES [CHAP. 5 mass of gas occupy at 40 "C, the barometric pressure being 745 torr and the level of mercury inside the tube 25 mm below that outside? Ans. 5.36. (a) 78 cm3; (b) 89 cm3 At a certain altitude in the upper atmosphere, the temperature is estimated to be - 100 "C and the densityjust lOV9 that of the earth's atmosphere at S.T.P. Assuming a uniform atmospheric composition, what is the pressure, in torr, at this altitude? Ans. 4.82x lO-' torr 5.37. At 0 "C the density of nitrogen at 1 atm is 1.25 kg/m3. The nitrogen which occupied 1 500 cm3 at S.T.P. was compressed at 0 "Cto 575 atm and the gas volume was observed to be 3.92cm3, in violation of Boyle's law. What was the final density of this nonideal gas? Ans. 478 kg/m3 5.38. The respiration of a suspension of yeast cells was measured by observing the decrease in pressure of gas above the cell suspension. The apparatus, shown in Fig. 5-8,was arranged so that the gas was confined to a constant volume, 16.0 cm3, and the entire pressure change was caused by uptake of oxygen by the cells. The I Density = 1.034 gc 3 /m Temp. = 37" C Fig. 5-8 CHAP. 51 MEASUREMENT OF GASES 75 pressure was measured in a manometer the fluid of which had a density of 1.034 g/cm3. The supply of fluid was adjustable so that the level on the closed side was kept constant. The entire apparatus was immersed in a thermostat at 37C. In a 30-min observation period the fluid in the open side of the manometer dropped 37 mm. Neglecting the solubility of oxygen in the yeast suspension, compute the rate of oxygen consumption by the cells in cubic millimeters of 0 2 (S.T.P.) per hour. Ans. 5.39. 105 mm3/h A mixture of N2, NO, and NO2 was analyzed by selective absorption of the oxides of nitrogen. The initial volume of the mixture was 2.74 cm3. After treatment with water, which absorbed the NOz, the volume was 2.02 cm3. A ferrous sulfate solution was then shaken with the residual gas to absorb the NO, after which the volume was 0.25 cm3. All volumes were measured at barometric pressure. Neglecting water vapor, what was the volume percentage of each gas in the original mixture? An$. 9.1% NZ, 64.6% NO, 26.3% NO2 5.40. A handball with an internal volume of 60 cm3 was filled with air to a pressure of 1.35 atm. A dishonest player filled a syringe to the 25-cm3mark with air at 1.OO atm ancl injected it into the handball. Calculate the pressure inside the tampered handball, assuming its volume did not change. Ans. 1.77 atm 5.41. A 250-mL flask contained krypton at 500 torr. A 450-mL flask contained helium at 950 torr. The contents of the two flasks were mixed by opening a stopcock connecting them. Assuming that all operations were carried out at a uniform constant temperature, calculate the final total pressure and the volume percent of each gas in the mixture. Neglect the volume of the stopcock. Ans. 789 torr, 22.6% Kr 5.42. A glass vacuum tube was sealed at the factory at 750C with a residual pressure of air of 4.5 x 10-7 torr. Then a metal "getter" was actuated to consume all the oxygen (which is 21% by volume of air). What was the final pressure in the tube at 22 "C? Ans. 1.03 x 10-7 torr 5.43. The vapor pressure of water at 80 "C is 355 torr. A 100-mL vessel contained water-saturated oxygen at 80 "C, the total gas pressure being 760 torr. The contents of the vessel were pumped into a 50 mL vessel at the same temperature. What were the partial pressures of oxygen and of water vapor, and what was the total pressure in the final equilibrated state? Neglect the volume of any water which might condense. Ans. 810 torr, 355 torr, 1 165 torr 5.44. A gas sample was collected in an apparatus similar to that in Fig. 5-7 except that the confining liquid was water. At 17 "(2 the volume of gas was 67.3 cm3, the barometric pressure was 723 torr, and the water level in the bulb was 210 mm below that in the collection tube. Later that day the room warmed to 34C and the barometer went up to 741 torr. The experimeter slowly adjusted the bulb to equalize the levels. What was the new volume reading? The vapor pressure of water is 14.5 torr at 1'7 "C and 39.9 torr at 34 "C. The density of mercury is 13.6 times that of water. Ans. 70.4 cm3 This page intentionally left blank Chapter 6 The Ideal Gas Law and Kinetic Theory AVOGADRO'S HYPOTHESIS Avogadro's hypothesis states that equal volumes o all gases under the same conditions of temperature f and pressure contain the same number of molecules. Thus 1 L (oir mL or m3 or other unit of volume) of oxygen contains the same number of molecules as 1 L (or mL, etc.) of hydrogen or of any other gas, provided the volumes are measured under the same conditions. Avogadro's hypothesis enables us to determine the relative masses of the molecules (molecular masses) of gases. The logic is shown in Example 1. EXAMPLE 1 At standard conditions, 1 L of oxygen weighs 1.43 g and 1 L of carbon monoxide weighs 1.25 g. By Avogadro's hypothesis, 1 L of carbon monoxide (S.T.P.) contains the same number of molecules as 1 L of oxygen (S.T.P.). Hence a molecule of carbon monoxide must weigh 1.25/1.43 times as much as a molecule of oxygen. Accordingly, if we take the molecular mass of oxygen as 32, then the molecular mass of carbon monoxide is 1 -x 32 = 28 I .L3 -3,r 1.43 When applied precisely, the gas density method may be used for atomic mass determinations, particularly for the lighter elements. Even crude gas density experiments may be used, in conjunction with chemical composition data and known atomic masses to establish the molecular mass, and hence the molecular formula of a gaseous compound. EXAMPLE 2 A hydride of silicon that has the empirical formula SiH3 was found to have an approximate gas density at S.T.P. of 2.9 g/L. By comparison with oxygen, whose molecular mass and density are known, the molecular mass of the hydride is then -x 32 = 65 Although this approximate molecular mass might be in error by as much as 10 percent, it is sufficiently accurate to allow assignment of the molecular formula Si2H6(molecular mass 62.2) and to rule out SiH3 (3 1.l), Si3H9 (93.3), and higher multiples of the empirical formula unit mass. 2.9 1.43 MOLAR VOLUME If 1 mol of any gas has the same number of molecules, N A , as 1 mol of any other gas (Chapter 2), and if equal numbers of molecules correspond to equal volumes at S.T.P. (Avogadro's hypothesis), then 1 mol of any gas has the same volume at S.T.P. as 1 mol of any other gas. This standard molar volume has the value 22.414 L. Because of the deviations from ideal behavior (and thus from Avogadro's hypothesis) shown by real gases, the actual observed molar volume of a gas at S.T.P. be slightly different from 22.414 L, usually lower. In the rest of this chapter, the rounded value 22.4 L will be used for all real gases. IDEAL GAS LAW Let us apply the combined gas law (Chapter 5) to 1 mol of ideal gas, using a subscript zero to denote standard conditions. ---- PV T PoVo - (1 atm)(22.4 L/mol) = 0.082 1 L - atm K-' mol-' To 273 K 77 78 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 The quantity R = 0.082 1 L - atm - K-' - mol-' is known as the universalgas constant. For n moles of ideal gas at the same temperature and pressure, the volume would be n times as great. Thus P V / T = nR, or PV = nRT This is the ideal gas law; it should be memorized by the student, along with the numerical value of R.The above value of R is used when P is in atmospheres, V is in liters, T is in kelvins, and n is in moles. When SI units are used for P and V (pascals and cubic meters), then R = 8 . 3 1 4 5 J K-' must be used. The mass, in grams, of gas present is given by w=nd where A is the molar mass in grams per mole; or by w=dV - mol-' where d is the gas density, in grams per liter if V is in liters. Hence the alternative forms of the ideal gas law: PV= ( 2 ) R T P = ($)RT For an ideal gas, V is proportional to n when P and T are fixed. Hence the awkward concept of volume percentage (or fraction) discussed in Problem 5.12 can now be replaced with mole percentage, or mole fraction, where each gas is assumed to occupy the entire volume of the mixture but is at its own partial pressure. GAS VOLUME RELATIONS FROM EQUATIONS A chemical equation representing the reaction or production of two or more gaseous substances directly indicates the volumes of the gases participating in the reaction when T and P are fixed. The volumes are related to the numbers of molecules indicated in the equation and may be evaluated without reference to the reacting weights of the gases. For example: 4NH3 (gas)+ 3 0 2 (gas) -+2 N 2 (gas) 4 molecules 4 mol 4 x 22.4 L 4L 4 ft3 3 molecules 3 mol 3 x 22.4 L 3L 3 ft3 2 molecules 2 mol 2 x 22.4 L 2L 2 ft3 + 6 H 2 0 (vapor) 6 molecules 6 mol 6 x 22.4 L 6L 6 ft3 The above interpretation is valid so long as all the water remains in the vapor state under the conditions of temperature and pressure specified. At S.T.P. the water would be condensed to a negligible volume of liquid, and so 7 L of reactants would produce only 2 L of products rather than 8 L. GAS STOICHIOMETRY INVOLVING MASS In chemical reactions involving gases, the amounts of some reactants or products may be specified in units of mass. These problems are solved by the same methods as in Chapter 4, except that the number of moles of gas may be determined from its volume, temperature, and pressure using the ideal gas law. As alway we turn to the balanced equation for the relative numbers of moles of all reactants and products. CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 79 EXAMPLE 3 Carbon dioxide can be removed from the recirculated air aboard a spaceship by passing it over lithium hydroxide. 2LiOH(s) + C02(g) -+Li2C03(s) H20(g) + Calculate the number of grams of LiOH consumed in the above reaction when 100 L of air containing 1.20% CO*, at 29C and 776 torr, is passed through. nco, = (0.012 0)( 100 L)[776 torr/(760 torr/atm)] = 0.049 4 mol 0.082 1 L . atm K-' mol--' 302 K IzL~OH= 2(nco2) = 2(0.0494 mol) = 0.098 8 mol Amount of LiOH = (0.098 8 mo1)(23.9 g mol-') =I 2.36 g BASIC ASSUMPTIONS OF THE KINETIC THEORY OF GASES The ideal gas law can be derived solely from theoretical principles by making a few assumptions about the nature of gases and the meaning of temperature. Such a derivation can be found in any physical chemistry textbook. Herewith, though, is a listing of th.e basic assumptions. A gas consists of a very large number of molecules which are in a state of continual random motion. A molecule has negligible size. Pressure is a consequence of the force of molecular collisions on a measuring surface, such as the container walls. Collisions between molecules or between a molecule and an inert surface are perfectly elastic, with no change in the total kinetic energy of the gas molecules. No forces are exerted between molecules except through collisions. Therefore, between collisions, a molecule travels in a straight line at constant speed. The average kinetic energy per gas molecule, is independent of the nature of the gas and is directly proportional to the temperature. Here, the kinetic energy is defined by the standard physical formula in terms of the molecular mass, m, and the molecular speed, U. This statement may be taken as a more exact definition of temperature than the qualitative one given in Chapter 1. A list of some common energy units is given in Table 6-1. Table 6-1. Some Common Energy Units Unit joule (SI) calorie kilocalorie Calorie, nutritionists' British thermal unit 1 Unit Symbol Definition J cal kcal Cal Btu m 2 .kg . sP2 = N . m 4.184 J 103 cal 103 cal 252 cal= 1054 J* * Approximate definition. PREDICTIONS OF THE KINETIC THEORY (a) A mechanical argument based on Assumptions 1, 2, and 3 above shows that for a gas composed of N molecules PV = @q;mu2)avg] (b) Using Assumption 4, the distribution of speeds over the assemblage of molecules is predicted in closed mathematical form, known as the Maxwell-Boltzmann distribution. Figure 6- 1 shows, for 80 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 0.5 1 .o 1.5 2.0 2.5 3.0 3.5 4.0 4.5 -- U kmls Fig. 6-1 hydrogen, some plots of the fraction of molecules having speeds close to a given value U, as a function of U, and at two different temperatures. The speed for which a distribution curve has its maximum is called the most probable speed, ump.It is slightly smaller than the average speed, uavg. Still another speed, slightly larger than uaVg,is the root-mean-square speed, urns; it is defined as that speed for which a molecule's kinetic energy would be equal to the average kinetic energy over the whole sample, i.e., For the Maxwell-Boltzmann distribution: (c) From the results of (a) and (b),and using N = nNA, and N A m = A, we can derive the ideal gas law. The agreement of the above with the empirically determined ideal gas law validates the important definition of temperature in assumption 4, above. ( d ) The frequency of molecular collisions with a unit area of container wall is predicted as Z= TART)'/^ NAP Now, the rate at which a gas effuses (flows out) through a small hole in the container into a vacuum is exactly the rate at which the molecules would collide with a wall area equal to the area of the hole. Hence, from the above equation, the effusion rates of two gases, both at the same pressure and temperature, are in the ratio 112 z 2 CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 81 where 4 is the molar mass. This explains Graham's luw o eflusion, the experimental finding f that the rates of effusion of gases at equal pressures and temperatures are inversely proportional to the square roots of their densities. (By the ideal gas law, gas density is proportional to molar mass.) (e) The inverse dependence of average velocity and of effusion rate on the square root of molar mass shows up in the treatment of other transport phenomena, such as dzflusion, thermal conduction, and nonturbulentflow, although the theory is not as exact in these phenomena, since it must be augmented by a detailed consideration of the nature of intermolecular collisions, a process involving some nonideal aspects of gases. Problem 6.60 treats an important case of diffusion, the process in which one gas penetrates another on the way to equalization of concentrations throughout a gas mixture. The assumption will be made in this book that the relative diffusion rates of two gases are inversely proportional to the square roots of their molar masses, as is the case more exactly for effusion. Solved Problems VOLUMES AND MOLAR MASSES OF GASES 6.1. Determine the approximate molar mass of a gas if 560 cni3 weighs 1.55 g at S.T.P. P V = ($)RT wRT - (1.55 g)(0.082 1 L . atm - K-' . mo1--')(273 K) = 62.0 g . mol-l A=PV (1 atm)(0.560 L) Another Method. Molar mass = weight of 1 L at S.T.P. x number lof liters at S.T.P. per mole = (-) (22.4 L/mol) = 62.0 g/rnol This second method is faster if the data happen to be given at S.T.P. 6.2. At 18 "C and 765 torr, 1.29 L of a gas weighs 2.71 g. Calculate the approximate molar mass of the gas. For variety in calculational procedure, first convert the given gas data to moles instead of relying on a formula for A as in Problem 6.1. P V = nRT n = - pv RT 765 P=atm 760 T = (18 + 273) K = 291 K n= ($Jatm)(1.29 L) = 0.0544 mol (0.082 1 L - atm K-' . mol-')(291 K) Then A! = 2.71 g/0.0544 mol = 49.8 g/mol. 82 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 6.3. Determine the volume occupied by 4.0 g of oxygen at S.T.P. Molar mass of oxygen is 32. Volume of 4.0 g 0 2 at S.T.P. = number of moles in 4.0 g O2 x standard molar volume = (a) (22.4 L/mol) = 2.8 L 6.4. What volume would 15.0 g of argon occupy at 90 "C and 735 torr? wRT - (15.0 g)(0.082 1 L aatm . K-' mol-')(363 K) V== 11.6L AP (39.9 g/mol)(% atm) Another Method. At S.T.P. this much argon would occupy 1` = 15'0 (22.4 L/mol) = 8.4 L (39.9 g/mol) This problem can be completed by the usual procedure for converting from S.T.P. to an arbitrary set of conditions. v2 = (2)(2) 363 K = (8*4L)(731() (G) 760 torr = 11.6 6.5. Compute the approximate density of methane, CH4, at 20C and 5.00 atm. The molar mass of methane is 16.0. d=-= RT &P (16.0 g m01-~)(5.00atm) = 3.33 g/L (0.082 1 L . atm . K-' . mo1-')(293 K) 16.0 g Another Method. Density = mass of 1 mol volume of 1 mol = 3.33 g/L 6.6. Dry air consists of approximately 21 ?40 0 2 , 78% N2, and 1O`/ Ar by moles. What is the average (i.e. "apparent") molar mass of dry air? Calculate the density of dry air at S.T.P. Compute the mass of 22.4 L of air at S.T.P., i.e., 1 mol, which contains 0.21 molO2, 0.78 mol NZ, and 0.01 mol Ar. (0.21 mo1)(32.0 g/mol) + (0.78 mo1)(28.0 g/mol) + (0.01 mo1)(39.95 g/mol) = 29 g The mass of 22.4 L of a gas at S.T.P. would be its molar mass if it were a pure gaseous substance. Hence 29 g/mol appears to be the molar mass of air. Its density is - 1.29 g/L 29 g = 22.4 L 6.7. An organic compound had the following analysis: C = 55.8%, H = 7.03%, 0 = 37.2%. A 1.500-g sample was vaporized and was found to ocupy 530 cm3 at 100C and 740 torr. What is the molecular formula of the compound? CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 83 The approximate molar mass, calculated from the gas density data, is 89.0. The empirical formula, calculated from the percentage composition data, is C2H30, and the empirical formula unit mass is 43.0. The exact molar mass must therefore be (2)(43.0) = 86.0, since this is the only integral multiple of 43.0 which is reasonably close to the approximate result of 89. The molecule: must therefore contain twice the number of atoms in the empirical formula, and the molecular formula must be C4H6O2. Another Method. Instead of calculating the empirical formula, we can use the: composition data to calculate the number of moles of atoms of each element in 89 g of compound. n(C) = (0.558)(89 g) = 4.1 12.0 g/mol n(H) = (0.070 3)(89 g) := 6.2 1.01 g/mol n(0) = (0.372)(89g) = 2.1 16.0 g/mol These numbers approximate the numbers of atoms in a moleciile, the small deviations from integral values resulting from the approximate nature of the molar mass measurement. The molecular formula, C4HsO2, is obtained without going through the intermediate evaluation o:f an empirical formula. 6.8. Mercury diffusion pumps may be used in the laboratory to produce a high vacuum. Cold traps are generally placed between the pump and the system to be evacuated. These cause the condensation of mercury vapor and prevent mercury from diffusing back into the system. The maximum pressure of mercury that can exist in the system is the vapor pressure of mercury at the temperature of the cold trap. Calculate the number of mercury vapor molecules per unit volume in a cold trap maintained at - 120 "C. The vapor pressure of mercury at this temperature is 1 0 - l ~ torr. n P ( 10d6/760) atm = 1.0 x Moles per liter = - = -= V RT (0.0821 L.atm.K-'.mo1-')(153K) IO-~O mol/L Molecules per liter = (1 .O x 1Od2' mol/L)(6.0 x 1023molecules/mol) = 6 x 103 molecules/L or 6 molecules/cm3. REACTIONS INVOLVING GASES In each of the next four problems all gases are measured at the: same temperature and pressure. 6.9. What volume of hydrogen will combine with 12 L of chlorine to form hydrogen chloride? What volume of hydrogen chloride will be formed? The balanced equation for this reaction is H ~ ( ~ L +s C ~ ~ ( ~ -+ Z S ) z ) L 2HCl(gas) (1 molecule) ( 1 molecule) ( 2 molecules) The equation shows that 1 molecule H2 reacts with 1 molecule Clz to form 2 molecules HC1. But, by Avogadro's hypothesis, equal numbers of molecules of gases under the same conditions of temperature and pressure occupy equal volumes. Therefore the equation also indicates that 1 volume of H2 reacts with 1 volume C12 to form 2 volumes of HCl (gas). Thus 12 L of H2 combines with 12 L of C12 to form 2 x 12 L = 24 L of HCl. 6.10. What volume of hydrogen will unite with 6 ft3 of nitrogen to form ammonia? What volume of ammonia will be produced? NZ(gaS) + 3H3(gas) -, 2NH3(ga~) The equation indicates that, since only gases are involved, 1 volume N2 reacts with 3 volumes H2 to form 2 volumes NH3. Therefore 6 ft3 N2 reacts with 3 x 6 := 18 ft3 H2 to form 2 x 6 = 12 ft3 NH3. 84 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 6.11. Sixty-four liters of NO is mixed with 40 L of completion: 0 2 and the following reaction allowed to go to 2NO(gas)+ 02(gas) -+ 2N02(gas) Calculate the total volume of gas after completion of the reaction. Since the volume of oxygen required is only half that of the NO, (64) = 32 L of 0 2 will be consumed and 40 - 32 = 8 L will be in excess. The volume of NO2 formed is the same as that of the NO reacted. The final volume of gas then will be 64 L of NO2 plus 8 L of excess 0 2 or a total of 72 L. 4 6.12. What volume of 0 2 at S.T.P. is required for the complete combustion of 1 mol of carbon disulfide, CS2? What volumes of CO2 and SO2 at S.T.P. are produced? cs2(liquid)+ 302 (gas) -+ CO;!(gas) + 2s02 (gas) (1 mol) (3 mol) (1 mol) (2 mol) The equation shows that 1 mol CS2 reacts with 3 molO2 to form 1 mol CO2 and 2 mol S02. One mole of a gas at S.T.P. occupies 22.4 L. Therefore Volume of 3 mol O2 at S.T.P. = 3(22.4) = 67.2 L 0 2 Volume of 1 mol CO2 at S.T.P. = l(22.4) = 22.4 L CO2 Volume of 2 mol SO2 at S.T.P. = 2(22.4) = 44.8 L SO2 6.13. How many liters of oxygen, at standard conditions, can be obtained from 100 g of potassium chlorate? 2KC1O3 ( d i d ) -+ 2KC1+ 302 (gas) (2 mol) (3 mol) Molar Met hod. The equation shows that 2 mol KC103 gives 3 mol used for the number of moles. n(KC103) = 02. As in previous chapters, the symbol n will be 100 g = 0.816 m01 KC103 122.6 g/mol n ( 0 2 ) = in(KC103) = j(0.816) = 1.224 rnol O2 Volume of 1.224 mol O2 at S.T.P. = (1.224 mol 02)(22.4L/mol) = 27.4 L O2 Another Method. The equation shows that 2 mol KC103[2(122.6) = 245.2 g] gives 3 molar volumes 02[3(22.4 L) = 67.2 L]. Then 245.2 g KC103 gives 67.2 L O2 1 g KC103 and Note: at S.T.P. at S.T.P. L) = 27.4 L 0 2 gives -L 0 2 67.2 245.2 gives loo(& 100 g KC103 at S.T.P. It was not necessary in either method of solution to calculate the mass of oxygen formed. CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 85 6.14. What volume of oxygen, at 18 "C and 750 torr, can be obtained from 100 g of KC103? This problem is identical with Problem 6.13, except that the 27.4 L of 0 2 at 0 "C and 760 torr must be converted to liters of 0 2 at 18C and 750 tom. Volume at 18C and 750 torr = (27.4 L) [- 273 K Another Method. When we are not concerned with S.T.P., calculate the volume directly from the number of moles. nRT V=-= P (1.224 mol)(0.082 1 L - atm K-' mol-')(291 K) = 29.6 L $j atm 6.15. How many grams of zinc must be dissolved in sulfuric acid in order to obtain 500 cm3 of hydrogen at 20C and 770 torr? Zn (solid) H2SO4 + ZnS04 ( I mol) + + H2(gas) ( 1 rnol) First find the number of moles of PV n(H2) = RT 1 H2 using the ideal gas law. atm)(O.500 L) -- 0.021 1 mol H2 (0.082 1 L . atm . K-' mol-')(293 K) (a Next use the factor-label method to complete the problem, obtaining the appropriate molar ratio from the balanced equation. 0.021 1 mol H2 x (1 mol Zn/l mol H2) x (65.4 g Zn/l mol Zn) = 1.38 g Zn 6.16. A natural gas sample contains 84% (by volume) CH4, 110% C2H6, 3% C3H8, and 3% N2. If a series of catalytic reactions could be used for converting all the carbon atoms of the gas into butadiene, C4H6,with 100% efficiency, how much butadiene could be prepared from 100 g of the natural gas? Molar masses: CH4 = 16, C2H6 = 30, C3H8= 44, N2 = 28, C4H6 = 54. The volume percent of a gas mixture is the same as the mole percent. 100 rnol mixture = 84 rnol CH4 + 10 mol C2H6 + 3 mol C3H8 + 3 mol N2 = 84( 16 g CH4) = + lO(30 g C2H6) + 3(44 g C3H8) + 3(28 g N2) 1 860 g natural gas The number of moles of C in 100 mol mixture is 84( 1) 1 mol (54 g) C4H6,113 mol C gives + lO(2) + 3(3) = 113 mol C. Since 4 rnol C gives (ymOl )(54 g/mOl) Then and 1860 g natural gas yields 100 g natural gas yields 1 530 g C4H6 1 530 g C4H6 +&(I 530 g) = 82 g C4H6 6.17. A reaction mixture for the combustion of SO2 was prepared by opening a stopcock connecting two separate chambers, one having a volume of 2.125 1, filled at 0.750 atm with SO2 and the other having a volume of 1.500 L filled at 0.500 atm with 02; both gases were at 80 "C. (a) What were the mole fraction of SO2 in the mixture, the total pressure, and the partial pressures? ( b ) If the mixture was passed over a catalyst that promoted the formation of SO3 and then was returned to the original two connected vessels, what were the mole fractions in the final mixture, and what 86 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 was the final total pressure? Assume that the final temperature is 80 "C,and that the conversion of SO2 is complete to the extent of the availability of 0 2 . PV n ( S 0 , ) = -= RT (0.750 atm)(2.125 L) = 0.055 0 rnol (0.082 1 L . atm K-' . mol-')(353 K) (0.500 atm)( 1.500 L) = 0.025 9 rnol (0.082 1 L eatm . K-' . mo1-')(353 K) PV 40,) = -= RT Each mole fraction is evaluated by dividing the number of moles of the component by the total number of moles in the mixture. Let x be the symbol for mole fraction. x(S02) = n ( S 0 2 )+ n(0,) nW2) - 0.055 0 mol (0.055 0 + 0.025 9) rnol - -- 0.055 0 - 0.680 0.080 9 ~ ( 0 2 = - 0.320 ) z= 0.025 9 0.080 9 Note that the mole fraction is dimensionless and that the sum of the mole fractions is unity. The total pressure prior to the reaction can best be evaluated by using total volume (2.125 L + 1.500L = 3.625 L) and total number of moles, n, (0.080 9 mol). nRT (0.0809 mol)(0.082 1 L . atm K-' . mo1-')(353 K) = 0.647 atm P(t0t) = -= V (3.625 L) Partial pressures in a mixture are easily computed by multiplying the respective mole fractions by the total pressure. This statement can be proved as follows: P(X> - n(X)RT/V P(t0t) - nRT/V Thus, P ( S 0 2 ) = x ( S 0 2 ) x P(tot) = (0.680)(0.647 atm) = 0.440 atm - 4x1 = x(X) n whence P(X) = x(X)P(tot) P ( 0 2 ) = x(Oz) x P(tot) = (0.320)(0.647 atm) = 0.207 atm The sum of the partial pressures must of course equal the total pressure. The chemical equation for the reaction is 2s02 + 0 2 +2s03 All substances are gases under the experimental conditions. The number of moles of O2 required for stoichiometric conversion is half the number of moles of SO2; but the number of moles of O2 available, 0.025 9, was less than half of 0.055 0. Therefore all the 0 2 was used up and an excess of SO2 remained. For the quantities of gases participating or formed in the reaction. 40,) = 0.025 9 mol n ( S 0 2 )used = 2 x 40,) = (2)(0.0259 mol) = 0.051 8 mol The product would be n ( S 0 3 ) = n(SOz)used = 0.051 8 mol so that the residue would be (0.055 0 - 0.051 8) mol = 0.003 2 mol SO2.After the reaction is completed, x ( S 0 3 )= n(S03) 0.051 8 mol n(S0,) + n ( S 0 2 ) - (0.051 8 + 0.003 2) mol n (S 0 2 ) - 0.003 2 mol n ( S 0 , ) + n ( S 0 2 ) - 0.0550 rnol . -- 0.051 8 0.055 0 0.942 x(SO*)= = 0.058 n(tot)RT (0.0550 mo1)(0.082 1 L - atm . K-' - mo1-')(353 K) P(t0t) = ____ = 0.440 atm V 3.625 L CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 87 Note that the final total pressure is the same as the initial partial pressure of S02. This happens because the only gases present at the end are SO2 and SO,. Because each mole of SO3 was made at the expense of 1 mol of S02, the sum of the numbers of moles of the two gases had to equal the initial number of moles of SO2. The final total pressure is less than the initial total pressure because the consumption of the oxygen caused a decrease in the total number of moles of gas. KINETIC THEORY 6.18. (a) Show how the value of R in L . atm. K-' . mol-' can be derived from the value in SI units. (b) Express R in calories. (4 R = 8.314 5 J . K-' . mol-' = (8.314 5 N - m . K-' . mol-') 1 atrn (1.013 x 105'N.m-2 ) (10'1 m3 dm') (E) dm3 = 0.082 1 L atm . K-' - - mol-' R = (8.3145 J K-' .mol-') (411iy - = 1.987 cal - K-' .mol-' J) 6.19. Calculate the root-mean-square velocity of H2 at 0C. e;lJ = (3RT ) 'I2= r(8.314 5 J K-' . mol.-')(273 K) (2.016 g/mol)(l kg/'lOOO g) = 1.84 x 103 m/s = 1837 (J/kg)'I2 = 1.84 x 103 or 1.84 km/s. 6.20. Compute the relative rates of effusion of H2 and CO2 through a fine pinhole. Supplementary Problems VOLUMES, MOLAR MASSES, AND REACTIONS OF GASES 6.21. If 200 cm3 of a gas weighs 0.268 g at S.T.P., what is its molar mass? Ans. 30.0 6.22. Compute the volume of 16 g of nitrous oxide, N20, at S.T.P. Ans. 8.1 L 6.23. What volume will 1.216 g of SO2 gas occupy at 18 "C and 755 torr? A ~ S .456 c1n3 88 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 62. .4 A chemist wished to determine the molar mass of a liquid compound but feared it would decompose if heated. So he injected (via a syringe) a 0.436 g sample of the liquid into a 5.00 L flask of argon at 17 "C to which an open tube manometer was attached. The liquid vaporized completely in the large flask, causing the difference in Hg levels to change from 16.7 mm to 52.4 mm. Calculate the molar mass of the liquid. Ans. 44.2 g/mol 6 2 . Compute the weight of 2.65 L of carbon dioxide gas at 34 "C and 1.047 atm pressure. .5 Ans. 4.84 g 62. .6 Determine the density of H2S gas at 27C and 2.00 atm. Ans. 2.77 g/L 62. .7 Find the molar mass of a gas whose density at 40 "C and 785 torr is 1.286 kg/m3. Ans. 32.0 62. .8 A collapsed balloon and its load weighs 216 kg. Calculate to how many cubic meters it should be inflated with hydrogen gas in order to launch it from a mountain top at - 12 "C and a pressure of 628 torr. The density of air at these conditions is 1.11 g/L. Ans. 210 m3 62. .9 One of the methods for estimating the temperature of the center of the sun is based on the ideal gas law. If the center is assumed to consist of gases whose average molar mass is 2.0, and if the density and pressure are 1.4 x 103kg/m3 and 1.3 x 109 atm, calculate the temperature. A m . 2.3 x 107 K 63. .0 An electronic vacuum tube was sealed off during manufacture at a pressure of 1.2 x 10-5 torr at 27 "C. Its volume is 100 cm3. Compute the number of gas molecules remaining in the tube. Ans. 3.9 x lOI3 molecules 6 3 . One of the important criteria of hydrogen as a vehicular fuel is its compactness. Compare the numbers .1 of hydrogen atoms per cubic meter available in (a) hydrogen gas under a pressure of 14.0 MPa at 300 K; (b)liquid hydrogen at 20 K at a density of 70.0 kg/m3; ( c ) the solid compound DyCo3Hs, having a density of 8200 kg/m3 at 300 K, from which all of the hydrogen may be made available for combustion. Ans. (a) 0.68 x 1028atoms/m3; (b) 4.2 x 1028atoms/m3; (c) 7.2 x 1028atoms/m3 6 3 . An empty steel gas tank with valve weighed 125 lb. Its capacity is 1.5 ft3. When the tank was filled with .2 oxygen to 2000 lbf/in2 absolute at 25 "C, (a) what percent of the total weight of the full tank was 02? Assume the validity of the ideal gas law. (One atmosphere of pressure equals 14.7 lbf/in2.) Pure oxygen gas is not necessarily the most compact source of oxygen for confined fuel systems because of the weight of the cylinder necessary to confine the gas. Other compact sources are hydrogen peroxide and lithium peroxide. The oxygen-yielding reactions are 2H202 -+ 2H20 +0 2 2Li202-+ 2Li20 + 0 2 (b) Rate 65% by weight solution of H202 in H20, and ( c ) rate pure Li202, in terms of % of total weight which is "available" oxygen. Neglect the weights of the containers. Compare with part (a). Ans. (a) 12% (b) 31% ( c ) 35%. Both (b) and ( c ) are more efficient than (a), on a weight percent basis. 63. .3 A 0.331 g sample of pure ammonium carbonate, (NH4)2C03.H20, was placed in an evacuated 30.0 cm3 pyrolysis tube with a pressure rating of 45.0 atm. It was heated at 250C until it decomposed completely: (NH4),C03.H 2 0 -+2NH3 + CO2 + 2 H 2 0 CHAP. 6 1 THE IDEAL GAS LAW AND KINETIC THEORY 89 (a) Was there any liquid water in the tube? (b) Did the tube explode? (The vapor pressure of water at 250C is 39.2 atm.) Ans. (a) No, P(H20) = 8.30 atm (b) No, P(tota1) = 20.7 atm 6 3 . An iron meteorite was analyzed for its isotopic argon content. The amount of 36Arwas 0.200 mm3 (S.T.P.) .4 per kilogram of meteorite. If each 36Ar atom had been formed by a single cosmic event, how many such events must there have been per kilogram of meteorite? Ans. 5.4 x lOI5 6.35. Three volatile compounds of a certain element have gaseous densities calculated back to S.T.P. as follows: 6.75, 9.56, and 10.08 kg/m3. The three compounds contain 96.0% 33.9%, and 96.4% of the element in question, respectively. What is the most probable atomic mass of the element? Ans. 72.6, although the data do not exclude 72.6/n, where n i s a positive integer. 6.36. Chemical absorbers can be used to remove exhaled CO2 of space travelers in short spaceflights. Li20 is one of the most efficient in terms of absorbing capacity per unit weight. If the reaction is Li20 + CO2 + Li2C03 what is the absorption efficiency of pure Li20 in liters CO2 (S.T.P.) per kilogram? Ans. 752 L/kg 6 3 . A mixture of methane, CH4, and ethane, C2H6, exerted a pressure of 2.33 atm when confined to a certain .7 vessel at a pressure, P,and temperature, T . It was mixed with excess oxygen and burned completely to CO2 and H20. After removing all the H 2 0 and excess oxygen, the CO2 was returned to the original vessel where its pressure was 3.02 atm, when measured at the same pressure, P, and temperature, T , as the original mixture. Calculate the mole fractions of the gases in the original mixture. Ans. x(CH4) = 0.704, x(C2H6) = 0.296 6 3 . Exactly 500 cm3 of a gas at S.T.P. weighs 0.581 g. The composition of the gas is as follows: C = 92.24%, .8 H = 7.76%. Derive its molecular formula. Ans. C2H2 6.39. A hydrocarbon has the following composition: C = 82.66%, H Ans. = 17.34%. The density of the vapor is 0.2308 g/L at 30C and 75 torr. Determine its molar mass ancl its molecular formula. 58.2, CdH10 6 4 . How many grams of oxygen are contained in 10.5 L of oxygen measured over water at 25 "C and 740 torr? .0 Vapor pressure of water at 25 "C is 24 torr. Ans. 12.9 g 64. .1 An empty flask open to the air weighed 24.173 g. The flask was filled with the vapor of an organic liquid and was sealed off at the barometric pressure and at 100C. At room temperature the flask then weighed 25.002 g. The flask was then opened and filled with water at room temperature, after which it weighed 176 g. The barometric reading was 725 mmHg. All weighings were done at the temperature of the room, 25 "C.What is the molar mass of the organic vapor? Allow for the buoyancy of the air in the weighing of the sealed-off flask, using 1.18 g/L for the density of air at 25 "C arid 1 atm. Ans. 213 gjmol 6 4 . An ambitious student set out to design the proverbial lead balloon. He assumed a spherical shape of radius, .2 r , and a skin weight of 5.00 g /cm2. The balloon was to be filled with He at the ambient temperature of 25 "C and pressure of 711 torr. At these conditions the density of air is 1.10 g/L. For a sphere of radius, r, the 90 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 surface (skin) area is 47rr2 and the volume is 47rr3/3. Calculate how large the radius must be to get the balloon off the ground. How much must it weigh? Ans. r = 158 m; weight = 1.82 x 107 kg, including the skin and the He. 6.43. A 50 cm3 sample of a hydrogen-oxygen mixture was placed in a gas buret at 18 "C and confined at barometric pressure. A spark was passed through the sample so that the formation of water could go to completion. The resulting pure gas had a volume of 10 cm3 at barometric pressure. What was the initial mole fraction of hydrogen in the mixture (a) if the residual gas after sparking was hydrogen, (b) if the residual gas was oxygen? Ans. (a) 0.73; (b) 0.53 6.44. How much water vapor is contained in a cubic room 4.0 m along an edge if the relative humidity is 50 percent and the temperature is 27 "C? The vapor pressure of water at 27 "C is 26.7 torr. The relative humidity expresses the partial pressure of water as a percentage of the water vapor pressure. Ans. 0.82 kg 6.45. A batch of wet clothes in a clothes dryer contains 0.983 kg of water. Assuming the air leaves the dryer saturated with water vapor at 48 "C and 738 torr total pressure, calculate the volume of dry air at 24 "C and 738 torr required to dry the clothes. The vapor pressure of water at 48 "C is 83.7 torr. Ans. 1.070 x 104 L 6.46. Ethane gas, C2H6, burns in air as indicated by the equation 2C2H6+ 702 -+ 4co2 + 6H20. Determine the number of (a) moles of CO2 and of H20 formed when 1 rnol of C2H6is burned, (b)liters of 0 2 required to burn 1 L of C2H6, (c) liters of CO2 formed when 25 L of C2H6 is burned, ( d ) liters (S.T.P.) of CO2 formed when 1 mol of C2H6 is burned, (e) moles of CO2 formed when 25 L (S.T.P.) of C2H6 is burned, (f) grams of CO2 formed when 25 L (S.T.P.) of C2H6 is burned. Ans. (a) 2 mol CO2, 3 mol H20; (6) 3.5 L; (c) 50 L; ( d ) 44.8 L; (e) 2.23 mol; ( f ) 98.2 g 6.47. Pure nitrogen can be prepared by the decomposition in aqueous solution of ammonium nitrite, NH4N02: NH4N02 -+ N2 + 2H20 If 56.0 mL of Nz was collected over water at 42 "C and a total pressure of 778 torr, what mass of NH4N02 must have been decomposed? The vapor pressure of water at 42 "C is 61 torr. Ans. 0.131 g 6.48. Lithium reacts with hydrogen to produce the hydride, LiH. Sometimes the product is contaminated with unreacted lithium metal, and the extent of the contamination can be measured by measuring the amount of hydrogen gas generated by reacting a sample with water: LiH + H 2 0 -+ LiOH + H2 2Li + 2 H 2 0 + 2LiOH + H2 A 0.205 g sample of contaminated LiH yielded 561 mL of gas measured over water at 22C and a total pressure of 731 torr. Calculate the percent by weight of lithium metal in the sample. The vapor pressure of water at 22C is 20 torr. Ans. 37% 6.49. A mixture of 5.00 g of water, 5.00 g of methanol (CH30H), and 5.00 g of ethanol (C2H50H)was put into a large sealed vessel (previously evacuated) and heated until all traces of liquid had evaporated. If the total pressure was 2.57 atm what was the partial pressure of the ethanol? Ans. 0.514 atm CHAP. 61 THE IDEAL GAS LAW AND KINETIC THEORY 91 6.50. Fifty grams of aluminum is to be treated with a 10% excess of H2SO4. The chemical equation for the reaction is 2A1+ 3H2S04+ A12(S04), + 3H2 (a) What volume of concentrated sulfuric acid, of density 1.80 g/cm3 and containing 96.5% H2SO4 by weight, must be taken? (b) What volume of hydrogen would be collected over water at 20C and 785 torr? Vapor pressure of water at 20 "C is 17.5 torr. Ans. (a) 173 cm3; (b) 66.2 L 6.51. A 0.750-g sample of solid benzoic acid, C7H602, was placed in a 0.500-L pressurized reaction vessel filled with O2 at 10.0 atm pressure and 25C. To the extent of the availability of oxygen, the benzoic acid was burned completely to water and C02. What were the final mole fractions of CO2 and H20 vapor in the resulting gas mixture brought to the initial temperature? The vapor pressure of water at 25 "C is 23.8 torr. Neglect both the volume occupied by nongaseous substances and the solubility of CO2 in H20. (The water pressure in the gas phase cannot exceed the vapor pressure of water, so most of the water is condensed to the liquid.) Ans. x(CO2) = 0.213, x(H20) = 0.0032 6.52. Two gases in adjoining vessels were brought into contact by opening a stopcock between them. One vessel measured 0.250 L and contained NO at 800 torr and 220 K; the other measured 0.100 L and contained O2 at 600 torr and 220 K. The reaction to form N2O4 (solid) exhausts the limiting reactant completely. (a) Neglecting the vapor pressure of N2O4, what is the pressure anti composition of the gas remaining at 220 K after completion of the reaction? (b) What weight of N2O4 is formed? Ans. (a) 229 torr NO; (b) 0.402 g N2O4 6.53. The industrial production of NH3 by use of a natural gas feedstock can be represented by the following simplified set of reactions: CH4 + H20 -+ CO + 3H2 (1) 2CH4 + O2 CO + H20 N2 + 3H2 --+ + 2CO + .4H2 CO2 + + H2 (2) (3) (4) 2NH3 By assuming (i) that only the above reactions take place, plus the chemical absorption of CO2, (ii) that natural gas consists just of CH4, (iii) that air consists of 0.80 mole fraction N2 and 0.20 mole fraction 02, and (iv) that the ratio of conversion of CH4 by processes ( I ) and (2) is controlled through admitting oxygen for reaction (2) by adding just enough air so that the mole ratio of N2 to H2 is exactly 1:3, consider the overall efficiency of a process in which 1200 m3 (S.T.P.) of natural gas yields 1.00 metric ton of NH3. (a) How many moles of NH3 would be formed from each mole of natural gas if there were complete conversion of the natural gas subject to the stated assumptions? (b) What percentage of the maximum yield calculated in (a) is the actual yield? Ans. (a) 2.29 mol NH3/mol CH4; (b) 48 percent KINETIC THEORY 6.54. Calculate the ratio of (a) , U Ans. (a) 1.22; (b) 1.13 to U, , , (b) uaVgto umP. 6.55. At what temperature would N2 molecules have the same average speed as He atoms at 300 K? Ans. 2100 K 6.56. At what temperature would the most probable speed of CO molecules be twice that at O"C? Ans. 819C 92 THE IDEAL GAS LAW AND KINETIC THEORY [CHAP. 6 6.57. 6.58. Show that the ideal gas law can be written P = $c, where E is the kinetic energy per unit volume. What is the ratio of the average molecular kinetic energy of UF6 to that of He, both at 300 K? Ans. 1.000 6.59. What is the kinetic energy of a mole of CO2 at 400 K (a) in kilojoules (b) in kilocalories Ans. (a) 4.99 kJ; (b) 1.192 kcal 6.60. Uranium isotopes have been separated by taking advantage of the different rates of diffusion of the two isotopic forms of UFs. One form contains uranium of atomic mass 238, and the other of atomic mass 235. What are the relative rates of diffusion of these two molecules if Graham's law applies? Ans. UF6 with 235Uis faster by a factor of 1.004. 6.61. The pressure in a vessel that contained pure oxygen dropped from 2000 torr to 1 500 torr in 47 min as the oxygen leaked through a small hole into a vacuum. When the same vessel was filled with another gas, the pressure dropped from 2000 torr to 1500 torr in 74 min. What is the molar mass of the second gas? Ans. 79 6.62. A large cylinder of helium filled at 2 000 lbf/in2 has a small thin orifice through which helium escaped into an evacuated space at the rate of 3.4 millimoles per hour. How long would it take for 10 millimoles of CO to leak through a similar orifice if the CO were confined at the same pressure? Ans. 7.8 h 6.63. Calculate the average speed of a nitrogen molecule in air at room temperature (25 "C).Hint: use SI units for R and for molar mass, and u will come out in m/s. Ans. 475 m/s 6.64. How does the average speed of a nitrogen molecule in air compare to that of an oxygen molecule in air? (Air is approximately 20% oxygen and 80% nitrogen by moles.) Ans. The nitrogen molecules move at 1.07 times the speed of the oxygen molecules. (Nitrogen is 7% faster.) 6.65. You learned that in a mixture each gas exerted a partial pressure in proportion to its mole fraction. Yet in a box of air, the average N2 molecule, because of its greater speed hits the walls more frequently than the average 0 2 molecule, suggesting that nitrogen might contribute more than 80% of the pressure. Explain the apparent discrepancy. Ans. Although the average O2 molecule collides with the wall less frequently (by a factor of 1/1.07), it which is 1.07 times greater than that of the average N2 molecule (see Problem carries a momentum (mu), 6.64). The force of an impact depends on the momentum. Chapter 7 Thermochemistry HEAT Heat is a form of energy. Other forms of energy, such as mechanical, chemical, and electric energy, tend in natural processes to be transformed into heat energy. When any other kinds of energy are transformed into heat energy, or vice versa, the heat energy is exactly equivalent to the amount of transformed energy. All substances require an input of heat energy to raise their temperature (other variables remaining constant), and will yield the same amount of heat when they are cooled to their initial temperature. A solid requires an input of heat to melt or sublime; likewise a liquid needs an input of heat to evaporate. Upon reversal of such a phase change, the same amount of heat is released. The units most commonly used in expressing the quantity of heat are listed in Table 6-1. The use of the SI unit, the joule, based on quantities from mechanics, emphasizes the interconvertibility of the various forms of energy. Chemists have long used the calorie anti the kilocalorie, but they now generally prefer the joule and kilojoule. Engineers have been the principal users of the Btu. HEAT CAPACITY The heat capacity of a body is the amount of heat required to raise the temperature of that body 1 K. The heat capacity itself may depend on temperature. For pure substances, frequent use is made of such quantities as the molar heat capacity (heat capacity per mole) and the spec@ heat capacity (heat capacity per unit mass, sometimes called simply the spec@ heat). The average specific heat capacity of water is 1.OO cal/g K = 4.184 kJ/kg . K and the deviations from this average are all less than 1 percent between the freezing and boiling points. + CALORIMETRY An object undergoing a temperature change without a chemical reaction or change of state absorbs or discharges an amount of heat equal to its (average) heat capacity times the temperature change. Heat exchange = (heat capacity) x (temperature change) The heating or cooling of a body of matter of known heat capacity can be used in calorimetry, the measurement of quantities of heat. ENERGY AND ENTHALPY When a system absorbs heat, part of the absorbed energy may be used for doing work, such as lifting a weight, expanding against the atmosphere, or operating a battery, and part is stored within the system itself as the energy of the internal motions of the atoms and molecules themselves, as energy associated with rearrangements of the atoms that occur in chemical reactions, and as the energy of interaction among the atoms and molecules. This stored portion is known as the internal energy, E . The amount of heat absorbed by any system undergoing a modification such as an increase in temperature, a change in physical state, or a chemical reaction, depends somewhat on the conditions under which the process occurs. Specifically, the amount of heat absorbed is exactly equal to the increase in E if no work is done 93 94 THERMOCHEMISTRY [CHAP. 7 by the system. This would be the case in an ordinary chemical reaction, not linked to a battery, carried out in a closed reactor vessel so that no expansion against the outside atmosphere occurred. The increase in E can be represented by A E . (The A symbol for a difference is borrowed from mathematics. A E is the difference in E accompanying the process, defined as the final value of E minus the initial value.) Most chemical reactions, processes of heating or cooling, and changes of physical state are carried out in open, rather than closed, containers. Because of expansions or contractions in volume that might occur in the process, work might be done by the reacting system on the surroundings, or vice versa. In such a case, the principle of conservation of energy requires that the amount of heat absorbed must adjust itself to provide for the small but significant amount of this work. A new function, the enthalpy, H , can be defined which is related simply to the heat flow in an open, or constant-pressure, vessel. By definition, H = E P V . The amount of heat absorbed in a constant-pressure process is exactly equal to A H , the increase in H . In summary, if q is the amount of heat absorbed by the system of interest from its surroundings, + q (at constant volume) = A E q (at constant pressure) = A H These equations are exact in the absence of coupling to work-generating devices such as batteries. Any of the terms in these equations may have either sign. An endothermic process is one for which q is positive, i.e., heat is absorbed by the system of interest. An exothermic process is one for which q is negative, i.e., heat is evolved. Most of the thermochemical problems in this book will concern A H . Even though we may not know the absolute value of E or H , the above equations provide the experimental basis for measuring changes in these functions. ENTHALPY CHANGES FOR VARIOUS PROCESSES Change of temperature of a substance If a substance of heat capacity C is heated or cooled through a temperature interval A T , q =CAT provided, as is henceforth assumed, that C is independent of temperature. Subscripts are often used to designate a heat capacity measured at constant pressure, C,, and at constant volume, C,.Particularly, A H = CpAT The quantities C and H are extensive, i.e., proportional to the amount of material undergoing the process. We shall use a lowercase symbol to refer to a unit mass. Thus c, cp, and c, will denote specific heat capacities; and, for a sample of material of mass w , c = cw Where the subscript is omitted in the problems of this book, C, will be implied. Change of phase of a substance The heat that must be absorbed to melt a substance is sometimes called the latent heat o fusion, f q(fusion). For the melting of ice at O'C, for example, the process may be written: O'C, 1 atm H20(s)-+ H 2 0 ( 1 ) Here (s) and ( 1 ) are used to designate the solid and liquid states, respectively. AH = q(fusion) = (80 cal/g)(l8.0 g/mol) = 1.44 kcal/mol = 6.02 kJ/mol CHAP. 71 THERMOCHEMISTRY 95 Similarly, A H for a vaporization is equal to the latent heat of vaporization carried out at constant temperature and pressure. The latent heat of vaporization of water at 100C and 1 atm is 540 cal/g or 9.72 kcal/mol = 40.7 kJ/mol. A H for a sublimation process is equal to the latent heat for conversion of a substance from the solid to the gaseous state. Chemical reaction A AH-value assigned to a chemical equation defines the enthalpy change occurring when the number of moles of each reactant consumed is equal to its coefficient in the balanced equation. C(graphite) + 02(g) H2(g) + I ~ ( s ) + CO,(g) 2HI(g) A H = -393.51 kJ + A H = +52.72 kJ In the above reactions, 393.51 kJ is liberated when 1 mol of gaseous CO2 is formed from graphite and oxygen, and 52.72 kJ is absorbed in the formation of 2 mol of gaseous HI from hydrogen and solid iodine. The standard enthafpy o formation of a substance, AH;, is the enthalpy change accompanying the f formation of one mole of the substance from its elements, each being in its normal stable state at the designated temperature and 1 bar. For example, at 25 "C the standard state of H2, 02, or N2 is the C12, pure gas at 1 bar pressure; the standard state of bromine or mercury is the liquid; the standard state of iron, sodium, or iodine is the solid; the standard state of carbon is graphite, which is more stable than diamond. For ions in aqueous solution the standard state is a. hypothetical one molal solution (see Chapter 12 for meaning of "molal") with no interactions between ions. With reference to the examples in the preceding paragraph, the standard enthalpy of formation of C02(g) is -393.51 kJ/mol and of HI(g) is +26.36 kJ/mol (half of the +52.72 kJ enthalpy increase on forming two moles of HI). The superscript O is used in AH?, and will be used elsewhere, to indicate the standard condition of pressure, 1 bar, and, for dissolved substances, of concentration, 1 molal. Table 7-1 lists some values of AH? at 25 "C. Note the absence of entries for H2(g), Iz(s), C( graphite), and all other elements in their standard states. By definition their AH7-values are zero, as is that of H+(aq>. The standard state had been 1 atm for many decades before it was changed to 1 bar. Consequently most available tabulations are based on 1 atm. Fortunately, since A H is not strongly dependent on pressure, and since 1 bar is very close to 1 atm, the numerical values show little or no differences and they can be overlooked so far as this chapter is concerned. RULES OF THERMOCHEMISTRY The internal energy and the enthalpy of a system depend only on the state of the system, as specified by the external parameters such as pressure and temperature. (For example, recall from Chapter 6 that the kinetic energy contribution to E for an ideal gas is uniquely determined by the temperature.) When the system goes from an initial to a final state, A E and A H depend only on those two states, and are independent of the path taken between them. This path-independence implies two important rules of thermochemistry. 1. A E and A H for reverse processes are exactly the negatives of the values for the corresponding forward processes. EXAMPLE 1 A H for the fusion of a mole of ice is 1 440 cal/mol, since it is found experimentally that 1440 cal is absorbed in the melting of 1 mol at constant temperature, 273 K, and at constant pressure, 1 atm. Then A H for freezing is -1 440 cal/mol; this amount of heat must be transferred from the water to the surroundings in order to freeze it. 2. If a process can be imagined to occur in successive stages, A H for the overall process is equal to the sum of the enthalpy changes for the individual steps. (This rule is called Hess' Law o f Constant Heat Summation .) 96 THERMOCHEMISTRY [CHAP. 7 Table 7-1. Standard Enthalpies of Formation at 25C Substance AH"fkca1. mol-' -400.50 -304.20 +26.73 +0.45 -220.9 -47.96 -65.85 +13.3 +8.8 -26.42 -94.05 -14.2 -151.6 -235.80 -288.5 -45.3 -39.95 15.49 - 184.36 -21.3 - 197.0 -23.9 0.0 -8.70 -22.06 -64.8 +6.30 Ati;/kJ . mol-' Substance AH:/kcal. mol-' AH;/kJ . mol-' - 1 675.7 + -1 272.8 +111.84 +1.88 -924.7 -200.7 -275.5 +55.6 +36.8 - 110.53 -393.51 - 59.4 -634.3 -986.6 -1 206.9 -190 -167.16 +64.8 -771.36 -89.1 -824.2 - 100.0 0.0 -36.38 -92.3 1 -271.1 +26.36 -41.61 -57.80 -68.32 -44.88 -4.93 -9.5 +14.92 - 104.42 -95.06 -103.6 -24.21 -44.6 -143.1 +12.10 +21.45 +8.60 +2.19 -4.66 +34.1 -54.97 -76.4 -68.6 -89.6 - 133.48 -70.94 -94.58 -36.78 -174.10 -241.81 -285.83 -187.8 -20.6 -39.7 $62.4 -436.9 -397.7 -433.5 -101.3 - 186.6 -598.7 +50.63 +89.75 +35.98 +9.16 -19.50 +142.7 -229.99 -319.7 -287.0 -374.9 -558.48 -296.8 1 -395.72 - 153.89 It is impossible to measure accurately the heat liberated when C burns to CO, because the combustion cannot be stopped exactly at the CO stage. We can, however, measure accurately the heat liberated when C burns to CO2 (393.5 kJ per mole), and also the heat liberated when CO burns to CO2 (283.0 kJ per mole of CO). The enthalpy change for the burning of C to CO is determined by treating algebraically the two experimentally determined thermochemical equations. If two chemical equations are added or subtracted, their corresponding enthalpy changes are, by Hess' law, to be added or subtracted. Thus EXAMPLE 2 2C(gruphite) + 202(g) -+ 2C02(g) g) 2CO(g) + 02( -+ 2C02(g) AH" = (2 mole)(-393.5 kJ/mol) = -787.0 kJ AH" = (2 mol)(-283.0 kJ/mol) = -566.0 kJ Subtracting the second equations (both chemical and enthalpic) from the first, and transposing -2CO to the right, we have 2C(gruphite) + 02(g) -+ 2CO(g) AH" = -221.0 kJ Thus, AH?, the standard enthalpy of formation of C O ( g ) ,is evaluated as -221.0 kJ = -110.5 kJ/mol 2 mol CO EXAMPLE 3 The heat of sublimation of a substance is the sum of the heats of fusion and vaporization of that substance at the same temperature. CHAP. 71 THERMOCHEMISTRY 97 EXAMPLE 4 It follows from Hess' law that The enthalpy change of any reaction is equal to the sum of the enthalpiesof formation of all products minus the sum of the enthalpies of formation of all reactants, each AH, being multiplied by the number of moles of the substance in the balanced equation. Thus, for the reaction PCls(g) we have + H 2 0 (g) + POC13(g ) + 2HC1(g) The AH" listing for (1) is AHY for POC1,; for (2), twice AHY of HCl; for (3) and (4), the negative of AH; for PCls and H20, respectively. The Ahl;-values were all taken from Table 7-1. Note the use of fractional coefficientsin the formation equations of some of the compounds; this is not uncommon in thermochemistry. This example shows how a relatively small number of tabulated enthalpies of formation suffice for the calculation of the enthalpy change for many chemical reactions. SCOPE OF THERMOCHEMICAL CALCULATIONS The result of Example 4 above applies, strictly speaking, only to the standard states of the substances involved. (It is beyond the scope of this text to extend the calculations to other conditions of temperature and pressure. Such calculations are straightforward and are covered in courses in Physical Chemistry.) Calculations of AH" are nonetheless extremely useful because A H is not very sensitive to conditions (except in the extreme) and A H " provides us with a valid qualitative assessment of the reaction energy in most situations. For example, the result of Example 4 tells us to beware of a violent reaction when you open a PC15 container in moist air! The AH" calculation is important also because it is the first step in calculating A H at other conditions, and in calculating AGO (see Chapter 16), which in turn determines spontaneity, and equilibrium states. Solved Problems HEAT CAPACITY AND CALORIMETRY 7.1. (a) How many joules are required to heat 100 g of copper ( c = 0.389 J/g - K) from 10 "C to 100C? (6) The same quantity of heat as in (a) is added to 100 g of aluminum ( c = 0.908 J/g K) at 10 "C. Which gets hotter, the copper or aluminum? (4 AH = C A T = (0.389 J/g K)( 100 g)[(100 - lO)K] = 3 500 J * (b) Since the specific heat capacity of copper is less than that of aluminum, less heat is required to raise the temperature of a mass of copper by 1 K than is required for an equal mass of aluminum. Hence the copper is hotter. 98 THERMOCHEMISTRY [CHAP. 7 7.2. One kilogram of anthracite coal when burned evolves about 30 500 kJ. What amount of coal is required to heat 4.0 kg of water from room temperature (20C) to the boiling point (at 1 atm pressure), assuming that all the heat is available? AH(heating the water) = C A T = (4.184 kJ/kg. K)(4.0 kg)[(100 - 20)K] = 1 339 kJ Amount of coal required = 339 kJ = 0.044kg = 44 g 30500 kJ/kg 73 .. A steam boiler is made of steel and weighs 900 kg. The boiler contains 400 kg of water. Assuming that 70 percent of the heat is delivered to boiler and water, how much heat is required to raise the temperature of the whole from 10 "C to 100 "C? Specific heat capacity of steel is 0.1 1 kcal/kg. K. AH(for heating) = C(tota1)AT = [C(boiler) + C(water)]AT = [(0.11)(900)kcal/K + (l.O0)(400)kcal/K](90K) = 44900 kcal Input required 44900 kcal 0.70 64 000 kcal 7.4. Exactly three grams of carbon were burned to CO2 in a copper calorimeter. The mass of the calorimeter is 1 500 g, and the mass of water in the calorimeter is 2 000 g. The initial temperature was 20.0 "C and the final temperature 31.3 "C. Calculate the heat of combustion of carbon in joules per gram. Specific heat capacity of copper is 0.389 J/g - K. q(ca1orimeter) = C(tota1)AT = [C(Cu) + C(H20)]AT = [(0.389 J/g = * K)(1500 g) + (4.184 J/g * K)(2000 g)][31.3- 20.O)Kl 1.012 x 105 J 1.012 x 10' J = 3.37 x 104 J/g 3.00 g Heat value of carbon = 7.5. A 1.250-g sample of benzoic acid, C7H602, was placed in a combustion bomb. The bomb was filled with an excess of oxygen at high pressure, sealed, and immersed in a pail of water which served as a calorimeter. The heat capacity of the entire apparatus was found to be 10 134 J/K, including the bomb, the pail, a thermometer, and the water. The oxidation of the benzoic acid was triggered by passing an electric spark through the sample. After complete combustion of-the sample, the thermometer immersed in the water registered a temperature 3.256 K greater than before the combustion. What is AE per mole of benzoic acid combusted in a bomb-type calorimeter? Assume that no correction must be made for the sparking. (This is a constant-volume process.) q(acid) = -q(calorimeter) = -(10 134 J/K)(3.256 K) = -33.00 kJ AE(combustion) = -33.00 kJ q( acid) = -3223 kJ/mol number of moles acid - (1.250 g)/( 122.1 g/mol) 76 .. A 25.0-g sample of an alloy was heated to 100.0 "C and dropped into a beaker containing 90 g of water at 25.32 "C. The temperature of the water rose to a final value of 27.18 "C. Neglecting heat loses to the room and the heat capacity of the beaker itself, what is the specific heat of the alloy? CHAP. 71 THERMOCHEMISTRY 99 Heat loss by alloy = heat absorbed by water (25.0 g)(c)[(lOO.O- 27.2)KI = (90 g)(4.184 J/g * K)[(27.18 - 25.32)KI from which c = 0.385 J/g - K. 7.7. Determine the resulting temperature, t , when 150 g of ice at 0 "C is mixed with 300 g of water at 50 "C. 1. Consider the heat absorbed by the ice and by the water from it. AH(fusion) = (80 cal/g)( 150 g) = 1.20 x 104cal AH(heating 150 g from 0 "C to final temperature) =: C A T =: (l.OOcal/g~K)(150g)[(t-O)K] 2. Now consider the change in enthalpy of the hot water. A H = C A T = (1.00 cal/g. K)(300 g)[(t - 50)KI where, presumably, t < 50, consistent with a loss of heat from the hot water. The sum of the AH's must equal zero since heat is assumed not to leak into or out of the total system treated in (1) and (2). 1.20 x 104 150t 300(t - 50) = 0 3. + + from which t = 6.7 "C. Note: If the amount of ice were 200 g rather than 150 g the above procedure would have yielded the result, t = -2 "C. This answer is impossible because the final temperature cannot lie outside the range of the initial temperatures. In this case the result means that there is insufficient hot water to melt all the ice. The final temperature must therefore be 0 "C. You can calculate that there is 12.5 g of unmelted ice present. 7.8. How much heat is given up when 20 g of steam at 100C is condensed and cooled to 20"C? The heat of vaporization of water at 100C is (40.7 kJ/mol) (-)1 molg 18.02 = 2.26 kJ/g AH(condensation) = -(mass) x (heat of vaporization) = -(20 g)(2.26 kJ/g) = -45.2 kJ AH(coo1ing) = C A T = (4.184J/g - K)(20 g)[(20 - 100)Kl = -6.7 kJ AH(tota1) = -45.2 - 6.7 = -51.9 kJ The amount of heat given up is 51.9 kJ. 7.9. How much heat is required to convert 40 g of ice (c = 0.5 cal/g - K) at -10C to steam (c = 0.5 cal/g - K) at 120"C? AH(heating ice from - 10C to ice at 0C) = C A T = (0.5 ca.l/g. K)(40 g)(10 K) = 0.2 kcal AH(me1ting ice at 0C) = (mass) x (heat of fusion) = (40 g)(80 cal/g) = 3.2 kcal AH(heating water from 0 "C to 100C) = C A T = (1 .OO cal/g - K)(40 g)( 100 K) = 4.0 kcal AH(vaporization to steam at 100 "C) = (mass) x (heat of vaporization) = (40 g)(540 cal/g) = 21.6 kcal AH(heating steam from 100 "C to 120 "C) = C A T = (0.5 cal/g - K)(40 g)(20 K) = 0.4 kcal AH( total) = (0.2 + 3.2 + 4.0 + 21.6 + 0.4)kcal = 29.4 kcal 7.10. What is the heat of vaporization of water per gram at 25 "C and 1 atm? We can write the thermochemical equation for the process: 100 THERMOCHEMISTRY [CHAP. 7 H2W) + H2O(g) AH" can be evaluated by subtracting AH? of reactants from AH; of products, as tabulated in Table 7-1. AH" = AH;(products) - AH;(reactants) = -241.81 - (-285.83) = 44.02 kJ The enthalpy of vaporization per gram is 44*02kJ/mol = 2.44 kJ 18.02 g/mol Note that the heat of vaporization at 25 "C is greater than the value (2.26 kJ/g) at 100 "C. THERMOCHEMICAL EQUATIONS 7 1 . The thermochemical equation for the combustion of ethylene gas, C2H4, is .1 AH" = -1 410 kJ C2H4(d + 302(g> 2CO2(d + 2H20V) + Assuming 70 percent efficiency, how many kilograms of water at 20C can be converted into steam at 100C by burning 1 m3 of C2H4 gas measured at S.T.P.? (l m3)(1Oo0 L/m3) = 44.6 mo] n(C2H4) = 22.4 L/mol AH(l m3) = n(C2H4)x AH( 1 mol) = (44.6 mol)(-1410 kJ/mol) = -6.29 x 104 kJ The useful heat is then (0.70)(6.29 x 104 kJ) = 4.40 x 104 kJ. For the overall process, consider two stages: H20(1,20 "C) H20(1,100 "C) + H20(E, 100 "C) H2O(g, 100 "C) A H = (4.184 kJ/kg - K)(80 K) = 335 kJ/kg ---+ A H = (40.7 kJ/mol)/(0.018 02 kg/mol) = 2 259 kJ/kg AH(tota1) = 2594 kJ/kg The mass of water converted, w , is then equal to the amount of heat available divided by the heat requirement per kilogram w(HZ0) = 4.40 x 104 kJ = 17.0 kg 2 594 kJ/kg 7 1 . Calculate OH" for reduction of ferric oxide by aluminum (thermite reaction) at 25C. .2 We must start with a balanced equation for the reaction. We may then write in parentheses under each formula the enthalpy of formation, taken from Table 7-1, and multiply each enthalpy by the corresponding number of moles in the balanced equation. Remember that AH; for any element in its standard state is zero, by definition. (n AH;)/kJ: Then AH" of the reaction is given by AH" = (sum of n AH; of products) - (sum of n AH; of reactants) = -1 675.7 - (-824.2) = -851.5 kJ 2Al+ Fe203 2(0) 1(-824.2) + 2Fe+ A1203 2(0) 1(- 1 675.7) This is AH" for the reduction of 1 mol Fez03. 7 1 . AHY for N (g) (not the standard state of nitrogen) has been determined as 472.7 kJ/mol, and for .3 0 (g) as 249.2 kJ/mol. Calculate AH" for the hypothetical upper-atmosphere reaction: + - N (g>+ 0 (g) NO (g> CHAP. 71 THERMOCHEMISTRY 101 ( n AH?)/kJ: where AH? for NO ( g ) has been taken from Table 7-1. Then OH" = 89.7 - (472.7 + 249.2) -632.2 kJ -632.2 kJ = = -151.1 kcal 4.184 kJ/kcal 7.14. Calculate the enthalpy of decomposition of CaC03 into CaO and CO;?. ( n AH;)/kcal: CaC03 (s)-+ CaO (s) -288.5 - 15 1.6 AH" = (-151.6 - 94.0) - (-288.5) = 42.9 kcal= 179.5 kJ This is the enthalpy change for the decomposition of 1 mol CaCO,. A positive value signifies an endothermic reaction. 7.15. (a) Calculate the enthalpy of neutralization of a strong acid by a strong base in water (b) The heat liberated on neutralization of HCN (weak acid) by NaOH is 12.1 kJ/mol. How many kilojoules are absorbed in ionizing 1 mol of HCN in water? (a) The basic equation for neutralization is as follows: (nAH?)/kJ: Hf (aq) 0 + OH- (Llq) -230.0 Thus AH" = -285.8 - (-230.0) = -55.8 kJ. (b) The neutralization of HCN (aq) by NaOH (aq) may be thought of as the sum of two processes, ionization of HCN (aq) and neutralization of H+ (aq) with OH- (aq). [Since NaOH is a strong base, NaOH (aq) implies complete ionization, and a separate thermochemical equation for the ionization need not be written.] We may therefore construct the following thermochemical cycle: H+ (aq) OH- (aq) + H20 ( I ) SUM + + HCN (aq) + H+ (aq) + CN- (aq) HCN (aq) OH- (aq) + H2O ( I ) + CN- (aq) AH" (experimental) is negative because heat is liberated on neutralization. From the principal of additivity, x + (-55.8) = - 12.1 or The ionization process is endothermic to the extent of 43.7 kJ/mol. 7.16. The heat evolved on combustion of acetylene gas, C2H2, at 25 "C is 1 299.1 kJ/mol. Determine the enthalpy of formation of acetylene gas. The complete combustion of an organic compound involves the formation of CO2 and H20. (n AH:)/kJ: Thus C2H2 ( g ) + $ 0 2 ( 8 ) 2co2 ( g ) + H20 (4 X 0 2(-393.5) -285.8 - 1 299.1 = [2(-393.5) + (--285.8)] - x Solving, x = AH; for C2H2 ( g )= 226.3 kJ/mol. 7.17. How much heat will be required to make 1 kg of CaC2 according to the reaction given below? + - N ( d+0 ( g ) 472.7 249.2 NO ( g ) 89.7 + CO2 ( g ) -94.0 - + - H20 ( I ) -285.8 AH" (ionization) = x AH" = -55.8 kJ AH" (experiment) = -12.1 kJ x = 43.7 kJ AH" = -1 299.1 kJ + - 102 THERMOCHEMISTRY [CHAP. 7 (nAHy)/kJ: CaO (s) -634.3 + 3C (s) 0 -+ AH" = -(59.4 + 110.5) - (-634.3) CaC2 (s) -59.4 + CO (g) -110.5 kJ = +464.4 This is the heat required to make 1 mol CaC2; 1 kg CaC2 will require. (6 4 . : ~ ~ ~ a (464.4~kJ/mol) ~ ~ o l ) = 7 245 kJ 7.18. How many kilojoules of heat will be evolved in making one mole of H2S from FeS and dilute hydrochloric acid? ( n AH?)/kJ: FeS (s) + 2H' (aq)-+ Fe2+ (aq) - 100.0 0 -89.1 + H2S (g) -20.6 Since HCl and FeC12 are strong electrolytes, chloride can be omitted from the balanced equation. A P = -(89.1 + 20.6) + 100.0 = -9.7 kJ/mol H2S Supplementary Problems HEAT CAPACITY AND CALORIMETRY 7.19. How many calories are required to heat each of the following from 15 "C to 65 "C: (a) 1.0 g water, (b) 5.0 g Pyrex glass, (c) 20 g platinum? Specific heat capacity of Pyrex glass, 0.20 cal/g - K; of platinum, 0.032 cal/g. K. A m . (a) 50 cal; (b) 50 cal; (c) 32 cal 7.20. The combustion of 5.00 g of coke raised the temperature of 1.OO kg of water from 10 "C to 47 "C. Calculate the heating value of coke. Ans. 7.4 kcal/g = 3 1 kJ/g 7.21. Assuming that 50 percent of the heat is useful, how many kilograms of water at 15 "C can be heated to 95 "C by burning 200 L of methane, CH4, measured at S.T.P.? The of combustion of methane is 891 kJ/mol. Ans. 11.9 kg 7.22. The heat of combustion of ethane gas, C2H6 is 1 561 kJ/mol. Assuming that 60 percent of the heat is useful, how many liters of ethane measured at S.T.P. must be burned to supply enough heat to convert 50 kg of water at 10C to steam at lOO"C? Ans. 3 150 L 7.23. A 45.0-g sample of an alloy was heated to 90.0 "C and then dropped into a beaker containing 82.0 g of water at 23.50 "C. The temperature of the water rose to a final 26.25 "C. What is the specific heat capacity of the alloy? Ans. 0.329 J/g . K = 0.079 cal/g . K 7.24. If the specific heat capacity of a substance is h cal/g - K, what is its specific heat in Btu/lb. O F ? Ans. h Btu/lb * O F CHAP. 71 THERMOCHEMISTRY 103 7.25. Determine the resulting temperature when 1 kg of ice at 0 "C is mixed with 9 kg of water at 50 "C. Heat of fusion of ice is 80 cal/g = 335 J/g. Ans. 37C 7.26. How much heat is required to change 10 g of ice at 0 "C to steam at 100 "C?Heat of vaporization of water at 100C is 540 cal/g=2259 J/g. Ans. 7.2 kcal=30 kJ 7.27. A styrofoam coffee cup serves as an inexpensive calorimeter for measurements that do not require high accuracy. To 25.0 mL of water in such a cup at 24.33 "C a 1.00 g portion of KCl (s) was added. It dissolved completely after a short period of gentle stirring (using the thermometer). The minimum temperature reached was 22.12 "C. Estimate A H " for the heat of solution of KCl in kJ/mol. You may assume the solution has the same heat capacity as the water and that the heat capacity of the cup itself and of the thermometer is negligible. H " , sol'n = +17.2 kJ/mol 7.28. In an ice calorimeter, a chemical reaction is allowed to occur in thermal contact with an ice-water mixture at 0 "C. Any heat liberated by the reaction is used to melt some ice; the volume change of the ice-water mixture indicates the amount of melting. When solutions containing 1.00 millimole each of AgN03 and NaCl were mixed in such a calorimeter, both solutions having been precooled to O"C, 0.20 g of ice melted. Assuming complete reaction in this experiment, what is A H for the reaction Ag' + Cl- t AgCl? Ans. -67 kJ = -16 kcal 7.29. A 15.3 g portion of a certain organic liquid at room temperature, 26.2"C, was dumped into the reaction chamber of an ice calorimeter wherein it presumably cooled to 0.0 "C. The rise in water level indicated that 3.09 g of ice had melted. Calculate the specific heat of the liquid. = 0.616 cal/g. K = 2.58 J/g . K THERMOCHEMICAL EQUATIONS 7.30. What is the heat of sublimation of solid iodine at 25"C? Ans. 14.92 kcal/mol I2 = 62.4 kJ/mol I2 7.31. Is the process of dissolving H2S gas in water endothermic or exothermic? To what extent? Ans. exothermic, 4.6 kcal/mol = 19.1 kJ/mol 7.32. How much heat is released on dissolving 1 mol of HC1 (g) in a large amount of water? (Hint: HCl is completely ionized in dilute solution.) Ans. 17.9 k.cal= 74.8 kJ 7.33. The standard enthalpy of formation of H (g) has been determined to be 218.0 kJ/mol. Calculate A H " in kilojoules for the following two reactions: (4H ( d + Br (8) Ans. (U) + HBr ( g ) (4H ( d + 13r2 (4 + HBr ( g ) + Br (d -366.2 kJ; (b) -142.6 kJ 7.34. Determine A H " of decomposition of 1 mol of solid KC103 into solid KCl and gaseous oxygen. Ans. -10.9 kcal = -45.5 kJ 104 THERMOCHEMISTRY [CHAP. 7 7.35. The heat released on neutralization of CsOH with all strong acids is 13.4 kcal/mol. The heat released on neutralization of CsOH with H F (weak acid) is 16.4 kcal/mol. Calculate AH" of ionization of H F in water. Ans. -3.0 kcal/mol 7.36. Find the heat evolved in slaking 1 kg of quicklime (CaO) according to the reaction CaO (s)+ H 2 0 ( I ) Ans. --f Ca(OH), (s) 282 kcal= 1 180 kJ 7.37. The heat liberated on complete combustion of 1 mol of CH4 gas to CO2 (g) and H20 ( I ) is 890 kJ. Determine the enthalpy of formation of 1 mol of CH4 gas. Ans. -75 kJ/mol 7.38. The heat evolved on combustion of 1 g of starch, (C6H1005)x,into CO2 (g) and H20 ( I ) is 17.48 kJ. Compute the standard enthalpy of formation of 1 g of starch. Ans. -5.88 kJ 7.39. The amount of heat evolved in dissolving CuS04 is 73.1 kJ/mol. What is A H ; for SO:- (aq)? Ans. -909.3 kJ/mol 7.40. The heat of solution of CuS04. 5H20 in a large amount of water is 5.4 kJ/mol (endothennic). Calculate the heat of reaction for CUSO4.5H2O (S) CUSO4 (S) f 5H2O ( I ) Use data from Problem 7.39. + Ans. 78.5 kJ (exothermic) = 18.8 kcal 7.41. The heat evolved on combustion into CO2 (g) and H20 ( I ) of 1 mol C2H6 (g) is 1559.8 kJ, and for combustion of 1 mol C2H4 ( g ) is 1410.8 kJ. Calculate A H of the following reaction: C2H4 ( g ) + H2 ( g ) ---$ C2H6 (8)- Ans. 7.42. -136.8 kJ The solution of CaC12 6H20 in a large volume of water is endothermic to the extent of 14.6 kJ/mol. For the reaction CaC12 (s) Ans. 82.4 kJ/mol (exothermic) + 6 H 2 0 ( I ) -+ CaC12 . 6 H 2 0 (s) A H = -97.0 kJ. What is the heat of solution of CaC12 (anhydrous) in a large volume of water? 7.43. Hydrolysis is an important organic reaction in which a large organic molecule utilizes a water molecule to split into two smaller molecules. Since the same number and kind of chemical bonds are present in both reactants and products, one might expect only a small energy change. Calculate A H " for the gas phase hydrolysis of dimethyl ether based on Table 7-1 and the additional heat of formation values: CH30CH3 (g), -185.4 kJ/mol; CH30H (g), -201.2 kJ/mol. The reaction is: CH30CH3 ( g ) + H20 Ans. (d ---$ 2CH30H ( g ) 7.44. A H " = +24.8 kJ The commercial production of water gas utilizes the reaction C (s) + H20 (g) H2 (g) + CO (8). The required heat for this endothermic reaction may be supplied by adding a limited amount of oxygen and burning some carbon to carbon dioxide. How many grams of carbon must be burned to CO2 to provide enough heat for the water gas conversion of 100 g of carbon? Neglect all heat losses to the environment. + Ans. 33.4 g CHAP. 71 THERMOCHEMISTRY 105 7.45. The reversible reaction Na2S04 10H20-+ Na2S04+ 10H20 a A H = +18.8 kcal goes completely to the right at temperatures above 32.4"C, and remains completely on the left below this temperature. This system has been used in some solar houses for heating at night with the energy absorbed from the sun's radiation during the day. How many cubic feet of fuel gas could be saved per night by the reversal of the dehydration of a fixed charge of 100 lb Na2S04 lOH20? Assume that the fuel value of the gas is 2000 Btu/ft3. Ans. 5.3 ft3 7.46. + An important criterion for the desirability of fuel reactions for rockets is the fuel value in kilojoules, either per gram of reactant, or per cubic centimeter of reactant. Compute both of these quantities for each of the following reactions: (b) 2LiBH4 (s) + KC104 (s) -+ Li20 (s) B203 (s) KCl (s) 4H2 ( g ) (c) 6LiAlH4 (s) 2C(N02)4 ( I ) 3A1203 (s) 3Li20 (s) 2CO2 (g) 4N2 (g) ( d ) 4HN03 ( I ) + 5N2H4 ( I ) 7N2 ( g )4- 12H20 ( g ) (e) 7N204 ( I ) + C9H20 ( I ) 9 c o 2 ( g ) + 10H20 ( g ) + 7N2 ( g ) (f 4ClF3 (I)+(CH3)2N2H2 (I) 2CF4 ( g ) + N2 (8)+4HC1 ( g )+4HF ( g ) (4 N2H4 ( I ) + 2H202 (0 N2 (8)+ 4H20 ( g ) + + + + + + + Use the following density values: N2H4 (I), 1.01 g/cm3; H202 (0, 1.46 g/cm3; LiBH4 (s), 0.66 g/cm3; KC1O4 (s), 2.52 g/cm3; LiAlH4 (s), 0.92 g/cm3; C(N02)4 ( I ) , 1.65 g/cm3; HN03 ( I ) , 1.50 g/cm3; N204 ( I ) , 1.45 g/cm3; C9H20 ( I ) , 0.72 g/cm3; C1F3 ( I ) , 1.77 g/cm3; (CH3)2N2H2 (I), 0.78 g/cm3. In computing the volume of each reaction mixture, assume that the reactants are present in stoic hiometric proportions. Ans. (a) 6.4 kJ/g, 8.2 kJ/cm3; (b) 8.3 kJ/g, 12.4 kJ/cm3; (c) 11.4 kJ/g, 14.6 kJ/cm3; ( d ) 6.0 kJ/g, 6.0 7.5 kJ/cm3; (e) 7.2 kJ/g, 8.9 kJ/cm3; (f) kJ/g, 9.1 kJ/cm3 7.47. An early-model Concorde supersonic airplane consumed 4 700 gallons of aviation fuel per hour at cruising speed. The density of the fuel was 6.65 pounds per gallon and A H of combustion was -10500 kcal/kg. Express the power consumption in megawatts (1 MW = 106 W = 106 J/s) during supersonic cruising. Ans. 173 MW 7.48. Two solutions, initially at 25.08 "C, were mixed in an insulated bottle. One consisted of 400 mL of a weak monoprotic acid solution of concentration 0.200 mol/L. The other consisted of 100 mL of a solution having 0.800 mol NaOH/L. After mixing, the temperature rose to 26.25 "C. How much heat is evolved in the neutralization of one mole of the acid? Assume that the densities of all solutions are 1.00 g/cm3 and that their specific heat capacities are all 4.2 J/g K. (These assumptions are in error by several percent, but the subsequent errors in the final result partly cancel each other.) Ans. 31 kJ/'mol 7.49. An old type of miner's lamp burns acetylene, C2H2, which is made in situ by dropping water onto calcium carbide, CaC2. The lamp designer is concerned about how hot the calcium carbide chamber will get. Calculate the heat produced, in kilojoules per liter (S.T.P.) of C2H2 generated. (See Problem 7.16 for additional data.) CaC2 (s) Ans. 5.77 kJ/L ( A H " = -129.3 kJ/mol) 7.50. (a) How does the heat of combustion of the C2H2 compare to the heat generated in making it from the calcium carbide? (b) If one adds the generating reaction to the combustion reaction, the overall reaction of the lamp becomes: CaC2 (s) H20 ( I ) $ 0 2 Ca(OH), (s) 2C02 (g) + - + + 12H2 ( g ) + + 2H20 ( I ) -+ Ca(OH), (s) + C2H2 (8) + + + + 106 THERMOCHEMISTRY [CHAP. 7 Calculate AH" for the above reaction and compare it to the results of the separate calculations made in Problems 7.49 and 7.50(a). Ans. (a) Heat of combustion = 58.0 kJ/L, ( A H = -1 299.1 kJ/mol). This is about ten times the result of Problem 7.49. (b) AH" = -1 428.4 kJ/mol, exactly the sum of the previous calculations (-129.3 - 1299.1). Chapter 8 Atomic Structure and the Periodic Law ABSORPTION AND EMISSION OF LIGHT Historically important in the development of modern atomic theory was the recognition that although polyatomic molecules show more or less broad bands of absorption and emission in the visible and ultraviolet regions of the spectrum, the characteristic light absorption or emission by individual atoms occurs at fairly narrow lines of the spectrum, which correspond to sharply defined wavelengths. The line spectrum of each element is so uniquely characteristic of that element that atomic spectroscopy can be used for precise elementary analysis of many types of chemically complex materials. The exploitation of spectral information requires a recognition of two aspects of light, its wave character and its particle character. Wave character of light A light beam is associated with an electromagnetic disturbance, which is propagated periodically in the direction of the beam. The distance between two successive maxima in the intensity of the disturbance is called wavelength and is usually given the symbol lambda, A. Thefrequency is the number of such maxima passing a given point per second and is usually given the symbol nu, U . The common unit for frequency is the hertz, having the symbol Hz, where 1 Hz:= 1 s-' . The product of the wavelength and the frequency is equal to the velocity of propagation, usual.1~ designated by c. c = xu The velocity of light in vacuum is the same for all wavelengths, ;!.998 x 10' m/s, so that a simple inverse proportionality exists between wavelength and frequency. The velocity of light in the atmosphere is reduced by less than one part in a thousand below the value for vacuum, so that the above value is acceptable for most laboratory applications. Another term is often used, the wave number, V , defined as 1/X, or as u / c . The most common unit for wave number is cm--'. Particle character of light The energy of light is absorbed, emitted, or converted to other forms of energy in individual units, or quanta (singular, quantum). The unit itself, often referred to as the particle of light, is called aphoton. The energy of a photon is proportional to the frequency: E = hu = (6.626 x 10-34J . s)u The universal proportionality constant, h, is called Planck's constant. Although visible light, to which the human eye is sensitive, is confined to the wavelength range 400 to 700 nm, the above discussion refers to radiation of all wavelengths. Chemists in their studies of matter routinely use radiation whose wavelength ranges from 0.1 nm (x-rays) to several centimeters (microwaves). INTERACTION OF LIGHT WITH MATTER A major step toward the understanding of atomic structure was Bohr's explanation of the hydrogen spectrum. His postulates were the following: 1. The electron of the hydrogen atom revolves around the nucleus in stable circular orbits. 2. In each stable orbit the electrostatic attraction between the negatively charged electron and the positively charged nucleus provides the centripetal force needed for the circular motion of the electron. The energy of the atom is the sum of the potential energy of electrostratic 107 108 ATOMIC STRUCTURE A N D THE PERIODIC LAW [CHAP. 8 interaction between the nucleus and the electron and the kinetic energy of motion of the electron. 3. Only certain stable orbits are allowed, those for which the angular momentum of the electron is an integer, n, times the constant h/27r. (This is the same h that occurs in the equation for the energy of a photon of light.) 4. An electron can move from one stable orbit to another only by absorbing or releasing an amount of energy exactly equal to the difference between the energies of the two orbits. If this energy is absorbed or released as light, a single photon of absorbed or emitted light must account for the required energy difference, so that where A E is the difference between the energies of the final and initial orbits. This theory of Bohr satisfactorily accounted for the observed series of spectral lines for hydrogen. The predicted orbit energies in the simplest form of the theory are given by me4z2 E(n) = -8cih2n2 where m and e are the mass and charge of the electron c0 is the permittivity of free space, and 2 is the atomic number of the nucleus (1 in the case of hydrogen). Note that the energy is negative with respect to the state in which the electron and nucleus are very widely separated, which is the zero energy state. In SI units, rn = 9.1095 x 10P3'kg e = 1.602 x lO-I9 C c0 = 8.854 x C2/N.m2 The predicted allowed wave numbers observed in the spectrum are: where nl < n2 in absorption and n1 > n2 in emission. The quantity R,defined purely in terms of universal constants, R = - me 84h3c is called the Rydberg constant. It has the value 109 737 cm-' . (The observed value for 'H is 0.06% lower because of the reduced mass effect. In accordance with a refinement of the simple Bohr theory, R increases with increasing nuclidic mass toward a maximum of 109737 cm-'.) The predicted radii of the orbits are: n2 h20 r(n) = -ao where a. = z xme2 For hydrogen ( Z = l), the predicted value of the first Bohr orbit (n = 1) is a. = 5.29 x 10-" m = 0.529 The Bohr theory, even when elaborated and improved, applies only to hydrogen and hydrogen-like species such as He' and Li2+; it explains neither the spectra of atoms containing even as few as two electrons nor the existence and stability of chemical compounds. The next advance required an understanding of the wave nature of matter. 4 A. CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 109 PARTICLES AND WAVES De Broglie proposed that not only does light have the dual properties of waves and particles but also particles of matter have properties of waves. The wavelength of those particle waves is given by h mv where m and v are the mass and velocity of the particle. Planck's constant is so small that the wavelengths are in an observable range only for particles of atomic or subatomic mass. An experimental confirmation of the de Broglie relation for a beam of uniformly energetic electrons was followed by a theoretical development of quantum mechanics, or wave mechanics, which reproduced Bohr's successful prediction of the stable energy levels of the hydrogen atom. Furthermore, this powerful new formalism is applicable in principle to many-electron atoms and to many-atom molecules. The Bohr postulates are replaced by the Schrodinger equation, which must be solved by the methods of partial differential equations. The equation has mathematical similarities to descriptions of physical waves, and the arbitrary introduction of integers in the Bohr theory receives justification in quantum mechanics in the requirement that wavelike solutions of the equation must be continuous, finite, and single-valued. The steady-state solutions of the wave equation correspond to states of fixed energy, as in the Bohr theory, and in many cases, including the hydrogen atom, of fixed angular momentum. Many other descriptive properties, like the location of an electron, are not definitely fixed but are represented by probabilities of distribution over a range of numerical values. In the hydrogen atom the electron is not confined to a two-dimensional orbit but is represented by a wave that extends over three-dimensional space. The wave amplitude indicates the varying probability of finding the electron at different locations with respect to the nucleus. A=- Orbitals A solution to the Schrodinger equation for an electron must satisfy three quantum conditions corresponding to the three dimensions of space. Each quantum condition introduces an integer, called a quantum number, into the solution. A separate solution, describing a probability distribution of finding the electron at various locations, exists for each allowed set of the three quantum numbers. Such a solution is called an orbital; it is similar to a hypothetical time-exposure photograph of an electron taken over a time interval large enough so that each region of space is represented by the weighted probability of finding the electron in that region. These three quantum numbers are usually designated as follows: 1. n, principal quantum number: This number almost exclusively determines the energy of the orbital in one-electron systems, and is still the principal determinant of energy in manyelectron systems. 2. I, angular momentum quantum number: This number defines the shape of the orbital and together with n defines the average distance of the electron from the nucleus. 3. ml, magnetic quantum number: This number determines the orientation of the orbital in space. In addition to the three dimensions in space describing the relative positions of the electron and the nucleus, there is a fourth dimension internal to the electron itself, related to the spin of the electron around an internal axis and characterized by the magnetic moment associated with this spin. The quantum number associated with electron spin is usually designated as m,. Each of the four quntum numbers may have only certain vtalues. (a) n may be any positive integer. Electrons having a given n-value are said to be in the same shell. Shells are designated by capital letters as follows: I n Designation I I 1 2 L 3 M 4 N 5 0 6 7 1 Q K P 110 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 (b) 1 may have any integral value from 0 to n - 1 . For historical reasons, orbitals with a given 1-value are designated by a characteristic letter. 1 0 s ~ ~~ 1 2 3 4 g ~ Designation ~ ~ p ~ d f An orbital is usually referred to by its principal quantum number, followed by a letter corresponding to its 1 value; e.g., Is, 2p, 3s. Electrons having a given n-value and a given 1-value are said to be in the same sub-shell. (c) ml may have any integral value from -I to +l. This rule gives the correct number of orbitals, 21 + 1, associated with a given (n, I ) combination. Thus there are three "p" orbitials, corresponding to ml = 1, 0, and - 1 . However, it is usually more convenient in chemistry to use a new set of three orbitals (oriented along the x, y , and z axes, respectively) which are obtained from the basic set by a mathematical mixing process. Similar procedures apply to a set of five "d" orbitals, and so forth. (d) m, may be either + $ or - 4. The probabilities of finding an electron at various distances r from the hydrogen nucleus are given in Fig. 8-1 for several (n, 1 ) combinations. The shapes of s, p, and d orbitals are shown in Fig. 8-2. An inspection of Fig. 8-1 shows a detail which emerges as an exact mathematical consequence of the theory: when 1 = n - 1, the most probable distance of the electron from the nucleus is exactly equal to the radius of the Bohr orbit, n2ao.In general, the average distance of the electron from the nucleus increases with increasing n. This same figure shows another interesting feature for all allowed I-values except the maximum for a given n; namely, the existence of minima of zero probability, corresponding to spherical nodal surfaces around the nucleus, at which the electron will not be found. Figure 8-2 shows that s n=3,1=O I I n = 2, I =1 I I I I n =3,1= 1 1 1 1 2 3 4 5 6 7 U 9 rlnao Fig. 8-1 CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 111 0 (a) s orbital X ( b ) p orbitals Y .X (c) d orbitals Fig. 8-2 Angular Dependence of Orbitals orbitals are spherically symmetrical around the nucleus. Each p orbital is concentrated along the + and - portions of one of the Cartesian axes, with a nodalplane of zero probability perpendicular to that axis at the nucleus. Of the five allowed d orbitals for a given n, only four have the same shape. Each of these four looks like a three-dimensional four-leaf clover, with the highest probability along the x and y axes in the case of the dX2-.~2 in between the axes in the case of the dXJ,, and dyz.Each of these orbitals has and d.yz, two perpendicular nodal planes through the nucleus. The fifth d orbital, 4 2 , has maxima in the +z and -z directions and a secondary concentration in the (x, y ) plane. This orbital has two conical nodal surfaces through the nucleus, one projecting above the (x, y ) plane and one below, separating the +z and -z lobes from the doughnut-shaped ring of probability concentrated in the (x, y ) plane. THE PAUL1 PRINCIPLE AND THE PERIODIC LAW The Pauli exclusion principle states that no two electrons in. an atom may have the same set of all four quantum numbers. This principle places the following limits on the number of electrons for the various (n, I ) orbital designations: Maximum Number of Electrons, Orbital 2(21+ 1) Because the Schrodinger equation cannot be solved exactly for polyelectron atoms, it has become the practice to approximate the electron configuration by assigning electrons to hydrogen-like orbitals, which are designated by the same labels as for hydrogen and have the same spatial characteristics (symmetry, nodes, etc.). t 112 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 The electrons for a given atom will preferentially occupy those orbitals having the lowest energy. The buildup (Aufbau) principle describes in detail the way in which each additional electron for each succeeding element of increasing atomic number is orbitally assigned. Although n is the principal determinant of orbital energy, the importance of 1 cannot be overlooked in polyelectron atoms. The reason for this is that the overall electron cloud tends to screen the positive charge of the nucleus from any test electron in question, and this screening works differently for different orbital types. In particular, the higher its 1 value, the more an electron shuns the region near the nucleus, the less it is attracted, and the higher it lies (i.e., closer to zero) on the energy scale. Such considerations lead to the following order in which orbital types are filled. 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 5d 4 < 6p < 7s < 6d g 5 f f Application of the Aufbau principle accurately predicts the configurations in their ground, or lowest energy state, of the lower Z elements (see problem 8.10 for an exception), which are the elements of greatest interest. Exceptions are more common among high Z elements in which the outer energy levels become very close to each other and subtler considerations become relevant. Discussion of electron configurations throughout this chapter will always refer to elements in their ground state. ELECTRON CONFIGURATIONS The position of an element in the periodic table reflects the configuration of its outermost electron shell. In groups I and I1 (1 and 2) these configurations are nsl and ns2, where n is the number of the period in which the element is located and the exponent of s is the number of s electrons in the n level, and in groups I11 to VIII (13 to 18) they are ns2np' to ns2np6. These two sets of elements (sometimes called s-block and p-block) comprise the main group elements. As for groups 3 to 12, the transition elements, the outer configurations in periods 4 and 5 are ns2(n - 1)d' to ns2(n- l)d", but there are some anomalies and the filling of the d shell does not proceed uniformly. In the transition elements of periods 6 and 7 the (n - 2)f orbitals are filled before the (n - 1)d's are completed. There are many anomalies in the course of filling thef shell. The extra 14 elements in period 6 are the lanthanides and in period 7 the actinides, and these 28 appear separately below the main body of the table. The outermost electrons, often called the valence electrons, are primarily responsible for the chemical properties of the elements. Thus elements in the same group will show strong similarities. Recall that the earliest periodic table preceded by many decades any knowledge of electron configuration but was constructed by arranging similar elements in the same group. Electrons below the valence electrons are arranged just as in the corresponding noble gas element (group VIII, or 18). For instance 21Schas a core of 18 electrons (arranged as in IsAr) enclosed within a valence shell of configuration 3d'4s2. (Note that the subscript preceding an atomic symbol may be used to denote the atomic number.) The total configuration then for Sc may be written: [Ar]3d'4s2. ATOMIC RADII The electron cloud around an atom makes the concept of atomic size somewhat imprecise. Nevertheless it is useful to refer to an atomic size or an atomic radius. Operationally, one can arbitrarily divide the experimentally determined distance between centers of two chemically bonded atoms to arrive at two radii, on the crude picture that the two bonded atoms are spheres in contact. If the bonding is covalent (see Chapter 9), the radius is called a covalent radius, if it is ionic, the radius is an ionic radius. The radius for a nonbonded situation may also be defined in terms of the distance of closest nonbonding approach; such a measure is called the van der Waals radius. These three concepts of size are illustrated in Fig. 8-3. The following generalizations are observed regarding atomic size: 1. Within a given group of the periodic table, the radius increases with increasing atomic number. This fact is related to the increased n of the outermost shell. CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 113 (a) Portion of NaCl crystal showing the ionic radii (i) (b) %o C1, molecules in contact in liquid chlorine showing the covalent radii (c) and the van der Waals radii (v) Fig. 8-3 2. Within a given period of the periodic table, the covalent radii generally decrease with increasing atomic number. This is related to the facts that (i) the size of an atom depends on the average distance of its outermost electron(s), (ii) there is no change in n of the outermost electron(s) within a given period, and (iii) there is increasing nuclear charge with increasing atomic number. 3. Ionic radii.of cations are fairly small compared to the covalent radii for the corresponding atoms since all the outermost electrons (highest n) are usually removed. Anionic radii are only slightly larger than the van der Waals radii for the corresponding atoms since the extra electron(s) have the same n. However, the covalent radii of these atoms are appreciably less since they are bonded to their neighbors by electron sharing. 4. Within a given group, ionic radii increase with increasing atomic number. Within a given period, cationic radii decrease with increasing atomic number as do anionic radii, but there is a discontinuous increase in going from the last cation in a period to the first anion. IONIZATION ENERGIES The Bohr formula for the hydrogen atom energy levels predicts that higher energy levels get closer and closer together and asymptotically approach a limit of zero as n --+ 00. At this limit the atom has been ionized. The minimum energy required to ionize an isolated gaseous atom may be determined spectroscopically, thermochemically, or electrically. In the electrical method, a measurement is made of the accelerating potential that will impart to a projectile electron (not the electron within the atom in question) an amount of kinetic energy just sufficient to dislodge the bound electron from its atom. Thus the ionization energy can be measured directly in electrical terms. The electron volt is the energy imparted to an electron accelerated by a potential difference of 1 V. 1 eV = (charge on electron) x (potential difference) = (1.602 2 x 10-19 C)( 1 V) = 1.6022 x lO-I9 J Note that 1 V = 1 J/C. Ionization energies (I.E.) have been measured for all the atoms. They are all positive, corresponding to an ionization process that is endothermic. Some interesting trends have been observed. 1. Within a given group of the periodic table, the first ionization energy decreases with increasing atomic number. This is related to the increase in atomic size and the decreasing hold which the nucleus has for the increasingly distant outermost electron. (This rule is not uniformly obeyed for the transition metals.) 114 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 2. Within a given period, the general trend is an increase in ionization energy with increasing atomic number. However, atoms just beginning a new subshell or the second half of a subshell may have a smaller ionization energy than the previous atom. 3. The ionization energy for each succeeding stage of ionization is always greater than for the preceding stage. For instance, the second ionization energy of magnesium is considerably greater than the first (about twice as great), because the residual Mg2+ has twice the electrostatic attraction for the removed electron as does Mg+. However, the second ionization energy of sodium is many times the first, because the second electron must be removed from the n = 2 shell rather than the n = 3 shell. ELECTRON AFFINITY Some free atoms can capture an extra electron to form a stable gaseous anion, particularly with elements having almost-completed p subshells. For example, Cl(g) + e-(g) -+ Cl-(g) AH = -3.61 eV Chlorine is said to have an electron affinity (E.A.) of 3.61 eV. In this book a positive electron affinity implies an exothermic electron attachment as in the above example. MAGNETIC PROPERTIES The magnetic properties of bulk matter depend on the properties of the individual atoms. We have noted that electron spin has a magnetic moment associated with it. Two electrons occupying the same orbital (same n, I , and ml) have their magnetic moments canceled out because the two values of m, correspond to equal but oppositely directed spin angular momenta and thus to equal but opposed magnetic moments. Atoms, ions, or compounds in which at least one orbital is singly occupied can thus have a net magnetic moment. Such a substance is said to be paramagnetic and is attracted into a magnetic field. The magnitude of the magnetic moment, and thus the number of unpaired electron spins, can be determined experimentally by measuring the force of attraction of an externally imposed magnetic field for the substance. Substances without unpaired electron spins do not have magnetic moments and are repelled by a magnetic field; they are called diamagnetic. The repulsion of diamagnetic substances is much smaller in magnitude than the attraction of paramagnetic substances. Magnetic measurements are an important experimental tool for determining the electron assignment into orbitals for atoms, ions, and compounds. Several rules have been developed for the detailed assignment of electrons within a subshell. 1. Electrons within a subshell for which I > 0 will tend to avoid pairing within the same orbital. This rule, known as Hund's rule, reflects the relatively greater electrostatic repulsion between two electrons in the same orbital within a subshell as compared with occupancy of two orbitals having differing ml-values. 2. Electrons in singly occupied orbitals tend to have their spins in the same direction so as to maximize the net magnetic moment. 3. From the electron configuration one can easily infer whether or not an atom (or ion) is paramagnetic. This conclusion applies only to the free individual atom and the student is cautioned not to draw conclusions about condensed states of matter of the element. Thus the free aluminum atom has one unpaired electron, but aluminum metal is diamagnetic, as is well known. CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 115 Solved Problems ENERGY RELATIONSHIPS 8.1. Determine the frequencies of electromagnetic radiation of the following wavelengths: (a) 0.10 nm, (b) 5000 ( c ) 4.4 pm, ( d ) 89 m, (e) 562 nm. A, The basic equation for all these problems is v = (2.998 x lO'm/s)/X. 2.998 x 10' m/s = 5.996 x 1014 HZ = 599.6 THz (5 000 A)(lO-'O m/A) U=- 3.00 x 10' m/s = 6.8 x 1013 Hz = 68 THz 4.4 x 1W6 m 3.00 x 10' m/s = 3.4 x 106HZ = 3.4 MHz 89 m 2.998 x 10' m/s = 5.33 x 1014 HZ = 533 THZ m 562 x lOW9 U = v= 8.2. (a) What change in molar energy would be associated with an atomic transition giving rise to radiation at 1 Hz? (b) For any given photon, what is the numerical relationship between its wavelength in nanometers and its energy in electron volts? (a) If each of N A atoms gives off one 1-Hz photon, AE = NA(hv) (6.022 x 1023mol-')(6.626 x 10-34J . s)(l s-') = 3.990 x 10-l' J . mol-' = Since AE and v are proportional, we may treat the ratio 3.990 x 10-l' J mol-' 1 Hz as a "conversion factor" between Hz and J mol-', provided we understand just what this "conversion" means. Thus, for 1-MHz radiation, AE = (106 Hz)(3.990 x 10-l' J - mol-' (b) First let us find the frequency equivalent of 1 eV from the Planck equation, and then find the wavelength from the frequency. c A=-= v 2.998 x 10' m - s-l = (1.2398 x 10-6 m)(109 nm/m) = 1239.8 nm 2.4180 x 1014 s-' X E = hc = 1 239.8 nm - eV Because of the inverse proportionality between wavelength and energy, the relationship may be written 8.3. In the photoelectric effect, an absorbed quantum of light results in the ejection of an electron from the absorber. The kinetic energy of the ejected electron is equal to the energy of the absorbed photon minus the energy of the longest-wavelength photon that causes the effect. Calculate the kinetic energy of a photoelectron produced in cesium by 400-nm light. The critical (maximum) wavelength for the photoelectric effect in cesium is 660 nm. + - Hz-') = 3.990 x lOw43 . mol-I 116 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 Using the result of Problem 8.2(b), hc hc Kinetic energy of electron = hu - huc,it = - - L i t - 1 240 nm . eV - 1 240 nm - eV = 1.22 eV 400 nm 660 nm 8.4. It has been found that gaseous iodine molecules dissociate into separate atoms upon absorption of light at wavelengths less than 499.5 nm. If each quantum is absorbed by one molecule of 12, what is the minimum input, in kJ/mol, needed to dissociate I2 by this photochemical process? NAhc (6.022 x 1023mol-')(6.626 x 10-34J . s)(2.998 x 108 m - s-') E(per mole) = NA(hu) = -= X 499.5 x I O - ~m = 239.5 kJ/mol 8.5. A beam of electrons accelerated through 4.64 V in a tube containing mercury vapor was partly absorbed by the vapor. As a result of absorption, electronic changes occurred within a mercury atom and light was emitted. If the full energy of a single incident electron was converted into light, what was the wave number of the emitted light? Using the result of Problem 8.2(6) U=-=-=- - 1 u h u X c hc - 4.64 eV = 0.003 74 nm-I = 37400 cm-I 1 240 nm . eV 8.6. An electron diffraction experiment was performed with a beam of electrons accelerated by a potential difference of 10 kV. What was the wavelength of the electron beam? We can use the de Broglie equation, taking the mass of an electron as 0.911 x 10-30 kg. The velocity of the electron is found by equating its kinetic energy, ;my2, its loss of electric potential energy, 10 keV. to $rnv2 = (104 eV)(1.602 x 10-19 J/eV) = 1.602 x 10-15 J = 1.602 x 10-15 kg - m2 s - ~ v= ( 2 x 1.602 x kg . m2 s - ~ 0.911 x 10-30kg = (35.17 x 10'4)1/2 m/s = 5.93 x 107 m/s Now the de Broglie equation gives A=-= h rnv 6.63 x 1 0 - ~J~. s (0.911 x 10-30kg)(5.93 x 107 m/s) kg . m s-' a - 1.23 x 10-I' kg m2 - s-' = (1.23 x 10-I' m)(109 nm/m) = 0.0123 nm The results calculated above are somewhat in error because of the law of relativity, which becomes more and more relevant as the velocity approaches the speed of light. For instance for a potential difference of 300 kV the velocity calculated as above would exceed c, an invalid result since no particle can have a velocity greater than the speed of light. CHAP. 8 1 ATOMIC STRUCTURE AND THE PERIODIC LAW 117 ATOMIC PROPERTIES AND THE PERIODIC LAW 87 .. The Rydberg constant for deuterium (2H) is 109707 cm-'. Calculate (a) the shortest wavelength in the absorption spectrum of deuterium, (b) the ionization energy of deuterium, and (c) the radii of the first three Bohr orbits. (a) The shortest-wavelength transition would correspond to the highest frequency and to the highest energy. The transition would thus be from the lowest energy state (the groundstate), for which n = 1, to the highest, for which n = 00. 1 = (0.911 52 x lO-' cm)(107 nm/cm) = 91.152 nrn 109 707 cm-I (b) The transition computed in (a) is indeed the ionization of the atom in its ground state. From the result of Problem 8.2(b), A= 1239.8 nm eV = 13.601 eV 91.152 nm This value is slightly greater than the value for 'H. (c) From the equation (2 = 1) I.E. = 9 r = n2ao = n2(5.29 x 10-" m) the radii are 1, 4, and 9 times ao, or 0.529, 2.116, and 4.76 The reduced-mass correction, involving an adjustment of 3 parts in 104, is not significant, and the a0 for the first Bohr orbit of 'H is a perfectly satisfactory substitution. A. 8.8. (a) Neglecting reduced-mass effects, what optical transition in the He+ spectrum would have the same wavelength as the first Lyman transition of hydrogen (n = 2 to n = l)? (b) What is the second ionization energy of He? (c) What is the radius of the first Bohr orbit for He+? (a) He' has only one electron. It is thus classified as a hydrogen-like species with Z = 2, and the Bohr equations may be applied. From the equation fi = RZ2 (i i) - the first Lyman transition for hydrogen would be given by The assumption regarding reduced-mass effects is equivalent to considering R for He' the same as for 'H. The Z2-term can just be compensated by increasing nl and n2 by a factor of 2 each. The transition in question is thus the transition from n = 4 to n = 2. (b) The second ionization energy for He is the same as the first ionization energy for He+, and the Bohr equations may be applied to the ground state of He+, for which Z = 2 and n = 1. fi=RZ2(-$-i) =R(2)2(A-$) =4R This result is 4 times the result for deuterium in Problem 8.7. Since fi is proportional to energy, the ionization energy will likewise be 4 times that for deuterium. I.E. (He') =4 x I.E. (2H) = 4 x 13.6 eV = 54.4 eV 118 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 8.9. (a) Write the electron configurations for the ground states of N, Ar, Fe, Fe2+,Pr3+. (b) How many unpaired electrons would there be in each of these isolated particles? (a) The atomic number of N is 7. The first (or K ) shell will contain its maximum of 2 electrons. The next 5 electrons will be distributed in the L shell, 2 filling the lower-energy s subshell and the remaining 3 the p subshell. A common notation is ls22s22p3 Another notational form, [He]2s22p3,shows only those electrons beyond the complement in the previous noble gas. The atomic number of Ar is 18. The order of filling is that of the successive K , L, and M shells. ls22s22p63s23p6 The atomic number of Fe is 26. Beyond the Ar structure the order of filling is 4s, then 3d, until 26 electrons are assigned in all. 1s22s22p63s23p63d64s2 or [Ar]3d64s2 Fe2+ has 24 electrons, two fewer than the neutral Fe atom. Although 4s is of lower energy than 3d for atomic numbers 19 and 20 (K and Ca), this order reverses for higher nuclear charges. In general, the electrons most easily lost from any atom are those with the largest principal quantum number. [Ar]3d6 Pr3+ has 56 electrons, 3 fewer than the neutral Pr atom. Beyond Xe, the previous rare gas, the order of filling for the next period of elements is 6s2,then one 5d electron, then the whole 4f subshell, then the rest of the 5d subshell, then the 6p subshell. There are frequent replacements of the first 5d assignment with an additional 4 f , or of one of the 6s with an additional 5d, but these irregularities are of no consequence in the assignment of electrons in Pr3+. The 3 electrons removed from the neutral atom to form the ion follow the general rule of removal first from the outermost shell (highest n) and then from the next-to-outermost (next highest n); in this case the 6s electrons are removed first and no 5d electrons would remain even if there were one in the neutral atom. I s22s22p63s23p63d' 04s24p64d "4f 5s25p6 or [Xe] f 4 (b) Completed subshells have no unpaired electrons; thus, it is necessary to examine only the electrons past a noble gas core. For N, the 2p is the only incomplete subshell. Following Hund's rule the three electrons in this subshell will singly occupy the three available orbitals; there will be 3 unpaired electrons. Ar has no incomplete subshells and thus no unpaired electrons. Fe has six 3d electrons in the only unfilled subshell. The maximum unpairing occurs with double occupancy of one of the available d orbitals and single occupancy of the remaining four. There will thus be 4 unpaired electrons. Fe2+ also has 4 unpaired electrons, for the same reason as Fe. Pr3+ has 2 unpaired electrons, the two 4f electrons that singly occupy 2 of the 7 available 4f orbitals. 8.10. Nickel has the electron configuration [Ar]3d84s2. How do you account for the fact that the configuration of the next element, Cu, is [Ar]3d1'4s? In the hypothetical procedure of making up the electronic complement of Cu by adding 1 electron to the configuration of the preceding element, Ni, one might have expected only a ninth 3d electron. For atomic number 19, the 3d subshell is decidedly of higher energy than 4s; thus potassium has a single 4s electron and no 3d electron. After the 3d subshell begins to fill, however, beginning with element 21 (after 4s has already been filled), the addition of each succeeding 3d electron is accompanied by a lowering of the average energy of the 3d level. This is because each succeeding element has an increased nuclear charge which is only partly screened from a test 3d electron by the additional electron in the same subshell. The energy of the 3d subshell CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 119 decreases gradually as the subshell undergoes filling and drops below the level of the 4 towards the end of s the transition series. Another factor is that the configuration 3d1'4s'has a spherically symmetrical distribution of electron density, a stabilizing arrangement characteristic of all filled or half-filled subshells. On the other hand, the configuration 3d94s2 a "hole" (a missing electron) in the 3d shell, destroying the symmetry and any extra has stabilization. 8.11. The ionization energies of Li and K are 5.4 and 4.3 eV, respectively. What do you predict for the ionization energy of Na? The first I.E. of Na should be intermediate between that of Li and K. The I.E. of Na should be close to the arithmetic average of the two, or 4.9 eV. (The observed value is 5.1 eV.) 8.12. The (first) ionization energies of Li, Be, and C are 5.4, 9.3, and 11.3 eV. What do you predict for the ionization energies of B and N? There is a general increasing trend of I.E. with increasing atomic number in a given period. Note, however, a larger increase in going from Li ( Z = 3) to Be (2 = 4) than in going from Be to C (2 = 6). The filling of the 2s subshell gives Be a greater stability than would be suggested by a smooth progression across the periodic table. The next element, B (2 = 5), would have an I.E. that represents a balancing of the two oppositely directed factors, an increase with respect to Be because of the increased 2 and a decrease because a new subshell is beginning to be filled in the case of B. One might guess that the I.E. for B might actually be less than for Be, and this turns out to be the case. The observed I.E. for B is 8.3 eV. The increase in going from Z = 5 to 2 = 6 is 3.0 eV. One expects the increase in going to N ( Z = 7) to be similar, bringing the I.E. of N to about 14.3eV. In fact because of the extra stability of the half-filled p subshell the I.E. is even greater; 14.5 eV is observed. 8.13. In the ionic compound KF, the K+ and F- ions are found to have practically identical radii, about 0.134 nm each. What do you predict about the relative covalent radii of K and F? The covalent radius of K should be much greater than 0.134nm and that of F much smaller, since atomic cations are smaller than their parent atoms, while atomic anions are bigger than their parents. The observed covalent radii of K and F are 0.20 and 0.06 nm respectively. 8.14. The single covalent radius of P is 0.11 nm. What do you predict for the single covalent radius of Cl? P and C1 are members of the same period. C1 should have a smaller radius in keeping with the usual trend across a period. The experimental value is 0.10 nm. 8.15. The first ionization energy for Li is 5.4 eV and the electron affinity of C1 is 3.61 eV. Compute A H for the reaction carried out at such low pressures that the resulting ions do not combine with each other. The overall reaction may be decomposed into two partial reactions (1) Li(g) -+Li+(g) + e-(g) (2) Cl(g) + e-(g) 4 AE = NA(I.E.) AE = NA( - E.A.) Cl-(g) where e- stands for an electron. Although AH for each of the above partial reactions differs slightly from AE (by the term p A V ) , the AH of the overall reaction is the sum of the two AE's (the overall volume change is zero). Since the values of1.E. and E.A. are given on a per-atom basis, the factor 6.02 x 1023atoms/ mol is required to obtain A H on the conventional per-mole basis. AH(tota1 reaction) = AE(1) + AE(2)= NA(I.E.- E.A.) = (6.02 x 1023)(1.8 eV)(1.60 x 10-19 J/eV) = 170 kJ 120 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 Supplementary Problems ENERGY RELATIONSHIPS 8.16. What is the wavelength in meters of the radiation from (a) a low-range TV station broadcasting at a frequency of 55 MHz, (b) an AM radio station at 610 kHz, and ( c ) a microwave oven operating at 14.6 GHz? Ans. (a) 5.5 m; (b) 492 m; ( c ) 0.0205 m 8.17. The critical wavelength for producing the photoelectric effect in tungsten is 260 nm. (a) What is the energy of a quantum at this wavelength in joules and in electron volts? (b) What wavelength would be necessary to produce photoelectrons from tungsten having twice the kinetic energy of those produced at 220 nm? Ans. (a) 7.65 x 10-19 J = 4.77 eV; (b) 191 nm 8.18. In a measurement of the quantum efficiency of photosynthesis in green plants, it was found that 8 quanta of red light at 685 nm were needed to evolve one molecule of 02. The average energy storage in the photosynthetic process is 469 kJ per mole of 0 2 evolved. What is the energy conversion efficiency in this experiment? Ans. 33.5 percent 8.19. O2 undergoes photochemical dissociation into one normal oxygen atom and one oxygen atom 1.967 eV more energetic than normal. The dissociation of O2 into two normal oxygen atoms is known to require 498 kJ/mol 0 2 . What is the maximum wavelength effective for the photochemical dissociation of 0 2 ? Ans. 174 nm 8.20. The dye acriflavine, when dissolved in water, has its maximum light absorption at 453 nm, and its maximum fluorescence emission at 508 nm. The number of fluorescence quanta is, on the average, 53 percent of the number of quanta absorbed. Using the wavelengths of maximum absorption and emission, what percentage of absorbed energy is emitted as fluorescence? Ans. 47 percent 8.21. The prominent yellow line in the spectrum of a sodium vapor lamp has a wavelength of 590 nm. What minimum accelerating potential is needed to excite this line in an electron tube containing sodium vapor? Ans. 2.10 V 8.22. 8.23. Show by substitution in the formula given in the text that ao, the radius of the first Bohr orbit for hydrogen, is 5.29 x 10-" m. What accelerating potential is needed to produce an electron beam with an effective wavelength of 0.0256 nm? Ans. 2.30 kV 8.24. In studies of electron spin resonance (esr) the energy differences between spin states are very small, of the order of 1 x 10-4 eV, compared with about 3 eV in visible spectroscopy. What wavelength of radiation is required for esr, and which of the types in question 8.16 does it resemble? Ans. 0.012 m, microwave region 8.25. What accelerating potential must be imparted to a proton beam to give it an effective wavelength of 0.0050 nm? Ans. 33 V CHAP. 81 ATOMIC STRUCTURE AND THE PERIODIC LAW 121 ATOMIC PROPERTIES AND THE PERIODIC LAW 8.26. All atoms with Z odd must have at least one unpaired electron. Can an even Z atom have unpaired electrons? If so, give examples from the first three periods. Ans. Yes; C , 0, Si and S 8.27. Which atoms in the first transition period ( Z = 21-30) are diamagnetic? Give their configurations. Ans. Zn only; [Ar]3d"4s2 8.28. The configuration of Cr differs from that derived by the Aufbau procedure. Deduce the actual configuration and explain the anomaly. Ans. Cr; [Ar]3d54s'. Aufbau would give 3d44s2, so an s shifts to d to gain the stability of the half-filled shell. Note that the actual configuration has complete spherical symmetry. 8.29. The configurations of negative ions follow the Aufbau rule. Write the electron configurations for H-, N3-, F-, S2-. Ans. H-: ls2 or [He]. N3-: [Ne]. F-: [Ne]. S2-: [Ar] 8.30. The Rydberg constant for Li2' is 109 729 cm-' . (a) What is the long-wavelength limit in the absorption spectrum of Li2' at ordinary temperatures (where all the ions are in their ground state)? (b) What would be the shortest wavelength in the emission line spectrum of Li2+ within the visible (400 to 750 nm) region? (c) What would be the orbital radius of the ground state of Li2+? ( d ) What is the ionization energy of Li2+? Ans. (a) 13.5 nm; (b) 415.4 nm ( n = 8 -+ n = 5); (c) 0.176 A; ( d ) 122.4 eV 8.31. The first ionization energy of Li is found experimentally to be 5.363 eV. If the L electron (n = 2) is assumed to move in a central field of an effective nuclear charge, Z e ~consisting of the nucleus and the other , electrons, by how many units of charge is the nucleus shielded by the other electrons? Assume that the ionization energy can be calculated from Bohr theory. Ans. ZeR= 1.26; since the nuclear charge is 3+, the effective shielding by the two 1s electrons is 1.74 charge units. 8.32. What are the electron configurations of Ni2+, Re3+, and Ho3+? How many unpaired electron spins are in each of these ions? Ans. Ni2+: [Ar]3d8, 2 unpaired spins. Re3+: [Xe]4fI45d, 4 unpaired spins. Ho3+: [Xe]4f1O, 4 unpaired spins. 8.33. Which shell (give its letter and n value) would be the first to have a g subshell? If you answered correctly you will note that two-thirds of the elements have electrons in that shell and higher. Why then are there no g electrons? Ans. 0, -- 5. In accord with their high 1 value ( ) g orbitals are too high in energy to receive any electrons n 4, via the Aufbau principle. 8.34. What would you predict for the atomic number of the noble gas beyond Rn, if such an element had sufficient stability to be prepared or observed? Assume that g orbitals are still not occupied in the ground states of the preceding elements. Ans. 118 122 ATOMIC STRUCTURE AND THE PERIODIC LAW [CHAP. 8 8.35. All the lanthanides form stable compounds containing the 3+ cation. Of the few other ionic forms known, Ce forms the stablest 4+ series of ionic compounds and Eu the stablest 2+ series. Account for these unusual ionic forms in terms of their electronic configurations. Ans. Ce4+ has the stable electronic configuration of the noble gas Xe. Eu2+,with 61 electrons, could have the configuration [Xe]4f7, with the added stability of a half-filled 4 subshell. f 8.36. For the gaseous reaction K + F -+ K+ + F-, A H was calculated to be 91 kJ under conditions where the cations and anions were prevented by electrostatic separation from combining with each other. The ionization energy of K is 4.34 eV. What is the electron affinity of F? Ans. 3.40 eV 8.37. The ionic radii of S2- and Te2- are 1.84 and 2.21 A , respectively. What would you predict for the ionic radius of Se2- and for P3-? Ans. Since Se falls in between S and Te, one expects an intermediate value; obseryd value is 1.98 A. Since P is just to the left of S a slightly larger value is expected; observed value is 2.12 A . 8.38. Van der Waals radii for S and C1 are 1.85 and 1.80 A respectively. What would you predict for the van der Waals radius of Ar? Ans. Movin4 toward the right in the same period the trend continues toward smaller size. The observed value is 1.54 A. 8.39. The first ionization energy of C is 1 1.2 eV. Would you expect the first ionization energy of Si to be greater or less than this amount? Ans. Less because Si is larger; the observed value is 8.1 eV. 8.40. The first ionization energies of Al, Si, and S are 6.0, 8.1, and 10.3 eV, respectively. What do you predict for the first ionization energy of P? value of about 9.2 eV for P would be halfway between the values for Si and S, but because of the stability of the half-filled shell an anomalously high energy might be required to remove an electron from P, perhaps higher than for S. The observed value is 10.9 eV. 8.41. Several experimenters have attempted to synthesize superheavy elements by bombarding atoms of the actinide series with heavy ions. Pending confirmation and acceptance of the results, some investigators in the early 1970's referred to elements 104 and 105 as eka-hafnium and eka-tantalum, respectively. Why were these names chosen? Ans. Mendeleev had used the prefix eka- (Sanskrit word for first) to name elements whose existence he predicted, applying the prefix to a known element in the same periodic group as the predicted element. His eka-boron, eka-aluminum, and eka-silicon were later discovered, confirmed, and named scandium, gallium, and germanium, respectively. Elements 104 and 105 are predicted to have electronic structures analogous to Hf and Ta, respectively. Chapter 9 Chemical Bonding and Molecular Structure The formulas of the chemical compounds are no accident. There is an NaCl, but no NaCl,; there is a CaF2, but no CaF. On the other hand, certain pairs of elements form two, or even more, different compounds, e.g. Cu20, CuO; N20, NO, N02. In the case of ionic compounds the relative number of positive and negative ions in a formula is governed simply by the rule of electrical neutrality. In covalent compounds, or within polyatomic ions (like NOT), structures are formed by covalent bonds (i.e., electron sharing). A hierarchy of covalent bonding theories exists, of which the simplest, the drawing of Lewis structures, is emphasized in this and in most elementary texts. IONIC COMPOUNDS Ionic bonding describes the combination between oppositely charged particles, or ions. The principal forces are the classical electrical forces operating between any two charged particles. The charges of the elementary ions can be understood in terms of the electronic structure of atoms. For example, elements whose atomic numbers are within 2 or 3 of a noble gas tend to form ionic compounds containing a cation (positive ion) or anion (negative ion) isoelectronic with (having the same number of electrons as) the neighboring noble gas atom. EXAMPLE 1 What is the empirical formula for lithium oxide? Lithium loses one electron to achieve the He structure and thus forms lithium ion, L i f . Oxygen gains two electrons to achieve the Ne structure and thus To forms oxide ion, 02-. preserve neutrality, the compound must be Li20. For the transition metals, the lanthanides, and the actinides, no such simple rule exists. If we accept the ionic charges as chemical facts, we can still easily write the empirical formulas for ionic compounds in conformity to the requirement that the compound as a whole be neutral. The names of the elementary cations are simply the names of the elements, e.g. Ca2+ calcium ion. The names of the monatomic anions are derived from the names of the elements by replacing the last (or last two) syllable(s) of the element names by the suffix -ide, e.g. chloride, oxide. When two cations exist for the same element, the accepted convention is to write the charge per atom in Roman numerals within parentheses after the name of the metal. (There is still some remnant of the older mode of differentiating the two states by use of the suffix -ous for the lower of the two charge states and -ic for the higher.) EXAMPLE 2 Name the two oxides of lead, PbO and Pb02. Because oxide is 2-, the lead must be 2+ in the first case and 4+ in the other. The names are lead(I1) oxide and lead(1V) oxide, pronounced "lead two oxide" and "lead four oxide," respectively. In the old ous-ic system, the old (Latin) prefixes were used, giving plumbous oxide and plumbic oxide, respectively. Table 9-1 lists a few of the more commonly encountered polyatomic ions, their formulas, names, and charges. These should be memorized. Although hard and fast rules cannot be given, the following guidelines are helpful. Compounds of metals with nonmetals (also metals with polyatomic anions) tend to be ionic. Compounds of nonmetals with nonmetals will be covalent. If some physical properties of the substance in question are known, hard, brittle, high-melting compounds can normally be classified as ionic, whereas liquids and gases are covalent compounds. Some compounds have an intermediate character and are difficult to classify. Textbooks may be consulted for more elaborate rules (e.g., rules based on electronegativity differences), that deal with such cases. Distinctive suffixes are used for the complex oxy-anions in a partly systematic way, -ate for the most common or most stable and -ite for the ion containing less oxygen. The prefix per- is added to an -ate 123 124 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 Table 9-1 Common Polyatomic Ions Ion Name ammonium hydroxide carbonate hydrogen carbonate cyanide acetate peroxide nitrate nitrite phosphate Ion Name sulfate sulfite hydrogen sulfate perchlorate chlorate chlorite hypochlorite chromate dichromate permanganate NH,+ OH- c0:- so;s0:HSOT ClO, ClO, ClO, HCOT CNC2H302 0;NO, NOT PO:- c10CrOiCr& MnOT name to indicate even more oxygen, and the prefix hypo- is added to an -ite name to indicate a lower oxygen content than in the -ite ion. COVALENCE The covalent force between atoms sharing two or more "bonding" electrons is related to the delocalization, or smearing out, of the electrons over a wider region of space in the bond than they occupy in the separated atoms. The sharing of electrons in a covalent bond brings the bonded atoms closer together than in the absence of bonding. Thus, the distance between the two H nuclei in H2, 74pm, is much less than in the sum of the van der Waals (nonbinding) radii of two H atoms, 240pm. This binding of the two atoms at close approach is reflected also in energetic considerations. The energy of a bonded pair of atoms is less than the sum of the energies of the separated atoms. The socalled bond energy is the magnitude of this energy lowering; from another veiwpoint, the bond energy is the amount of energy, AE, required to break a chemical bond into two nonbonded fragments. Bond formation is exothermic and bond rupture is endothermic. Covalent bond energies range from about 150 to 550 kJ/mol for single electron-pair bonds formed among elements in the first three periods at normal temperatures. Another normal feature of the single covalent bond is that the spins of the bond-forming electrons, unpaired in the separate atoms, become paired in bond formation. VALENCE-BOND REPRESENTATION The formulas of many covalent compounds, especially those involving only the elements of the first few periods of the periodic table, were brought within the framework of the octet rule, according to which a total of eight valence, or outer-shell, electrons, either shared or unshared, should be in the region of each atom beyond the first period. For hydrogen the desired number is two. Electrons that are shared between two atoms are the bonding electrons and are to be counted toward the octet (or duplet in the case of hydrogen) of each of the bonding atoms. A single covalent bond consists of a pair of shared electrons, a double bond has two shared pairs, and a triple bond has three shared pairs. Bond distances are shorter and bond energies are greater for multiple bonds than for single bonds. Structural formulas, such as shown in Fig. 9-1, represent the valence electron distributions in covalent molecules and ions. These structures are not meant to indicate actual bond angles in threedimensional varieties such as CH3C1, NH3, and NH:; they merely show the number of bonds connecting CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 125 v,C=O H-eC-H Methyl chloride Ammonia Ammonium ion Hydroxide ion Fig. 9-1 Carbon dioxide Acetylene the various atoms. In these so-called Lewis formulas, a line between two atoms represents a pair of shared electrons and a dot represents an unshared electron. Two lines constitute a double bond, and three lines constitute a triple bond. The total number of electrons shown in such a molecular structure is equal to the sum of the numbers of valence electrons in the free atoms: 1 for H, 4 for C, 5 for N, 6 for 0, and 7 for C1. For an ionic structure, one additional electron must be added to this sum for each unit of negative charge on the whole ion, as in OH-, and one electron must be subtracted from the sum for each unit of positive charge on the ion, as in NH:. The number of pairs of electrons shared by an atom is called its covalence. The covalence of hydrogen is always one. The covalence of oxygen is almost always one or two. The covalence of carbon is four in almost all its stable compounds. Thus each carbon is expected to form either four single bonds, a double bond and two single bonds, two double bonds, or a single and a triple bond. Although the octet rule is not a rigid rule of chemical bonding, it is obeyed for C, N, 0 and F in almost all their compounds. The octet is exceeded commonly for elements in the third and higher periods of the periodic table. Resonance Sometimes more than one satisfactory Lewis structure can be written and there is no reason to select one over another. In such cases a single structural formula is inadequate to represent a substance correctly, and more than one such diagram must be written. The true structure is then said to be a resonance hybrid of the several diagrams. EXAMPLE 3 Experiment has shown that the two terminal oxygens in ozone are equivalent; that is, they are equidistant from the central oxygen. If only one of the resonance diagrams in Fig. 9-2(a) were written, it would appear that one of the terminal oxygens is bonded more strongly to the central oxygen (by a double bond) than is the other (by a single bond) and that the more strongly bonded atom should be closer to the central atom. The hybrid of the two ozone structures gives equal weight to the extra bonding of the two terminal oxygen atoms. Similarly, the three resonance structures of carbonate in Fig. 9-2(b) are needed to account for the experimental fact that all three oxygens are equidistant from the central carbon. 2- ( a ) Ozone ( b ) Carbonate ion Fig. 9-2 The total bond energy of a substance for which resonance structures are written is greater than would be expected if there were only one formal Lewis structure. This additional stabilization is called resonance energy. It arises from the same principle that is responsible for covalent bond energy, the delocalization of electrons about the atoms forming the bond. As a result of resonance in ozone, for example, the electrons constituting the second pair of the double bond are delocalized around the 3 126 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 oxygen atoms. The writing of two or more resonance structures is a way of overcoming the inability of a single valence-bond structure to show this delocalization. Formal charge Although a molecule as a whole is electrically neutral, it is of interest to know whether there are local charges which can be identified with particular parts of a molecule, the algebraic sum of which would equal zero. In an ion, the algebraic sum would equal the charge of the ion as a whole. In one approximate method of apportioning charges within a molecule or ion, the shared electrons in a covalent bond are arbitrarily divided equally between the two atoms forming the bond. Unshared valence electrons on an atom are assigned exclusively to that atom. Each atom is then assigned a formal charge which is equal to the number of valence electrons possessed by that atom in the neutral free state minus the number of valence electrons assigned to it in the structure. These charges may be written near the atoms on the structural diagrams. EXAMPLE 4 Figure 9-3 shows a single resonance structure for ozone. The central oxygen is assigned just five electrons (two in the unshared pair plus half of the three pairs in the bonds); this atom, being one electron short of the complement of six in a free oxygen atom, is thus assigned a formal charge of + 1. The terminal oxygen connected by a single bond is assigned 7 (6 in the unshared pairs plus half of one pair in the bond); having one electron more than a neutral oxygen atom, this atom is assigned a formal charge of -1. The other terminal oxygen has no formal charge because six electrons are assigned to it (four in the unshared pairs plus half of the two shared pairs). In the other Lewis structure the central oxygen is also positive and the lower left oxygen is negative. In the real molecule then one expects to find an unequal distribution of charge within the molecule with the central oxygen shown at the top positive and the portion of the molecule shown at the bottom encompassing the two terminal oxygens negative. The molecule has a dipole moment (vide itzfra), which confirms this conclusion. +1 .. .. -1 .. Fig. 9-3 A rough rule useful in choosing one allowable Lewis structure over another is that structures which minimize formal charge separation are favored. Especially to be avoided are formal charges of magnitude greater than 1 and structures in which appreciable formal charges of the same sign are located on adjacent atoms. Dipole moments and electronegativity There are some experimental procedures that give information about the actual distribution of charges within a molecule (as distinct from the arbitrary assignment of formal charges). One such is the measurement of dipole moment. An electric dipole is a neutral object that has a positive charge of magnitude q and a separately located equal, but opposite, negative charge. The extent to which a dipole is aligned in an electric field is dependent upon, and thus a measure of, the dipole moment, defined as the product of q and the distance, d , separating the positive and negative charges. In a covalent diatomic molecule, the dipole moment would be expected to be zero if the bonding electrons were shared truly equally by the two atoms. This is indeed the case in molecules of the type X2, where two identical atoms are bonded. In the more general type, XY, two different kinds of atoms are bonded, and a dipole moment is usually observed. This is explained by hypothesizing that one of the atoms, say Y, has a greater attraction for the shared electrons in the bond that does X. Y is said to have a greater electronegativity than X. Electronegativity correlates with other atomic properties; in general, CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 127 atoms with high ionization energies and/or high electron affinities tend to have high electronegativities. The most electronegative elements in order of decreasing electronegativity are F>O>NrCI C is slightly more electronegative than H. Metals are less electronegative than nonmetals. Hybrid orbitals The assignment of electrons to the various atomic orbitals, as discussed in Chapter 8, concerned the electron distribution in the ground state (the state having the lowest energy) of an isolatedfree atom. From the following ground-state configurations of elements in the second period, - B C N o (He)2s22p (He)2s22p2 (He)2s22p3 ( ~ ~ 1 2 ~ ~ 2 ~ ~ one might predict that the maximum covalence of an element would equal the number of unpaired electrons, since an unpaired electron from each of the bonding atoms participates in the covalent bond formation that we have described. The maximum covalences of B, C, N, and 0 should then be 1, 2, 3, and 2, respectively. These predictions account for most of the known compounds and complex ions of N (e.g., NH3, NO:) and of 0 (e.g., H20, CH30H, HOOH), but not for the commonly observed trivalence of B (e.g., BF3), the quadrivalence of C (e.g., CH4, CC14, CH30H), or the occasional quadrivalence of N (e.g., NH;). We can explain the higher-than-predicted covalences of B and C by rearranging the electron configuration to an energy level higher than the ground state before considering compound formation. For example, B might go into the configuration (He)2s2p2,where there are three unpaired electrons available for the formation of three covalent bonds. Although a boron atom could achieve this preparatory state only if it gained the excitation energy needed to transform the atom from its ground state, the principle of overall energy minimization could still be obeyed if the bond energies of the two additional bonds that could be formed exceeded in magnitude this required excitation energy. Similarly, a carbon atom could be excited to the configuration (He)2s2p3, with four unpaired electrons, if the energy released in the formation of the two additional bonds exceeded in magnitude the energy required for excitation to the new electron configuration. The above explanation accounts for the number of covalent bonds formed by boron and carbon but not for the equivalence of these bonds. The difference in spatial character of s and p orbitals (Fig. 8-2), and in their energies, might suggest differences in the bonds they formed, as measured by bond energy, bond distance, or angles between bonds. Experiment shows, on the contrary, that all three bonds in BF3 are equivalent, that the angles between any two bonds are the same, and that in fact the three fluorine atoms lie at the corners of an equilateral triangle with the boron atom in the center. Similarly, in CH4 all four bonds are equivalent, the angles between any two bonds are the same, and the four hydrogen atoms lie at the corners of a regular tetrahedron (Fig. 9-4) with the carbon atom at the center. Fig. 9-4 128 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 The equivalence of the bonds in BF3, CH4, and similar compounds was first explained by Linus Pauling. To accommodate the surrounding atoms, s and p orbitals of a given atom may mix with each other, or hybridize. The mathematical formulations of the hybrid orbitals are simple linear combinations of the mathematical formulations of the separate s and p orbitals, and the geometrical description of each hybrid orbital is a kind of superposition of the mappings of the separate s and p orbitals. Pauling showed that when one s and threep orbitals are hybridized so as to give a maximum concentration of the electron probability distribution along some particular direction (so as best to form a bond along this direction), the four resulting so-called sp3 hybrid orbitals indeed point to the corners of a regular tetrahedron. The angle made by any two lines (i.e., bonds) connecting the center of a regular tetrahedron with corners is 109'28' (Problem 9.18). This is the observed situation in CH4, CC14, SiF4, and many other compounds of the Group IV elements. A similar hybridization of one s with two p orbitals leads to a set of sp2 hybrid orbitals which have their maximum concentrations along a set of lines in one plane, forming angles of 120" with each other. This is the observed situation in BF3. The angles are close to 120" in C2H4, in which each of the two carbon atoms forms a set of sp2hybrid orbitals (Fig. 9-5). Two other important types of hybrid orbitals are the sp type, in which the two orbitals point in directions leading to a 180" bond angle (Fig. 9-6), the situation in C2H2 and in the high-temperature form of BeC12, and the d2sp3type, in which the six orbitals point to the corners of a regular octahedron (Fig. 9-7), the situation in SF6 and in many coordination compounds. F ,,,"pC, 121" ,H H-C-H H H IXIP Fig. 9-5 Fig. 9-6 Fig. 9-7 MOLECULAR-ORBITAL REPRESENTATION In Lewis structures, valence electrons are described as being either unshared electrons localized on particular atoms or shared electrons assigned to the bonds linking particular pairs of atoms. In an alternative representation, all the valence electrons are assigned to molecular orbitals (MOs) appropriate to the molecule as a whole. Just as an atomic orbital is the mathematical solution of a Schrodinger equation describing the probability distribution of the various positions that an electron having a given set of quantum numbers may occupy around an atomic nucleus, so a molecular orbital describes the distribution of positions in a molecule available to an electron having a given set of quantum numbers. Molecular orbitals may be approximated by writing mathematical combinations of the atomic orbitals of the constituent atoms or they may be pictured, at least in qualitative terms, as geometrical combinations of the contributing atomic orbitals. Several simple rules may be stated for molecular orbitals . 1. The total number of molecular orbitals (MOs) equals the sum of the numbers of atomic orbitals (AOs) of the constituent atoms. 2. Each orbital can hold 0, 1, or 2 electrons, corresponding to the possibility of two different directions of electron spin and the application of the Pauli principle. 3. When several molecular orbitals of equal or almost equal energy exist, electrons tend to fill these orbitals so as to maximize the number of unpaired electron spins. The more nearly equal the energy levels, the greater is this tendency. (Hund's rule) CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 129 4. The directions in space describing the orientation of the orbitals, although arbitrary in the case of a free atom (Fig. 8-2), are related to the positions of neighboring atoms in the case of molecules or complex ions. 5. A molecular orbital is most likely to be composed of atomic orbitals of similar energy levels. 6. In diatomic molecules, or more generally in localized two-centered bonds, orbitals of the two atoms that may be combined to form a molecular orbital are those that have the same symmetry about the internuclear axis. (In constructing molecular orbitals that extend over three or more atoms, more complicated rules of symmetry must be considered, involving the symmetry of the extended nuclear framework.) 7. A two-centered orbital with high electron probability in the region between the nuclei is a bonding orbital. One with zero or low electron probability between the nuclei is an antibonding orbital. A stable chemical bond exists when the number of electrons in bonding orbitals exceeds the number in antibonding orbitals. Similar but more complex rules apply to orbitals that extend over more than two atoms. The directional properties of molecular orbitals are governed by quantum numbers analogous to the atomic quantum numbers I and ml. Greek letters are used in molecular orbital notation to designate increasing values of the I-type quantum number, o, 7r, 6, etc., analogous to the Latin letters s, p , d , etc., for atomic orbitals. Only o and 7r orbitals will be considered in this book. There are two kinds of CT orbitals, bonding and antibonding (Fig. 9-8). (An antibonding orbital is designated with a superscript asterisk applied to the Greek letter.) A bonding orbital has a region of electron overlap, or high probability, between the bonded atoms, and has a lower energy than either of the component atomic orbitals. An antibonding orbital has a nodal plane, or region of zero probability, between the bonding atoms and perpendicular to the bond axis; its energy is greater than that of either constituent atomic orbital. a: antibonding v s bonding Fig. 9-8 If the bond axis is designated by x, o bonds can be formed by combining any two of the following atomic orbitals, each of which has a region of high electron probability lying along the x axis and cylindrically symmetrical about it: s, p x , dx2-y2, or hybrid orbitals pointing along the x axis. If the bond axis is z , o orbitals may be formed from s, p z , dz2, or appropriate hybrids. If the bond axis is y , the component atomic orbitals for CT orbitals are s, p y , dX2-?2, or appropriate hybrids. A 7r orbital related to a bond in the x direction is characterized by having a zero value at all points along the x axis. 7rJ, orbitals, having their maximum probabilities in the +y and -y directions (above and below the xz plane), may be formed from p y atomic orbitals. Similarly, 7rz orbitals, having their maximum probabilities above and below the xy plane, may be formed from p z atomic orbitals. T orbitals may also be bonding or antibonding (Fig. 9-9). A dxy orbital may combine with a p y orbital, or a dxzwith a p z , to form 7r orbitals. Nonbonding atomic orbitals are those which do not interact with orbitals of other atoms for one of the following reasons: 1. The two atoms are too far apart for good orbital overlap (as in the case of nonadjacent atoms). 2. The energy of the nonbonding orbital is not close to that of any orbital on the adjacent atom (e.g., the 3s orbital of C1 is of much lower energy than the 1s orbital of H in HC1). 130 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 0 0 .rr, bonding Fig. 9-9 3. The nonbonding orbital is in an inner shell and would not overlap with an orbital even of the neighboring atom (e.g., the K electrons in F2). 4. The nonbonding orbital does not match its symmetry with any available orbital of the adjacent atom (e.g., the 3p,, of C1 does not have a symmetry match with 1s of H in HC1, where x is the bonding direction, and the 2p, orbital of H is too high in energy to enter the picture). A nonbonding orbital has the same energy as in the free atom. Electrons occupying nonbonding orbitals correspond to the unshared electrons in Lewis structures. A buildup principle analogous to that for atoms exists for molecules. The order of filling molecular orbitals from the valence shells in the case of homonucleur diatomic molecules, where x is the bond axis, is as follows: < a; < 7r, = 7rz < ap, < 7r; = 7r; < a;y The ordering may change for heteronucleur diatomic molecules, and for homonuclear molecules at the point where this complete set of orbitals is about half-filled. The bond order in a diatomic molecule is defined as one-half the difference between the number of electrons in bonding orbitals and the number in antibonding orbitals. The factor one-half preserves the concept of the electron pair and makes the bond order correspond to the multiplicity in the valence-bond formulation: one for a single bond, two for a double bond, and three for a triple bond. Fractional bond orders are allowed. z BONDING AND MULTICENTER ' ~ BONDS t An explanation for the bonding in ethylene, C2H4, is as follows. The basic framework is established by combining five, two-centered, bonding, a orbitals, four of which are made from a 1s orbital on a hydrogen and an sp2 orbital on a carbon, and one of which is made from one sp2 orbital on each of the carbons. (The combinations of these two-centered a orbitals are bonding molecular orbitals that extend over the whole molecular framework. These extended molecular orbitals, represented by the lightly shaded regions of Fig. 9-10, may be referred to as a-type because their electron density is concentrated mainly along the axes connecting pairs of adjacent atoms.) If the plane of each H2C-C group is designated as the xy plane, the px and py atomic orbitals are used to form the sp2 hybrid orbitals. The p z orbitals on the two carbons (the darkly shaded regions of Fig. 9-10) are then available for forming r/r, orbitals (as indicated by the dashed lines in Fig. 9-10, which represent the overlap of the darkly shaded orbitals). After filling of the five a-type bonding orbitals, the remaining electron pair (of the total of 12 valence electrons on the two carbons and four hydrogens) goes into the T, orbital. The two carbons are thus bonded partly by the electrons in the a-type framework, constituting the equivalent of a single bond between the two carbons (plus the equivalent of a single bond connecting each of the hydrogens to its adjacent carbon), and partly by the pair of 7rz electrons forming the second part of the double bond indicated in Fig. 9-5. The 7r bond prevents rotation about the C-C axis and constrains all six atoms in C2H4 to the same plane. In acetylene, C2H2,the a-type skeleton of three bonding orbitals (represented by the lightly shaded regions of Fig. 9-1 1) is formed from the 1s orbitals on the hydrogen atoms and the hybrid sp orbitals on the carbon atoms. The p.uatomic orbitals are used to form the hybrid CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 131 Fig. 9-10. Ethylene Fig. 9-11. Acetylene Fig. 9-12. Ozone orbitals which point along the bonding x direction, and the remaining p orbitals are free to form the 7ry and 7rz orbitals (represented by the dashed lines joining the darkly shaded regions of Fig. 9-1 1). The ten valence electrons (one from each hydrogen and four from each carbon) fill the three a-type orbitals and the 7r,,and ri2orbitals. The carbons are held together by the equivalent of a triple bond (one a bond and two k bonds), as indicated in Fig. 9-6. In ozone, 03, atoms are held together in the first instance by a a-type framework (represented by the the lightly shaded regions of Fig. 9-12) in the xy plane of the molecule, with its electron density concentrated mainly along the axes connecting pairs of nearest-neighbor oxygen atoms. The p _ orbitals of all three oxygen atoms have the same symmetry with respect to the plane of the molecule and can thus combine to form T orbitals. The p z orbital of the central oxygen atom overlaps with the p z orbitals of both the terminal oxygens. As a result, a 7r bonding orbital and a 7r* antibonding orbital extend over all three atoms in the molecule, and a 7r nonbonding orbital involves the two terminal atoms. Two electrons in ozone occupy the 7r bonding orbital, enveloping all three oxygen nuclei, with equal probability on the two sides of the molecule. This representation is an alternative to resonance, which was conceptualized to preserve the supposed sanctity of the octet rule. The 7r nonbonding orbital is also occupied in ozone. The occupied 7r bonding orbital is represented by the dashed lines joining the darkly shaded regions of Fig. 9-12. Multicentered 7r orbitals are involved in the molecular-orbital representation of most structures for which resonance must be invoked in the valence-bond representation. In a long chain of atoms bonded within a planar configuration, such as the plant pigment carotene, C40H56, in planar ring compounds or like naphthalene, CI0H8, each 7r orbital extends over many carbon atoms, all those in the basic molecular plane, since the overlap of the p _ orbital of any nonterminal carbon with those of its two or three neighbors allows for the buildup of long chains or rings of extensively overlapping electron probability distributions. SHAPES OF MOLECULES Bond lengths Bond lengths between a given pair of atoms are approximately constant from compound to compound if the order of the bond (single, double, or triple) is the same. If it is assumed that the length of a single covalent bond is the sum of the covalent radii of the two bonding atoms, then quick and fairly reliable estimaies of bond lengths can be obtained from a compilation such as Table 9-2. Double bonds are shorter. From the precisely measured lengths 154pm, 133pm, and 120pm, for H3C-CH3, H2C=CH2, and HC=CH, respectively, a rule of thumb has been developed to allow a 21 pm shortening for any double bond and 34pm shortening for any triple bond with respect to a single bond between the same atoms. In the case of resonance or multicentered 7r orbitals, a bond length is intermediate between the values it would have in the separate resonance structures or between the values 132 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 Table 9-2. Single-bond Covalent Radii C Si N P Sb 77 pm 117pm 70 pm 1 lOpm 141pm 0 S F I c 1 66 pm 104 pm 64 pm 99 pm 133pm Table 9-3 Substance CH3Cl (CH3)20 HZCO ICN Bond c-Cl Predicted Length Observed Length 177pm 143pm 122pm 116pm rc rc 77 + 9 9 = 176pm + + YCI = c-0 c=o C=N ro = 77 + 66 = 143pm rc+ro-21= 143 - 21 = 122pm rc 7 7 t 7 0 - 3 4 = 113pm + YN - 34 = it would have in the absence of n bonding and in the presence of a localized two-centered applications of this rule are shown in Table 9-3. T bond. Some VSEPR BOND ANGLES Starting with the Lewis structure it is possible to predict fairly accurately the bond angles in a molecule. The VSEPR method (acronym for Valence Shell Electron Pair Repulsion) focuses on some central atom and counts the number of atoms bonded to it plus the number of unshared pairs. (Multiple bonds count the same as single bonds.) This characteristic VSEPR number is the number of orbitals (each occupied by an electron pair) that must emanate from the central atom. The angles between them are determined by the principle that the electron pairs will repel each other as far as possible. The VESPR method is a simple unsophisticated technique that does not require explicit identification of the orbitals. Invoking the hybrid orbital technique results in the same angles, but in a formal manner linked to the mathematical treatments of bonding. VSEPR numbers, corresponding angles, and hybrid orbital sets are listed in Table 9-4. Deviations from the above angles arise from unshared pairs, which repel more strongly than shared pairs. For example, the VSEPR number is 4 for CH4, :NH3, and I-@H. The HCH angle in CH4 is a perfect tetrahedral angle (109"28'), but the HNH bond in :NH3 is compressed to about 107" by the Table 9-4 VSEPR Number 2 3 4 5 6 Nominal Bond Angles 180" 120" 109O28' 90, 120", 180" 90", 180" Hybrid Orbital Set SP SP2 SP3 dsp3 d2sp3 CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 133 unshared pair, and in W H the two unshared pairs compress the HOH angle further to 104.5'. Double bonds also contribute extra repulsion which compresses adjacent bond angles slightly (see Fig. 9-5). Deviations arise also if the various atoms around the central atom are very different in size. COORDINATION COMPOUNDS The electrons of an electron-pair bond need not be contributed by both of the bonding atoms, as is demonstrated in the formation of NH: by the addition of a proton to NH3. The Lewis structures for NH3 and NHQ are shown in Fig. 9-13; H has no electrons. Such a bond is often called a coordinate covalent bond, but is essentially no different from any other covalent bond. In this particular case, once the coordinate covalent bond is formed, it becomes indistinguishable from the other three N-H bonds; in fact, the structure is a regular tetrahedron. H-N-H + H H Fig. 9-13 I H-N-H H I I Coordinate covalence is the common bonding type in coordination compounds, in which a central metal atom or ion is bonded to one or more neutral or ionic ligands. A typical ligand, such as NH3, Cl-, or CO, has an unshared electron pair that forms the coordinate covalent bond by interacting with the unfilled orbitals of the central metal. The overall charge of a complex ion is the algebraic sum of the charge of the central metal and the charges of the ligands. A number of rules have been adopted internationally for the naming of coordination compounds. 1. If the compound itself is ionic, the cation is named first. 2. A complex ion or nonionic molecule carries the name of the central metal last, with its oxidation state (charge per atom) in parenthesized Roman numerals (or zero). (A detailed discussion of oxidation state is given in Chapter 11.) 3. Anionic ligands are named with the suffix -0,as in chloro, oxalato, cyano. 4. The number of ligands of a given type is given by a Greek prefix, like mono- (often omitted), di-, tri-, tetra-, penta-, hexa-. 5. If the name of the ligand itself contains a Greek prefix, the number of ligands is indicated by such prefixes as bis-, tris-, tetrakis-, for 2, 3, or 4, respectively, and the name of the ligand is enclosed in parentheses. 6. When the complex ion is anionic, the name of the metal is given in its Latin form with the suffix -ate. 7. Some neutral ligands are given special names, like ammine for NH3, aqua for H20, carbonyl for CO. 8. When several ligands occur in the same complex, they are named in alphabetical order (ignoring numerical prefixes). EXAMPLE 5 [Co(NH3)ciI(N03)3 Hexaamminecobalt(II1) nitrate Tetracarbonylnickel(0) Ni(CO), K[Ag(CN),I Potassium dicyanoargentate(1) )4c121 c 1 Tetraamminedichlorochromium(II1) chloride [Cr(NH3 [ C O ( N H ~ C ~ H , N H ~ ) ~ ]Tris(ethylenediamine)cobalt(III)bromide) B~~ The last formula may be abbreviated as [Co(en)3]Br3. 134 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 Structure, properties, and bonding In many coordination compounds, the ligands are arranged around the central metal in regular geometrical forms, such as the octahedron, the tetrahedron, or the square. In the formulas listed above, the brackets define the complex made up of the central metal and its ligands. These brackets are often omitted when there is no ambiguity as to the nature of the complex. Many of the compounds are colored. Some are paramagnetic because of the presence of unpaired electrons, while other complexes of the same metal are not paramagnetic. The molecular-orbital energy-level diagram, Fig. 9-14, is the key to the explanation of their properties. Following is a very brief derivation of this diagram. Beginning students who may find it difficult to follow the derivation should nevertheless be able to use it for problem solving. Here we shall consider only the six-coordinated octahedral complexes. The hybridization on a metal atom leading to octahedral bonding is d2sp3.If the bonding axes are x , y, and z , the two d orbitals that are used for the hybridization are those that point along one or more of these axes, dz2 and dX2-~2(Fig. 8-2). Each of the resulting six hybrid orbitals mixes with an orbital directed along a bonding axis on the ligand, such as a p or a tetrahedral hybrid, to form o and o* molecular orbitals. Each bonding o orbital is occupied by a pair of electrons. The full set of molecular orbitals for the complex may be constructed from the following atomic orbitals: all five d orbitals of the next-to-outermost shell of the metal, the s and the three p orbitals of the metal's outer shell, and one orbital from each of the six ligands pointed along a bond axis, such as a p or a hybrid orbital. The total number of participating atomic orbitals is 15, and the total number LIGANDS CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 135 of orbitals remaining after molecular-orbital formation must still be 15. The electrons to be accommodated in these orbitals include the six previously unshared pairs provided by the six ligands for bond formation to the metal, plus the valence electrons of the metal, the number depending on the particular oxidation state. A schematic molecular-orbital diagram is shown in Fig. 9-14, with the contributing metal atomic orbitals shown on the left and the ligand orbitals on the right. (A more complicated diagram would be needed if the ligands also had 7r orbitals that participate in the bonding to the metal.) The actual energy spacings will depend on the particular case, but the relative ordering of the ( n - l ) d , ns, and np metal orbitals (where n is the principal quantum number of the outermost shell) is general. Also it is common for the ligand orbitals to be at lower energy than the metal orbitals. Note that of the nine metal orbitals, only six contribute to primary bonding in the complex. These six are the same ones that can be hybridized to form the octahedral hybrids pointing along the fx, fy,and fz directions; namely, the s, the three p , and the 4 2 and dx2~-4.2. six metal orbitals contributing to bonding mix with the six The ligand orbitals to form six a-type bonding and six o*-type antibonding molecular orbitals delocalized over the whole complex. The bonding molecular orbitals have lower energies than the ligand orbitals, and the antibonding have higher energies, as is usual. Also as usual, the lower the energy of a bonding orbital, the higher that of its corresponding antibonding orbital. The three remaining metal d orbitals, not having o character with respect to the bond directions, remain nonbonding and their energy is unchanged to a first approximation. Several consequences can easily be seen from this diagram. 1. The 12 electrons supplied by the ligands can be thought to occupy and fill the six bonding otype orbitals. 2. The next-lowest-lying orbitals, available for the valence electrons supplied by the metal, are the three nonbonding orbitals (the dxy, dxz,and dyCorbitals which do not point toward the ligands). 3. Of the a*-typeantibonding orbitals, the lowest-lying are the two whose metal components are dX?-y2 and d,?, designated as 0 . The energy difference between this level and the nonbonding : level is denoted as A. (The symbol for this energy difference is not standard; various textbooks use X , 10Dq or &.) The a: and the nonbonding orbitals are referred to, respectively, as the eg and t2g sets. 4. The first three valence electrons of the metal will occupy the nonbonding t2gorbitals. 5. The next two valence electrons of the metal could occupy either the t2g or the es orbitals, depending on whether A is respectively greater than or less than the energy increase associated with pairing two electrons in the same orbital, in violation of rule 1 on page 114. If the tZgorbitals are preferentially filled, the complexes tend to have low spin and the d3 and d6 configurations are stabilizing, in accordance with the features of filled or half-filled equalenergy orbitals. If the eg orbitals are filled before the t2, orbitals are doubly occupied, the complexes tend to have high spin and the d5 and d" are stabilizing configurations. 6. The weak electron transitions responsible for the color of coordination complexes are correlated with the t2g to eg transition, of energy A. (The origin of the strong transitions responsible for the deep colors of some complexes of metals in high oxidation states, like Mn 0 7, and Cr&, has a different explanation.) With regard to the options in rule 5 , CN-, CO, and NO, are strong-field ligands that tend to increase A and promote low-spin complexes; OH- and Cl- are weak-Jield ligands that lead to smaller values of A and high-spin complexes. A more complete listing of ligands in order of their field strength is known as the spectrochemical series. ISOMERISM There are many examples in chemistry of two or more substances that have the same numbers of atoms of each element per molecule or ion but differ in the spatial arrangement of the atoms. Such 136 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 substances, known as isomers, may differ in their physical properties (like melting point, boiling point, density, and color) and in their chemical properties (reactivities toward other substances). Three different categories of isomerism are described below. Structural isomerism One way of describing a chemical compound or ion is to list for each atom the numbers of each kind of atom connected to it by covalent bonds. Isomers which differ in these listings are called structural isomers. EXAMPLE 6 As shown in Fig. 9-15, two different structures can be drawn from C4Hlo, both conforming to the octet rule for carbon. In n-butane, two of the carbons are bonded to one carbon each and two are bonded to two each. In isobutane, three of the carbons are bonded to one carbon each and the fourth to three carbons. n-Butane melts at -135C and boils at O"C, while isobutane melts at -145C and boils at -10C. Within any one molecule there is a continuous, practically unrestricted, rotation of the atoms about any C-C bond, so that all structures which on paper differ only by the angular position of an atom or group of atoms linked by a single bond are really the same, as shown in Fig. 9-16. These diagrams are all representations of the same substances, n-butane. In each, we can number the carbon atoms 1 to 4 as we move from one end of the molecule to the other through successive C-C bonds. We see that the kinds of neighbors of a given numbered carbon atom are the same in all three diagrams. H I I I I H-CC-C-CC-CC-H I l l 1 H H H n-Butane H H H H H-C-H H-C-C-C-H I H Fig. 9-15 I H " 1 7 I H Isobu tane l H l H H- C '-H I II H H H-C4-H I H Fig. 9-16 I Structural isomers also exist for some coordination compounds. One example is a set of compounds in which a ligand in one isomer may occupy a position outside the coordination sphere in another isomer. Thus [Co(NH3),Br]S04 and [Co(NH3),S04]Br are structural isomers, each recognizable by its own color and complete set of distinctive properties. CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 137 Geometrical isomerism In some isomers the listing of bonded atoms around each atom is the same, but the compounds differ because at least two atoms, bonded to the same or to adjacent atoms but not to each other, a,re at different distances in the several forms. Such isomers are called geometrical isomers. An important set of geometrical isomers occurs in compounds containing a carbon-carbon double bond. The extra strength of the double bond, together with the coplanarity of the atoms bonded to the doubly bonded carbons, which is forced by the T electron overlap of the doubly bonded carbons, restricts the motion of groups with respect to the double bond. Consequently, diagrams showing different angular positions represent different substances. EXAMPLE 7 In Fig. 9-17 structures (a) and (b) are structural isomers of each other, as are (a) and ( c ) , since in (a) both chlorines are attached to the same carbon and in (b) or (c) one chlorine is attached to each carbon. C I c /=\H 1 \ /H Cl\ /c1 (6 ) Cl H/c'c\H Fig. 9-17 H /-iC, \ /H Structures (b) and (c) are geometrical isomers of each other because they differ in the spatial arrangement of their atoms but not in the listing of the mere numbers of atoms of each kind to which each atom is bonded. All three structures therefore represent different substances. EXAMPLE 8 Geometrical isomerism in compounds with square symmetry is illustrated in Fig. 9-18. Because of the rigidity of the square planar arrangement in the Pt, N and C1 atoms, the two forms are distinctive. For example, the distance between the two chlorine atoms in (a) is greater than in (b). If this compound had been tetrahedral with Pt in the center, isomerism would not have occurred, because any corner of a regular tetrahedron is equidistant to the other three corners. NH3 c1 I Pt --cl 1 NH3 C- Pt -NHj 1 NH3 CI (6) Fig. 9-18 I I ( a) Identical atoms adjacent to each other are said to be cis, as in Fig. 9-17(b) and 9-18(b), and when they are opposite they are called trans, as in Figs. 9-17(c) and 9-18(a). In octahedral complexes, structures of the type MX4Y2may exist as geometrical isomers. The two Y atoms (or groups) can occupy either adjacent sites [Fig. 9-19(a)] or opposite sites [Fig. 9-19(b)]; only two isomers exist because there are only two different distances between corners of a regular octahedron. The cis and trans forms of Pt(NH&C14 are isomers of this type. Possibilities for geometrical isomerism occur for other octahedral formula types, like MX3Y3, or complexes with more than two different kinds of ligands. Optical isomerism Optical isomers, occurring in pairs, are distinct structures with groups bonded to a central atom in arrangements which are mirror images of each other. Among the important categories of optical isomers are compounds containing four different groups singly bonded to a given carbon atom 138 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 X X X (a) cis X (6) pans Fig. 9-19 X X W W Y Fig. 9-20 Fig. 9-21 (Fig. 9-20) and octahedral complexes with three different kinds of ligands or with several multivalent ligands (Fig. 9-21). BONDING IN METALS Molecular orbital theory has been invoked to explain the bonding in metallic crystals, such as pure sodium or pure aluminum. Each MO, however, instead of subtending just a few atoms as in a typical molecule, must in a metal cover the entire crystal (of perhaps 1020 atoms!). Following the rule that the number of MOs must equal the number of AOs combined, this many MOs must be so close on an energy level diagram that they form a continuous band of energies. Thus the theory is known as the "band" . theory. Consider the case of sodium metal. Of the 11 electrons, the 10 forming the neon core are localized around each Na nucleus, leaving one per atom to fill the MOs that pervade the crystal. If there were N atoms in the crystal N MOs could be formed by use of one 3s orbital from each. Although these MOs have varying amounts of bonding and nonbonding character, their energies form a continuum within the 3s band. In metals beyond Group I the picture becomes more complicated as both s and p orbitals are used to form the band of MOs, which then contains many more orbitals than the number of electron pairs available. The criterion for electrical conductivity, a property of all metals, is that the energy band be only partially filled. A simple alternative model, consistent with band theory, is the "electron sea" picture, illustrated in Fig. 9-22 for sodium. The circles represent the sodium ions, which occupy regular lattice positions (the second and fourth lines of atoms shown are in a plane below the first and third). The eleventh electron from each atom is broadly delocalized so that the space between sodium ions is filled with an "electron sea" of sufficient density to keep the crystal electrically neutral. The massive ions vibrate about their nominal positions in the electron sea, which holds them in place somewhat like cherries in a bowl of CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 139 Fig. 9-22 gelatin. This simple model successfully accounts for the unusual properties of metals such as electrical conductivity and mechanical toughness. (In many metals, particularly the transition elements, the picture is more complicated, with some electrons participating in local bonding in addition to the delocalized electrons.) Solved Problems FORMULAS 9.1. Write formulas for the following ionic compounds: ( a ) barium oxide, (b) aluminum chloride, (c) magnesium phosphate. ( a ) The formula is BaO since the +2 charge on one barium ion just balances the -2 charge on the oxide ion. (b) Three chloride ions, - 1 charge on each, are needed to balance the +3 charge of one aluminum ion. The formula is A1C13. ( c ) Since neither 2 (the positive charge on magnesium ion) nor 3 (-thenegative charge on phosphate) is an integral multiple of the other, the smallest number must be found that is a multiple of both. The number is 6. The formula, Mg3(PO&, shows 6 units of positive charge (3 x 2) and 6 units of negative charge (2 x 3), per empirical formula unit. 9.2. Name the following compounds: (a) Mg3P2, (b) Hg2(N03)2. (c) NH4Tc04. ( a ) This compound of a metal and nonmetal is clearly ionic and thus P3-must be named phosphide; hence the name magnesium phosphide. (b) Since the charge on two NO, (nitrate ions) is -2, the total cationic charge (Hg2) must be f2. Since the average charge per Hg is +1, the name is mercury(1) nitrate (or mercurous nitrate). ( c ) The charge on the anion must be - 1 in order to balance the +1 charge on the ammonium ion. Since Tc belongs to the same group of the periodic table as Mn, TcO, is analogous to MnO;, permanganate. The name is thus ammonium pertechnetate. 140 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 ( a ) The charge of (Mn04) must balance that of two Na', i.e., (Mn04)2-. (This ion is called mangunate and is different from permanganate.) (b) The ion in brackets must balance the charge of four H', i.e., [Fe(CN),I4-. ( c ) The charge of (P3OlO)must balance the charge of one N a + and two Cd2", i.e., (P3O1$-. (4 The charge of (B407)must balance the charge of two N a f , i.e., (B407)2-. (e) The charge of two (CoF,) ions must balance the charge of three Ca2+, i.e., ( C O F ~ ) ~ - . v> The charge of two (B03) ions must balance the charge of three Mg2+, i.e., (B03)3-. ( g ) The charge of (U02) ion must balance the charge of two Cl-, i.e., (U02)2+. (h) The charge of two (SbO) ions must balance the charge of SO$-, i.e., (SbO) '. 9.4. The formula of calcium pyrophosphate is Ca2P207.Determine the formulas of sodium pyrophosphate and iron(II1) pyrophosphate (ferric pyrophosphate). The charge of the pyrophosphate ion must be -4 to balance the charge of two Ca2'. We can then write Na4P207and Fe4(P207)3. 9.5. Write all possible octet structural formulas for ( a ) CH40, (b) C2H3F, (c) NT. ( a ) Since the valence level of hydrogen is saturated with two electrons, each hydrogen can form only one covalent bond. Thus hydrogen cannot serve as a bridge between C and 0. The only possibility of providing four bonds to the C is to have 3 H's and the 0 bonded directly to it. Only one structure is possible, as shown in Fig. 9-23. Note that the total number (14) of valence electrons in the structure is the sum of the numbers of valence electrons in the free component atoms: 4 (in C ) + 6 (in 0 ) + 4 (in 4 H). (b) The only way of providing four bonds for each C within the limitation of 18 valence electrons is to have a C = C bond, as in Fig. 9-24. The reader should determine by trial and error that no other structure is possible. H H-C-0-H H Fig. 9-23 (c) I I H H /-\F: \ /H Fig. 9-24 The total number of available valence electrons is 16 ( 5 in each of the 3 free N atoms plus 1 arising from the net ionic charge). From this it can easily be seen that a linear structure without multiple bonds would not satisfy the octet rule. For example, in Fig. 9-25, the available electrons provide octets for the terminal nitrogen atoms, but four less than an octet for the central one. The only way to have the same number of electrons provide octets for more atoms is to reduce the number of unshared electrons by four and increase the number of shared electrons accordingly, by forming either two double bonds or one triple bond. The possible linear structures are shown in Fig. 9-26. [:-.-:I- .. ++ [:N-N=N:]- ++ [:N=N-N:]- Incorrect structure (b) Fig. 9-26 Fig. 9-25 Two resonance structures are written with the triple bond because there is no reason why one of the terminal nitrogens in an isolated azide ion should be different from the other. A different type of CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 141 Incorrect structure Fig. 9-27 structure, involving a ring, also satisfies the octet rule. This structure (Fig. 9-27) is ruled out because the bond angles of 60" or less demanded by the structure require considerable strain from the normal angles of bonding, especially with a double bond in the ring. 9.6. Experimentally the azide ion, NT, is found to be linear, with each adjacent nitrogen-nitrogen distance equal to 116 pm. (a) Evaluate the formal charge at each nitrogen in each of the three linear octet structures depicted in Fig. 9-26. (b) Predict the relative importance of these three resonance structures in NY. (a) In structure (a) of Fig. 9-26, the central N is assigned of the four shared pairs, or four electrons. This number is one less than the number of valence electrons in a free N atom, and thus this atom has a formal charge of + l . Each terminal N is assigned four unshared electrons plus of the two shared pairs, or six altogether. The formal charge is thus -1. The net charge of the ion, -1, is the sum 2( - 1) + 1. In structures (b)and ( c ) of Fig. 9-26, the central N again has four assigned electrons with a resulting formal charge of +l. The terminal triply bonded N has 2 plus of 3 pairs, or a total of 5, with a formal charge of 0. The singly bonded terminal N has 6 plus of 1 pair, or a total of 7, with a formal charge of -2. The net charge of the ion, -1, is the sum of +1 and -2. (b) Structure (a) is the most important because it has no formal charge of magnitude greater than 1. For structure (a) the nitrogen-nitrogen bond distance is predicted to be (70 + 7 0 ) minus the double bond shortening of 21 pm, or 119pm. The observed bond length is 116pm. 4 9.7. The sulfate ion is tetrahedral, with four equal S - 0 structural formula consistent with these facts. distances of 149 pm. Draw a reasonable It is possible to place the 32 valence electrons (6 for each of the 5 Group VI atoms plus 2 for the net negative charge) in an octet structure having only single bonds. There are two objections to this formula (Fig. 9-28). (1) The predicted bond distance, rs ro = 104 66 = 170pm, is much too high. (2) The calculated formal charge on the sulfur, +2, is rather high. Nonetheless this structure appears widely in texts, and it is rationalized that the short bond length is the result of the strong attraction between the formally +2 sulfur and -1 oxygens. An alternative is to write resonance structures containing double bonds. A structure like that in Fig. 9-29 places zero formal charge on the sulfur and -1 on each of the singly bonded oxygens. The shrinkage in bond length due to double bond formation helps to account for the low observed bond distance. Other resonance structures (making a total of six) with alternate locations of the double bonds are, of course, implied. Structures like this, with an expanded valence level beyond the octet, are generally considered to involve d orbitals of the central atom. This is the reason that second-period elements (C, N, 0, F) do not form compounds requiring more than eight valence electrons per atom; there is no such thing as a 2d subshell. + + I :+SI +: I :ij: 2- :ii: 1'Fig. 9-29 :o: . Fig. 9-28 142 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 9.8. Draw all the octet resonance structures of- (a) benzene, C6H6, and (b) naphthalene, C10H8. Benzene is known to have hexagonal symmetry, and the carbon framework of naphthalene consists of two coplanar fused hexagons. Invoke bonding only between adjacent atoms. (a) The hydrogens are distributed uniformly, one on each carbon, to conform with the hexagonal symmetry (Fig. 9-30). This leaves three carbon-carbon bonds to be formed at each carbon to bring its total covalence to four. Alternating single and double bonds form the only scheme for writing formulas within all the above restrictions. Hydrocarbons containing planar rings in which every ring carbon forms one double bond and two single bonds are called aromatic hydrocarbons. A shorthand notation has been developed for writing aromatic structures by the use of polygons. A carbon atom is assumed to lie at each corner of the polygons. Carbon-carbon bonds are written in the polygons but carbonhydrogen bonds are not explicitly written. Figure 9-31 shows the same two structures as in Fig. 9-30. H-C' H-C, c II C %-H /C-H I H-C4 t ) C H--C\ I `CH H Fig. 9-30 I C ,C-H I1 H I Fig. 9-31 (b) The coplanarity of naphthalene indicates its aromatic character. Note in Fig. 9-32 that the two carbons at the fusion of the two rings reach their covalence of four without bonding to hydrogen. The shorthand notation of these structures is shown in Fig. 9-33. A C-H bond is assumed at each of those carbons possessing only three bonds within the carbon skeleton. H H H-C, H I I H I c H ,C\ I I c H ,C-H H II I I H I Fig. 9-32 Fig. 9-33 It is common practice to use the shorthand notation shown in Fig. 9-34 (a) and (b) to represent benzene and naphthalene, respectively. The circle emphasizes the delocalization of the electrons without the necessity of showing the resonance forms. CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 143 0 BONDING PROPERTIES 9.9. From the data of Problem 7.33, express the H-H H2 --+ bond energy in kJ/mol. 2H kJ/mol The bond energy is the energy needed to dissociate gaseous H2 into separated atoms. A H for this reaction is twice AH, for 1 mol of H. AH = 2(218 kJ/mol) = 436 9.10. Does Table 7-1 give sufficient data for the evaluation of the Br-Br bond energy? No; the energy of Br is given in the table with respect to the standard state for Br2, which is the liquid and not the gas. The bond energy is the energy required to dissociate individual Br2 molecules (i.e., gaseous Br2) into individual Br atoms. 9.11. Enthalpies of hydrogenation of ethylene (C2H4) and benzene (C6H6) have been measured, where all reactants and products are gases. C2H4 H2 C6H6 + 3H2 + + + C2H6 C6H12 A H = -137 kJ A H = -206 kJ Estimate the resonance energy of benzene. If C6Hhhad three isolated carbon-carbon double bonds, A H of hydrogenation would be close to three times A H of hydrogenation of C2H4, with one double bond, or -411 kJ. The fact that benzene's hydrogenation is less exothermic by 41 1 - 206 = 205 kJ means that benzene has been stabilized by resonance to the extent of 205 kJ/mol. 9.12. Estimate A H for the reaction: C2H5C1(g) C2H6k) + C12k) given the following average bond energies, in kJ/mol: C-C 348 C-H 414 C1-Cl 242 + - + HC1(g) C-Cl 327 H-Cl 43 1 It is first necessary to write structural formulas for all reactants and products to determine which bonds are broken and which are formed. H H H-C-C-H I I H H H Bonds broken C-H 414 C1-Cl 242 I / + C1-Cl 4 H-C-C-C1 H I H I I H / + H-Cl Bonds .formed C-Cl 327 H-Cl 431 144 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 Since bond breaking is endothermic (positive A H ) and bond formation is exothermic (negative A H ) , A H for the reaction is 414 + 242 - 327 - 431 = -102 kJ/mol (The accurate answer, calculated from AH; values, is - 1 13kJ/mol. The reason for the difference is that tabulated bond energies are averages over many compounds in which the particular bonds appear. In reality the energy required to break a bond in a particular compound depends mainly on the intrinsic character of that bond, but also to a slight extent on features of the molecule away from the bond.) 9.13. The dipole moment ( p ) of LiH is 1.964 x 10-29 C - m, and the interatomic distance between Li and H in this molecule is 159.6pm. Approximately what is the percent ionic character in LiH? Let us calculate the dipole moment of a hypothetical, 100 percent ionized Li+H- ion pair with a separation of 159.6pm, i.e., assuming a point charge at each nucleus. p(hypothetica1) = ( 1 electronic charge) x (separation) = (1.602 x 10-19 C)(1.596 x IO-" m) = 2.557 x 10-29C. m The approximate percent ionic character equals 100 percent times the fraction: actual dipole moment over the hypothe tical 1.964 x 10-29C.m = 76.8% ionic Percent ionic character = 100% x 2.557 x 10-29 C m 9.14. The dipole moments of SO2 and CO2 are 5.37 x 10-30 C - m and zero, respectively. What can be said of the shapes of the two molecules? Oxygen is considerably more electronegative than either sulfur or carbon. Thus each sulfur-oxygen and carbon-oxygen bond should be polar, with a net negative charge residing on the oxygen. If CO2 has no net dipole moment, the two C-0 bond moments must exactly cancel. This can occur only if the two bonds are in a straight line [Fig. 9-35(a)]. (The net moment of the molecule is the vector sum of the bond moments.) The existence of a dipole moment in SO2 must mean that the molecule is not linear, but bent [Fig. 9-35(b)]. + I = Sum 0 9.15. By the use of molecular-orbital considerations, account for the fact that oxygen gas is paramagnetic. What is the bond order in 02? Each oxygen atom has the ls22s22p4 configuration in the ground state. Aside from the K electrons of the two atoms in 0 2 , which are so deeply imbedded in their respective atoms as not to overlap with other electrons, the remaining 12 electrons (6 from each of the atoms) will fill the lowest of the available molecular orbitals as shown in Fig. 9-36. The bond axis is the x-axis. The electron configuration of O2 is: ( K electrons) d ,2 ; rz r 7; . a 07; 7; r * , , 1 1 The last two electrons went into the antibonding equi-energetic 7r* orbitals, one into 7r; and one into 7r:, so as to maximize electron spin in accordance with Hund's rule. These two unpaired electrons confer paramagnetism upon the molecule. Bond order - (number of electrons in bonding orbitals) - (number of electrons in antibonding orbitals) 2 \+ +/ + + CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 145 =s Oxygen AOs MOs Oxygen AOs Fig. 9-36. Electrons are indicated by arrows, orbitals by squares. K electrons are omitted. Note: The above energy level diagram is not relevant to the molecules B2 and C2, in which the order of the cPxand n,,z levels is reversed. 9.16. Explain the observations that the bond length in NT is 2pm greater than in N2, while the bond length in NO+ is 9pm less than in NO. The electron configurations should be written for the four molecules according to the buildup principle N2 ( K electrons) c~:cr;'c~~~~r~,~ ( K electrons) c~:c~;~cr~~r~,, ( K electrons) c ~ ~ ~ ; ' c r ~ ~ ~ r ; , ~ ~ ; , ' ~ ( K electrons) O ~ O J * ~ T ; , ~ ~ ; , ~ ~ 2 $ NO NO+ The computed bond orders for N2 and N2f are 3 and 2$, respectively. N2 therefore has the stronger bond and should have the shorter bond length. The computed bond orders for NO and N O + are 2f and 3, respectively. NO has the stronger bond and should have the shorter bond length. As opposed to ionization of N2, which involves the loss of an electron in a bonding orbital, ionization of NO involves the loss of an electron in an antibonding orbital. 9.17. Two substances having the same molecular formula, C4H80, were examined in the gaseous state by electron diffraction. The carbon-oxygen distance was found to be 143pm in compound A, and 124pm in compound B. What can you conclude about the structures of these two compounds? In compound A, the carbon-oxygen distance is the sum of the single-bond covalent radii of carbon and oxygen, indicating a single bond. 77 pm + 66 pm = 143 pm + 146 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 (6) Fig. 9-37 The oxygen must therefore be nonterminal. One such structure, the actual one selected for this experiment, is the heterocyclic compound tetrahydrofuran [Fig. 9-37 (a)]. In compound B, the carbon-oxygen distance is close to that predicted for a double bond, 122pm. The oxygen must therefore be terminal. One such structure, the actual one selected for this experiment, is 2-butanone [Fig. 9-37(b)]. SHAPES OF MOLECULES 9.18. Verify the value 8 = 109'28' for the central angles in a regular tetrahedron. A simple way of constructing a regular tetrahedron is to select alternating corners of a cube and to connect each of the selected corners with each of the other three, as in Fig. 9-38(a). Figure 9-38(b) shows triangle OAB, determined by the center of the cube, 0, which is also the center of the tetrahedron, and two corners of the tetrahedron, A and B. If P is the midpoint of AB, we see from right triangle OPA that -= 0 2 54"44' Fig. 9-38 9.19. The C-C single-bond distance is 154pm. What is the distance between the terminal carbons in propane, C3Hg? Assume that the four bonds of any carbon atom are pointed toward the corners of a regular tetrahedron. With reference to Fig. 9-38(b), two terminal carbons can be thought of as lying at A and B, while the Then central carbon is at 0. - AB = 2AP = 2 AU sin - = 2(154 pm)(sin 54'44') = 251 pm - (- ;) CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 147 9.20. Sulfur and chlorine combine in various proportions forming S2C12, SC12, and SC14. Draw the Lewis structures of these molecules and using VSEPR theory predict their shapes. Fig. 9-39 In S2C12 (a) each S has a VSEPR number of 4 so that each ClSS bond angle is about 109.5" (probably somewhat less because of extra repulsion from the unshared pairs on each sulfur), and the molecule cannot be straight. There is free rotation about the SS bond so that there is no fixed conformation of the molecule. In SC12 (b),the S atom has a VSEPR number of 4 also, so that the ClSCl angle is a little less than 109". The molecule is bent. In SC14 (c) the VSEPR number is 5, but one of the positions has an unshared pair. That must be one of the trigonal positions in the trigonal pyramid as in Fig. 9-40(a) in which there are two shared-unshared angles of 90", rather than an axial position (b), which has three shared-unshared angles of 90". The stick figure corresponding to Fig. 9-40(a) has been called a "see-saw," and also a "saw-horse," in which the axial ClSCl group represents a horizontal beam. (However, the unshared pair bends the ClSCl group slightly out of line.) c 1 9.21. What are the 0-N-0 bond angles in the nitrate and nitrite ions? Only one Lewis structure needs to be drawn to deduce the VSEPR number, which in both cases is 3 for the central N. (In NO, there are three a-bonded neighbors and no unshared pairs, and in NO, two neighbors and one unshared pair as shown in Fig. 9-41.) Thus the nominal bond angle is 120", exactly what is found in nitrate, NO,, since resonance makes all angles equal. In NOT, however, the unshared pair repels the bonding pairs more than they repel each other, forcing them closer to each other. We would expect a slightly smaller angle than 120"; 115" is found. '0 120" :.o. N . / \ 120" II 120" N v.-..2 NO,- ..o: >o;. U 115" NO,- Fig. 9-41 9.22. The P0Cl3 molecule has the shape of an irregular tetrahedron with the P atom located centrally. The Cl-P--CI angle is found to be 103.5". Give a qualitative explanation for the deviation of this structure from a regular tetrahedron. 148 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 The VSEPR number for the P atom is 4 so that the nominal bond angles are 109'28'. However, the Lewis structure for P0Cl3 shows a double bond between P and 0. (P is allowed to exceed the octet because of the availability of 3d orbitals.) The increased electron density in the P =0 bond would make the intrinsic repulsion between the P = O bond and a P-Cl bond greater than between two P-Cl bonds. Thus the C1- P -C1 angle is lowered and the C1- P =0 angle raised. 9.23. PCls has the shape of a trigonal bipyramid (Fig. 9-42) whereas IFS has the shape of a square pyramid (Fig. 9-43). Account for this difference. CI F I Fig. 9-42 Fig. 9-43 :F: :F: :F: Fig. 9-44 .. The Lewis structures of the singly bonded compounds, Fig. 9-44, reveal a VSEPR number of 5 in PCls, for which the trigonal bipyramid structure with 90, 120, and 180" angles is predicted (Table 9-4). However, the unshared pair on the iodine raises its VSEPR number to 6 so that the nominal bond angles are 90". The square pyramid structure of IF5 may be thought of as an octahedron (see Fig. 9-7) with the unshared pair pointing to the corner below the indicated horizontal plane. Because of the repulsion of the unshared pair one expects the I atom to be slightly below the base plane of the pyramid. 9.24. Which of the four carbon-carbon bond types in naphthalene would you predict to be the shortest? Refer to Fig. 9-33. The four different kinds of carbon-carbon bonds are represented by 1-2, 1-9, 2-3, and 9-10. (Every other carbon-carbon bond is equivalent to one of these four. For example, 6-7 is equivalent to 2-3, 7-8 is equivalent to 1-2, and so on.) The bond with the greatest double-bond character should be the shortest. Among the three resonance structures shown in Fig. 9-33, the frequency of double bonds for the various bond types is as follows: 2 in 2-2 [in (a)and (c)], 1 in 1-9 [in (b)],I in 2-3 [in (b)],and 1 in 9-10 [in (c)]. Bond 2-2 is expected to have the greatest double-bond character and thus the shortest length. This prediction is found to be true experimentally. The above four bonds are found to have lengths of 136.5, 142.5, 140.4, and 139.3pm, respectively. Note that the method of counting the number of resonance structures containing a double bond between a given pair of carbon atoms is very crude and cannot distinguish among the last three of the listed bond types, each of which shows a double bond in only one resonance structure. Even within the framework of the limited resonance theory, it would be necessary to know the relative weighting of each of the two equivalent structures (a) and ( b ) with the nonequivalent structure (c). CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 149 9.25. What is the shape of the triple iodide ion, I,? The Lewis structure reveals a VSEPR number of 5 for the central I atom, two bonded neighbors and three unshared pairs. To determine which corners of the trigonal bipyramid are occupied by the terminal I atoms, find the arrangement which maximizes the angles between the unshared pairs. The preferred arrangement [Fig. 9-45(a)] must be the one in which the unshared pairs are all at 120" because any other alternative [Fig. 9-45 (h) and (c)] would have two sets of pairs at 90". Thus the two terminal I atoms must occupy the axial positions, at 180" to each other, and the molecule is perfectly linear. I I Fig. 9-45 COORDINATION COMPOUNDS 9.26. Soluble compounds of the complex ion Co(NH3)F have a maximum in absorption of visible light at 437 nm. (a) What is the value of A for this complex ion, expressed in cm-', and what is the ion's color? (6) How many unpaired electrons would you expect this ion to have if it is considered low-spin, and how many if it is considered high-spin? A= (43;nm) - (E) = 22 900 cm-' The color of the ion is complementary to the light absorbed. Although the prediction of the color from the data is not absolute, since the color depends not only on the wavelength of the absorption maximum but also on the shape of the whole absorption band and on the color sensitivity of the human eye, fairly reliable conclusions can be drawn. The absorption, which peaks in the blue-violet region of the spectrum, would be expected to cover most of the blue region and part of the green. The color would be expected to be yellow. (b) The electron configuration of Co3+ is [Ar]3d6. The spin is due to unpaired d electrons. For low spin, the six d electrons would all be paired in the three t2, orbitals, and the spin would be zero. For high spin, the two eg molecular orbitals would be available as well (see Fig. 9-14). Four of the available orbitals would be singly occupied and one doubly occupied, in order to maintain the maximum number of unpaired electrons, four in this case. The A-value in this case is large enough to rule out high spin, and the ion is diamagnetic. 9.27. Predict the magnetic properties of (a) Rh(NH&+, and (b) CoFiThis problem can be approached by comparison with Problem 9.26. For analogous complexes of two different members of the same group in the periodic table, A increases with increasing atomic number. Since A for Co(NH3):' is already so high that the ion is low-spin, Rh(NH# would certainly be lowspin and diamagnetic. [The observed A for Rh(NH&" is 34 000 cm-' and the ion is diamagnetic.] (b) F- is a weak-field ligand, tending to form complexes with a low A-value, so that the ion would be expected to be high-spin, with four unpaired and parallel electron spins [compare with Problem 9-26(b)]. The measured A is 13 000 cm-', a low figure, and the ion is indeed paramagnetic. (a) 150 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 ISOMERISM 9.28. Write all the structural isomeric formulas for C4HgCl. The molecular composition resembles butane, C4HI0,with the exception that one H is replaced by C1. The formulas can be found by picking all the differently located hydrogens in the two isomeric formulas of C4HI0(Fig. 9-15). Note that the two end carbons of n-butane are identical and the two interior carbons are identical. Thus a chlorine substituted on the left-hand carbon would have given the same compound as (a) in Fig. 9-46, only viewed from a different end. Similarly, substitution on the carbon next to the left would have given a I I I I H-Cc-C-Cc-C--CI I I I I H H H H H H H H I H H-C, I H I H H-CC-CC-C-C-H I H I H I H I H-C H H C I H I I I I H I (b) Fig. 9-46 compound identical with (b). In isobutane, one isomer (C) has the C1 on the central C, and there is only one other isomer (4 because the three terminal carbons are identical. Because of the free rotation around the C-C single bond axis, it is immaterial which of the several hydrogens attached to a given C is substituted for; all positions become averaged in time anyway, because of the free rotation. 9.29. Write formulas for all the structural and geometrical isomers of C4H8. If the four carbons are in a row, there must be one double bond in order to satisfy the tetracovalence of carbon. The double bond occurs either in the center of the molecule or toward an end. In the former case, two geometrical isomers occur with different positions of the terminal carbons relative to the double bond; in the latter case, two structural isomers occur, differing in the extent of branching within the skeleton. Additional possibilities are ring structures without double bonds. Unlike N (Problem 9.5c), C can form threeatom rings. H / \ 1 H H / \ CH3 H3C / \ H H3C-CH2 H \ c=c / / \ H H2C-CH2 H Note the shorthand notation of grouping two or three hydrogens with the carbon to which they are bonded. It is understood, of course, that each hydrogen is bonded to the carbon of its group and that the carbon of the group is bonded to the adjoining carbon or carbons. CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 151 9.30. Which of the isomeric C4HgCl compounds would you expect to be optically active? Refer to Fig. 9-46. Compound (6) is the only one which would exist in optically active isomeric forms, because it is the only one which has a carbon atom (the one bonded to C1) which is bonded to four different groups. Every other carbon atom in this or the other structures is bonded to at least two hydrogen atoms or to two CH3 groups. 9.31. How many geometrical isomers could [Rh(py)3Cl,] have? (py is an abbreviation for the ligand pyridine.) There are only two possibilities, one [Fig. 9-47(a)] with the three chlorines occupying bonding positions cis to each other on one face of the octahedron and the three pyridines on the opposite face, and the other [Fig. 9-47(b)] in which two of the chlorines are tram to each other and two of the pyridines in turn are tram to each other. C I Cl Fig. 9-47 9.32. Some ligands are multifunctional; that is, they have two or more atoms that can bind to the central metal atom or ion. Each binding site occupies a different corner on the coordination surface. Ethylenediamine (abbreviated en) is such a ligand; the two binding atoms are nitrogens and the two binding sites must be cis to each other because of the shape and size of en. How many geometrical isomers of [Cr(en2)C12]-t should exist, and which isomer(s) might display optical activity? Two geometrical isomers exist, cis and tram (Fig. 9-48). Each en can be represented by an arc terminating at the two binding sites. By drawing other arrangements of the arcs while preserving the positions of the chlorines, one can see that (b)is a distinct mirror image of (a),whereas there is no mirror image of (c) except (c) itself. In other words, only the cis isomer can be optically active; the tram isomer is its own mirror image. Cl C I C I cis Fig. 9-48 152 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 BONDING IN METALS 9 3 . Explain why metals are usually lustrous (mirror-like). .3 In the band model there is a continuum of empty energy levels rather than discrete energy levels. Thus light quanta of all energies within a wide range of wavelengths will be absorbed equally, and then the energized electrons will re-emit the light when they fall back into their ground-state orbitals. This is the mechanism for reflection of light of all frequencies, which we call "luster." 9 3 . Predict how the Group I1 metals differ from Group I in density, melting point, and mechanical .4 strength. In any given period the Group I1 ions are smaller and may thus approach each other more closely. At the same time twice as many electrons are present in the electron sea. The closer approach and the much greater electrostatic interactions beween the 2 + ions and the sea of high negative charge density lead to greater density and much greater bonding energy, which in turn leads to much higher melting point and hardness. Actually going from Group I to Group I1 increases the density by a factor of about 2 and raises the melting point by hundreds of degrees. 9 3 . Metals usually feel cool to the touch compared with other materials because they are such good .5 conductors of heat. How can we explain this unusual thermal conductivity? In most materials heat is conducted by atom-to-atom transfer of vibrational motion from the hot end to the cold end. In metals the thermal energy is transferred primarily by the motion of the electron sea's free electrons, which are very mobile. Supplementary Problems FORMULAS 9.36. 9.37. Write formulas for the following ionic compounds: (a) lithium hydride, (b) calcium bromate, (c) chromium(I1) oxide, ( d ) thorium(1V) perchlorate, (e) nickel phosphate, (f)zinc sulfate. Ans. (a) LiH; (b) Ca(BrO&; (c) CrO; ( d ) Th(ClO&; (e) Ni3(P04)2; (f)ZnS04 9.38. Name the following compounds: (e) CaW04, cf) C0C03. (a) Mg(I0)2, (b) Fe2(S04)3, (c) CaMn04, (d) KRe04, Ans. (a) magnesium hypoiodite; (b) iron(II1) sulfate or ferric sulfate; (c) calcium manganate; ( ) potassium perrhenate; (e) calcium tungstate; (f)cobalt(I1) carbonate d 9.39. The formula of potassium arsenate is K3AS04. The formula of po tassium ferrocyanide [systematicallycalled potassium hexacyanoferrate(I1)l is &Fe(CN)6. Write the formulas of (a) calcium arsenate, (b) iron(II1) arsenate, (c) barium ferrocyanide, ( d ) aluminum ferrocyanide. Ans. (a) Ca,(AsO,),; (b) FeAs04; (c) Ba2Fe(CN)6; ( d ) A14[Fe(CN)6]3 CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 153 9.40. Draw Lewis structures for each of the following: (a) C2HCl, (b) C2H60, ( c ) C2H40, (d) NH30, (e) NO; (both oxygens terminal), (f)N204 (all oxygens terminal), ( g ) OF2. H Ans. (U) H-CeC-CI: ( b ) H,C-C-0-H i H H or I (c) H,C-C=O H I H,C-0-CH, H ( d ) H-4-0-H (9) :F-0-F: 9.41. Complete the following structures by adding unshared electron pairs when necessary. Then evaluate the formal charges. (Some of the following structures may be minor resonance forms.) (a) N = C - - C = N ( d ) Cl==C==N (6) N = = C = C = N (e) N = = N = O (c) C l - C G N ( f ) N-N-- 0 ci lo-fo] II (4 H-BR H-N\ H N I (1) H I I `B-H B H ,N-H II H-B H-N HN\ I `B/ H B-H N-H I I I Am. (a) all zero; (b) + 1 on one N (which does not have an octet), -1 on the other; ( c ) all zero; ( d ) + 1 on C1, - 1 on N; ( e ) - 1 on terminal N, + 1 on central N; (f') +1 on central N , -1 on 0; ( g ) all zero; (h) + I on doubly bonded C1, -1 on 0; (i) + I on N, -1 on singly bonded 0; ( j ) +1 on C1, - I on each singly bonded 0; ( k ) + I on each N, -1 on each B; ( I ) all zero. 9.42. Given that the formula for formaldehyde is CH20, one could draw three Lewis structures starting with the skeletons: H H-C-0--H, H-C-0, H There are 12 valence electrons of which six are used in each of the above skeletons. Use the other six to complete the octets about C and 0, determine the formal charges, and then decide which structure is correct. I or C-0-H. I 154 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 Clearly the middle structure is correct. 9.43. Three molecules of formaldehyde under appropriate conditions condense into one cyclic molecule. Draw its Lewis structure. The correct structure has alternating C and 0 atoms. H Ans. \ / C H :O .. / \.. 0: BONDING PROPERTIES 9.44. The chlorine-to-oxygen bond distance in ClO, is 144pm. What do you conclude about the valence-bond structures for this ion? Ans. Since the Cl-0 single-bond length estimated from Table 9-2 is 165pm, there must be considerable double-bond character in the bonds. 9.45. How many resonance structures can be written for each of the following aromatic hydrocarbons? Anthracene Phenanthrene Naph t hacene Consider double bonds only between adjacent carbons. Ans. ( a ) 4; (b) 5; (c) 5 9.46. The structure of 173-butadieneis often written as H2C=CH -CH =CH2. The distance between the central carbon atoms is 146pm. Comment on the adequacy of the assigned structure. Ans. The expected bond length is 77 + 77 = 154pm for a purely single bond. There must be non-octet resonance structures involving double bonding between the central carbons, such as 6~~-cH=cH-CH~. 9.47. The average C -C why? Ans. bond energy is 347 kJ/mol. What do you predict for the Si -Si single-bond energy and Because Si is a much bigger atom than C, there will be less orbital overlap (less electron sharing) and probably less than 300 kJ/mol bond energy. (b) Which of these species should have the shortest 9.48. ( a ) What are the bond orders for CN -,CN, and CN'? bond length? Ans. (a) CN-, 3; CN, 2;; CN', 2; (6) CN 9.49. Which homonuclear diatomic molecule(s) of second-period elements, besides 02, should be paramagnetic? A ~ s . B2 CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 155 9.50. Which supposed homonuclear diatomic molecules of the second-period elements should have zero bond order? Ans. Be2, Ne2 9.51. Dipole moments are sometimes expressed in debyes, where 1 debye = 10-l8 (esu of charge). cm The electrostatic unit (esu) of charge is defined by 1 C = 2.998 x 109 esu. What is the value of 1 debye in SI units? Ans. 3.336 x 10-30C . m 9.52. Where in space are the six electrons in benzene that we normally draw as a ring inside the hexagon? Ans. The six electrons occupy molecular orbitals, which are made by combining p-orbitals, all of which are directed perpendicular to the plane of the molecule but have no density in that plane. Hence the electron density can be depicted by two identical circular clouds, one above and one below the plane of the molecule. 9.53. The dipole moment of HBr is 2.60 x 10-30C . m, and the interatomic spacing is 141pm. What is the percent ionic character of HBr? Ans. 11.5% 9.54. The dipole moments of NH3, AsH3, and BF3 are (4.97,0.60, and 0.00) x 10-30 C . m, respectively. What can be concluded about the shapes of these molecules? Ans. NH3 and AsH3 are both pyramidal and BF3 is planar. From the dipole moments alone, nothing can be concluded about the relative flatness of the NH3 and AsH3 pyramids, because the electronegativities of N as As are different. 9.55. The As-Cl Ans. bond distance in AsC13 is 217pm. Estimate the single-bond covalent radius of As. 118pm. 9.56. The carbon-carbon double-bond energy in C2H4 is 615 kJ/mol and the carbon-carbon single-bond energy in C2Hs is 347 kJ/mol. Why is the double-bond energy appreciably less than twice the single-bond energy? cr orbital has greater electron overlap between the atoms because its component atomic orbitals are directed toward each other, whereas the component p orbitals making up the 7r orbital are directed perpendicularly to the internuclear axis, and have only side-to-side overlap. 9.57. Estimate A H for the reaction: CZHSC1(g) given the following bond energies in kJ/mol: HC1(g) -k C2H4(g) Ans. 9.58. +43 kJ bond given that A@ of HF(g) is -271 kJ/mol, and given the H-F 565 H-H 435 Estimate the bond energy of the F-F following bond energies in kJ/mol: Ans. 153kJ/mol 156 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 SHAPES OF MOLECULES 9.59. The platinum-chlorine distance has been found to be 232pm in several crystalline compounds. If this value applies to both of the compounds shown in Fig. 9-18, what is the chlorine-chlorine distance in (a) structure (a), (b) structure (b)? Ans. (a) 464pm, (6) 328pm 9.60. What is the length of a polymer molecule containing 1001 carbon atoms singly bonded in a line, if the molecule could be stretched to its maximum length consistent with maintenance of the normal tetrahedral angle within any C-C-C group? Ans. 126nm 9.61. A plant virus was examined by the electron microscope and was found to consist of uniform cylindrical particles 15.0nm in diameter and 300nm long. The virus has a specific volume of 0.73cm3/g. If the virus particle is considered to be one molecule, what is its molar mass? A ~ 4.4 . 10' ~ 9.62. Assuming the additivity of covalent radii in the C -C1 bond, what would be the chlorine-chlorine distance in each of the three dichlorobenzenes (Fig. 9-49)? Assume that the ring is a regular hexagon and that each C-Cl bond lies on a line through the center of the hexagon. The distance between adjacent carbons is 140pm. qcl c 1 (b) c1 gycl Fig. 9-49 Ans. (a) 316pm; (b) 547pm; (c) 632pm 9.63. Estimate the length and width of the carbon skeleton in anthracene [see Problem 9-43(a) for a diagram]. Assume hexagonal rings with equal carbon-carbon distances of 140pm. Ans. 730pm long, 280pm wide 9.64. BBr3 is a symmetrical planar molecule, all B- Br bonds lying at 120" to each other. The distance between Br atoms is found to be 324 pm. From this fact and given that the covalent radius of Br is 114pm, estimate the covalent radius of boron, assuming that the bonds are all single bonds. Ans. 73pm 9.65. What is the VSEPR number for the central atom in each of the following species? (Hint: Draw the Lewis structures first.) (a) SO^, (b) SO^, ( c ) SO:-, ( d ) SO:-, ( e ) S F ~ , (f)s:-. Ans. (a) 3; (b) 3; (c) 4; ( d ) 4; (e) 6; (f)4 9.66 What are the bond angles in each of the species in the preceding problem? Ans. (a) slightly less than 120"; (b) 120"; (c) slightly less than (f)slightly less than 109'28' 109"28'; (d) 109'28' ( e ) 90"; 9.67. For each of the following species give the VSEPR number, the general shape, and the bond angles. (a) XeF4, (b) Xe03, (c) XeF2. CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 157 Ans. (a) VSEPR number 6, square planar, 90'; (b) VSEPR number 4, trigonal pyramid with unshared pair at apex, slightly less than 109'28'; (c) VSEPR number 5 , linear, 180" 9.68. Which of the following molecules or ions is/are straight (i.e., all bond angles are 180)? (a) OF2, (b) HCN, (c) H2S, (4 CO2, ( e ) IF;. AnS. (b), (4, (4 9.69. Which of the following molecules or ions is/are bent? (a) BeC12, (b) HOC1, (c) HOT, ( d ) NHT, (e) NT. Ans. (4, (4, (4 9.70. Which of the following molecules or ions is/are flat (i.e., all atoms are in same plane)? (4 BF3, (h) Xe04, (4 NO,, (4 C2H2, (4 HN3. Ans. (a), (c), (d), (e); (6) is perfectly linear, (e) is bent at H end. 9.71. Which of the following molecules or ions is expected to have a dipole moment? (a) CH2C12, (b) cis-C2H2C12, (c) trans-C2H2C12, ( d ) CH2CC12, (e) SF4, (h) H2so4, (i) NH,f, ( j ) N2134, ( k ) NC13. AnS. (f)XeF4, ( g ) C2F4, (4, (b), (4, (4, (4, ( k ) 9.72. Disregarding any hydrogen atoms, which of the following molecules or ions would you expect to be flat? (a) CH,,CHCHCl, (b) HN03, (c) HZPOT, ( d ) SOC12, (e) C6H50H (i.e., hydroxybenzene). Ans. ( 4 9 (b), (4 COORDINATION COMPOUNDS 9.73. Ans. (a) pentaamminebromocobaIt(II1) sulfate (b) dichlorobis(ethylenediamine)chromium(III) chloride (c) tetrapyridineplatinum(I1) tetrachloroplatinate(I1) (d) potassium hexafluoronickelate(1V) (e) potassium carbonylpentacyanoferrate(I1) (f) cesium pentafluorotellurate(1V) 9.74. Write formulas for the following compounds, using brackets to enclose the complex ion portion: (a) triamminebromoplatinum(I1) nitrate (b) dichlorobis(ethylenediamine)cobalt(II) monohydrate, (c) pentaamminesulfatocobalt(II1) bromide, ( d ) potassium hexafluoroplatinate(IV), (e) tetraaquadibromochromium(II1) chloride, ( f ) ammonium heptafluorozirconate(1V). Ans. ( a ) [Pt(NH,.), Br]NO, ( b ) [ C ~ ( e n ) ~ C H~ l0 .l 2 (c) [CO( NH, ) .$SO4] Br ( e ) [Cr(H,0),Br2]C1 (4 K2[PtF61 (f (NH4)3IzrF71 9.75. A for IrCli- is 27 600 cm-I (a) What is the wavelength of maximum absorption? (b) What would you predict for the magnetic behavior of this ion? Ans. (a) 362nm; (b) diamagnetic 158 CHEMICAL BONDING AND MOLECULAR STRUCTURE [CHAP. 9 9.76. (a) What is the maximum number of unpaired electrons which a high-spin octahedral complex of the first transition series could possess in the ground state? (b) What elements could show this maximum, and in what oxidation states? Ans. (a) 5; (6) Mn(II), Fe(1II) 9.77. If the octahedrally complexing metal of the first transition series has a d' configuration (in its relevant oxidation state), for what values of i could magnetic properties alone distinguish between strong-field and weak-field ligands? Ans. 4,5, 6, 7 9.78. Both Fe(CN)z- and Fe(H20)2' appear colorless in dilute solutions. The former ion is low-spin and the latter is high-spin. (a) How many unpaired electrons are in each of these ions? (b) Why should both ions be colorless, in view of the apparent significant difference in their A-values? 4 Ans. (a) 0 in Fe(CN)%-, in Fe(H20)i'. ( h ) A for Fe(CN)z- is so large that the absorption peak is in the ultraviolet; A for Fe(H20):' is so small that its absorption peak is in the infrared. Practically no visible light is absorbed by either. 9.79. Hexaaquairon(II1) ion is practically colorless. Its solutions become red when NCS- is added. Explain. (Compare with Problem 9.78) Ans. H20 is not a strong-field ligand, as noted in Problem 9.78. NCS- has vacant T* orbitals that overlap the tZg orbitals of the metal and so can accept electron density from the metal. This "back-bonding" increases the strength of the metal-ligand bond and lowers the t2g energy level, thus making NCS- a strong-field ligand. A strong-field ligand causes an increase in A and thus a lowering of the wavelength of maximum d-d absorption from the near-infrared well into the visible region (actually into the blue-green). ISOMERISM 9.80. 9.81. Among the paraffin hydrocarbons (CnH2n+2, where n is an integer), what is the empirical formula of the compound of lowest molar mass which could demonstrate optical activity in at least one of its structural isomers? 9.82. How many structural and geometrical isomers can be written for the following without counting ring compounds: (a) C3H5CI, ( h ) C3H4C12, ( c ) C4H7C1, ( d ) C5H10? Ans. ( a ) 4; ( b ) 7; ( c ) 11; (d) 6 9.83. For the square coplanar complex [Pt(NH3)(NH20H)py(N0,)1+, how many geometrical isomers are possible? Draw them. Ans. There are three isomers. Any one ligand can be truns to any of the other three; the two ligands not trans to the first have their positions automatically fixed as trans to each other. 9.84. Predict whether [I~-(en)~]~' should exhibit optical isomerism. If so, prove by diagrams that the two optical isomers are not simple rotational aspects of the same compound. Ans. There are two optical isomers, which may be represented as in Fig. 9-50. CHAP. 91 CHEMICAL BONDING AND MOLECULAR STRUCTURE 159 P e n en I lin Fig. 9-50 9.85. en How many isomers are there with the formula [M en XY31, where M is the central metal, en is the bidentate ligand ethylenediamine, and X and Y are monodentate ligands? Explain in words how they differ. Ans. There are two geometric isomers, in one of which two of the Ys are trans to each other, and in the other all Ys are cis. BONDING IN METALS 9.86. In silicon there is a gap in energy between the band of bonding MOs and a band of an equal number of antibonding MOs, all derived from the 3s and 3p atomic orbitals. Is silicon a metallic electrical conductor? Explain. Ans. No. I n a crystal of N atoms there will be 4N MOs, 2N of them bonding. The 4N valence electrons will just fill these 2N bonding MOs; but metallic conductivity requires that the band be only partially filled. 9.87. How does a rise in temperature affect the electrical conductivity of a metal? Explain. rise in temperature lowers the conductivity because vigorous atomic motion disrupts the long-range order of the lattice, thus obstructing the MOs that pervade the whole crystal. 9.88. Most metals are ductile and malleable, in contrast to the brittleness of most other solids. Account for these properties in terms of the 'electron-sea model. Ans. Since the atoms are rather far apart and the electron sea offers little resistance to deformation, it does not require much energy to cause one layer of atoms to slide past another. Thus the crystal is deformed rather than shattered by an external stress. This page intentionally left blank Chapter 10 Solids and Liquids Solids cannot be understood as well as the much simpler units of matter, the atoms and molecules. There are some simplifying features of solids, however, which allow considerable insight into their nature. One of these features is the regularity in the structure of all crystalline solids, as a consequence of which an entire macrocrystal consists of a three-dimensional replication of a basic unit of atoms, ions, or molecules arranged in a fixed way. CRYSTALS The arrangement of the simpler particles in a crystalline array is called a lattice. Every lattice is a three-dimensional stacking of identical building blocks called unit cells. The properties of the entire crystals, including its overall symmetry, can be understood in terms of the unit cell. The description of crystals has been systematized by the mathematical finding that there is a rather small number of arrangements of identical points, 14 to be specific, fulfilling the requirement that the arrangement, extended by pure translation, can fill out all of three-dimensional space. These 14 arrangements are called Bravais lattices. In this book only the three Bravais lattices having cubic symmetry will be considered in detail: the simple cubic, the face-centered cubic, and the body-centered cubic. The corresponding unit cells are represented in Fig. 10-1. Actually, the lattice points (corners of the cubes and centers of the edges or faces) represent the centers of the atoms or ions occupying the lattice. The atoms or ions by themselves are not points but three-dimensional objects which usually are in contact with each other. The representations of these objects in Fig. 10-1 are purposely shrunk for clarity of visualization. The length of the cube edge is designated by a. A crystal having any of these three lattices can be thought of as a three-dimensional stack of unit cell cubes, packed face to face so as to fill all of the space occupied by the crystal. Crystal classes less symmetrical than the cubic have unit cells which may be thought of as more or less distorted cubes, the opposite faces of which are parallel to each other. (The name of such a general geometric shape is a parallelepiped.) Thus, crystals with hexagonal symmetry, like snow (ice), have unit cells that are prisms with a vertical axis perpendicular to a rhombus-shaped base, the equal edges of which are at 60" and 120" with respect to each other. A typical hexagonal unit cell is shown in Fig. 10-2. The length of the rhombus edge is designated by a, and the height of the unit cell by c. Note that a hexagonal prism can be formed by joining three unit cells. The bulk density of a perfect crystal can be computed from the properties of the unit cell. It is necessary to apportion the mass of the crystal among the various unit cells, and then to divide the mass W a W (a) Simple cubic ( b ) Face-entered cubic (c) Body-centered cubic Fig. 10-1. Unit Cells of Cubic Symmetry 161 162 SOLIDS AND LIQUIDS [CHAP. 10 D C 60" L4 Fig. 10-2. A Hexagonal Unit Cell Fig. 10-3. Stack of Eight Cubic Unit Cells apportioned to one unit cell by the volume of the unit cell. In computing the mass of a unit cell, it is necessary to assign to the cell only that fraction of an atom which lies in that cell alone. If the unit cell is cubic, we see from Fig. 10-3 that Mass per unit cell = (mass of atoms, like A , on corners of the unit cell) + $ (mass of noncorner atoms, like C, on unit cell edges) ++(mass of nonedge atoms, like B, on the faces of the unit cell) (mass of interior atoms, like G, within the unit cell) + since a corner atom is shared by eight unit cells, etc. Note that the formula holds whether or not all the atoms belong to the same species. It also holds for noncubic cells. Coordination number The coordination number of an atom in a crystal is the number of nearest-neighbor atoms. The coordination number is constant for a given lattice [see Problems lO.l(b) and 10.12(4]. Close packing There are two simple lattice structures that allow a high degree of packing of the atoms in a crystal. A structure in which identical spheres (the atoms) occupy the greatest fraction of the total space is called a close-packed structure. It can be achieved by packing on top of each other two-dimensional closepacked layers, in each of which every sphere is surrounded by a regular hexagonal arrangement of six other spheres, as in Fig. 10-4. In the figure, the large circles represent the spheres in one such layer, and the squares surround the centers of spheres in the adjoining superimposed layer. If the third and every odd-numbered layer is made up of spheres directly over the spheres in the first layer and if every evennumbered layer is made up of spheres directly over the spheres in the second layer, the structure is called hexagonal close-packed; the corresponding unit cell for the centers of the spheres is given in Problem 10.12. If instead the structure is a regular alternation of three kinds of layers, the third being made up of spheres whose centers are enclosed by the small circles in Fig. 10-4, the structure is cubic close-packed; its unit cell is just the face-centered cubic. The close-packed layers are perpendicular to a body diagonal of the unit cell [see the dashed triangles in Fig. 10-l(b)]. Both the hexagonal close-packed and the cubic close-packed lattices have the same 74 percent filling of space by spheres in contact [see Problem 10.1(4],and the two lattices also have the same coordination number, 12. CHAP. 101 SOLIDS AND LIQUIDS 163 Fig. 10-4 The void spaces, or holes, in a crystal can sometimes admit foreign particles of a smaller size. Thus an understanding of the geometry of these holes becomes important. In the cubic close-packed structure, the two major types of holes are the tetrahedral and the octahedral holes. In Fig. lO-l(b), tetrahedral holes are in the centers of the indicated minicubes of side a/2. Each tetrahedral hole has four nearestneighbor occupied sites. The octahedral holes are in the body center and on the centers of the edges of the indicated unit cell. Each octahedral hole has six nearest-neighbor occupied sites. CRYSTAL FORCES The strength of the forces holding crystalline substances together varies considerably. In molecular crystals, like CO2 and benzene, each molecule is almost independent of all the others and retains practically the same internal geometry (bond lengths and angles) that it has in the gaseous or liquid state. The crystal is held together by the relatively weak van der Waals forces, and the melting point is never very high. For substances that can undergo intermolecular hydrogen bonding, the intermolecular forces in the crystal can be great enough to impose a noticeable change in the molecular geometry. Hydrogen bonding is the attraction, largely electrostatic in nature, between the positive charge-carrying hydrogen of the polar bond of one molecule (or part of a molecule) and the negative charge-carrying atom of the polar bond of another molecular (or part of the same molecule). The most important polar bonds that allow hydrogen bonding are those between hydrogen and the most electronegative elements: fluorine, oxygen, nitrogen, and chlorine. Water is an example; the angle between the two 0-H bonds in the vapor is about 105' but equals the tetrahedral 109'28' in the crystal, in conformity to the crystalline, rather than molecular, spatial requirements. In metals, a special type of crystalline force comes into play, characterized by a largely nondirectional nature. Fixed bond angles do not play a major role in metals, and the stablest crystal structures for most elementary metals are those with the densest packing. Important exceptions are the Group I metals 164 SOLIDS AND LIQUIDS [CHAP. 10 Table 10-1 Ionic Radii Ion + Radius/pm 60 95 133 Ion Radius/pm 97 69 50 208 136 181 195 216 140 184 Ai3+ Mg2+ Ca2+ Sr2+ Ba2+ Zn2+ and iron, which have the body-centered cubic structure. In the covalent crystals, like diamond or silicon carbide, the crystal is held together by a three-dimensional network of covalent bonds, the mutual angles of which are determined largely by the covalent bonding requirements of the individual atoms. IONIC RADII The attractive forces operating in ionic crystals are mostly electrostatic in nature, the classical attraction between oppositely charged particles. To avoid the repulsion between similarly charged particles, ionic substances crystallize in structures in which a positive and a negative ion can come within touching distance, while ions of like charge are kept away from each other. In fact, the dimensions of most simple purely ionic crystals can be understood by assuming an ionic radius (Chapter 9) for each ion, valid for all compounds of that ion, and taking the smallest cation-anion separation to be the sum of the ionic radii of the cation and anion. Radii for some elementary ions are listed in Table 10-1. FORCES IN LIQUIDS The kinds of forces that bind atoms, ions, and molecules in liquids are the same as those discussed for solids. Although just above the melting point these forces are insufficient to restrain the atoms, ions, or molecules to their lattice positions, in most cases they are strong enough to prevent vaporization. Liquid metals and salts are not very familiar except in industrial technology. Since the forces are so strong the temperatures required for melting are high and the temperature range for the liquid state is very great. Most familiar liquids consist of molecular substances, e.g., water, benzene, bromine, which freeze to molecular solids. The molecules are bound to their neighbors by weak forces, the strongest of which, by far, are hydrogen bonds. Others are known collectively as van der Waals forces. All atoms and molecules are attracted to one another. If a molecule has a permanent dipole moment (Chapter 9), dipole-dipole attraction makes a significant contribution to the weak force. But even in the absence of permanent dipoles, weak forces known as London forces exist, due to transient dipoles of very short duration, as explained by Fritz London. Generally, the greater the atomic number of the atoms in contact and the greater the area of contact, the greater are the London forces between molecules. The strength of the weak forces is revealed by the volatility of the substance, the greater the force, the higher the boiling point. Thus among the noble gases there is a steady increase in boiling point as atomic number increases and the London force increases. Comparing SiC14(b.p. 57.6"C) to PC13(b.p. 75.5"C) we see the contribution of the dipole-dipole attractions in the latter case. The contributions of + + Li Na K+ cs 169 126 65 Cd2+ Ni2+ AP+ HF- c1Br102S2- 99 113 135 74 CHAP. 101 SOLIDS AND LIQUIDS 165 dipole-dipole attractions and hydrogen bonding are very important in comparing organic compounds, as illustrated by C2H6(b.p. -89"C, London forces only), CH3F(b.p. -78"C, dipole), and CH30H(b.p. 65"C, H bonding). Among the compounds of formula CnHzn+2the straight-chain compound always has a boiling point higher than any of its branched-chain isomers, because of the increased area of contact of such a molecule with its surroundings. Solved Problems CRYSTAL DIMENSIONS 10.1. Metallic gold crystallizes in the face-centered cubic lattice. The length of the cubic unit cell [Fig. 10-l(b)] is a = 407.0pm. (a) What is the closest distance between gold atoms? (b) How many "nearest neighbors" does each gold atom have at the distance calculated in (a)? (c) What is the density of gold? (6) Prove that the packing factor for gold, the fraction of the total volume occupied by the atoms themselves, is 0.74. (a) Consider a corner gold atom in Fig. 10-l(b). The closest distance to another corner atom is a. The distance to an atom at the center of a face is one-half the diagonal of that face, i.e., which is less than a Thus the closest distance between atoms is -= 287.8 pm 407.0 Jz (b) The problem is to find how many face-centers are equidistant from a corner atom. Point A in Fig. 10-3 may be taken as the reference corner atom. In that same figure, B is one of the face-center points at the nearest distance to A . In plane ABD in the figure there are three other points equally close to A : the centers of the squares in the upper right, lower left, and lower right quadrants of the plane, measured around A . Plane ACE, parallel to the plane of the paper, also has points in the centers of each of the squares in the four quadrants around A . Also, plane ACF, perpendicular to the plane of the paper, has points in the centers of each of the squares in the four quadrants around A . Thus there are 12 nearest neighbors in all, the number expected for a close-packed structure. The same result would have been obtained by counting the nearest neighbors around B, a facecentered point. For the face-centered cubic structure, with eight corners and six face-centers, (c) Mass per unit cell = (8m) f (6m) = 4m + where m is the mass of a single gold a!om, 197.0U. Converting to grams, m = (197.0 U) = 3.27 x 10-22 g (6.023 x 1023 U and ' ) (d) The reverse of this type of calculation can be used for a precise determination of Avogadro's number, provided the lattice dimension, the density, and the atomic mass are known precisely. Since the atoms at closest distance are in contact in a close-packed structure, the closest distance between centers calculated in (a), a/&, must equal the sum of the radii of the two spherical atoms, From (c), there are four gold atoms per unit cell. Then, 2r. Thus, r = a / a / 2 = 166 SOLIDS AND LIQUIDS [CHAP. 10 Volume of 4 gold atoms = 4(:7rr3) and Packing fraction = = 4(:~) (-&) =3Jz lra3 volume of 4 gold atoms - 7ra3/3& volume of unit cell 7r Note that the parameter U for the gold unit cell canceled, so that the result holds for any cubic closepacked structure. The metallic radius calculated above, Y = ~ r / 2 ~= 143.9pm, is different from both " an ionic radius and a covalent radius. 10.2. Show that the tetrahedral and octahedral holes in gold are appropriately named. Find the closest distance between an impurity atom and a gold atom if the impurity atom occupies (a) a tetrahedral hole, (b) an octahedral hole. How many holes of each type are there per gold atom? Examine Fig. 10-l(b), and imagine a hole in the center of the upper left front minicube. This hole is equidistant from the four occupied corners of the minicube, the common distance being half a body diagonal of the minicube, or These four occupied corners define a regular tetrahedron (see Problem 9.18), and the center of the tetrahedron is the point equidistant from the corners, which we showed is the location of the hole. This justifies the name "tetrahedral hole." As the unit cell contains 8 tetrahedral holes (one in each minicube) and 4 gold atoms (Problem lO.l), there are 8 + 4 = 2 tetrahedral holes per gold atom. Now consider the hole at the center of the unit cell of Fig. 10-l(b). This hole is equidistant from the centers of all six faces of the unit cell, all of which are the nearest occupied sites to the hole. These six points are the vertices of an eight-faced figure; the faces are congruent equilateral triangles (whose edges are face diagonals of the minicubes). Such a figure is a regular octahedron, and the hole is at its center; hence the name "octahedral hole." The distance between the hole and a nearest-neighbor atom is a/2. A similar proof can be made for an octahedral hole on the center of an edge of the unit cell in Fig. lO-l(b) if we note that the actual crystal lattice consists of a three-dimensional stack of unit cells, as in Fig. 10-3. Each such edge-center hole is shared by four unit cells, and there are 12 edges in a cube, so that the number of octahedral holes per unit cell is 1$.$(12) = 4 The ratio of octahedral holes to gold atoms is thus 4 to 4, or 1. Note the competing advantages of tetrahedral and octahedral holes for housing impurities or second components of an alloy. If the crystal forces, whatever their nature, depend mostly on interactions between nearest neighbors, the octahedral hole has an advantage in having more nearest neighbors to interact with (6, as opposed to 4 . However, the tetrahedral hole has a shorter nearest-neighbor ) distance (afi/4 = 0.433a, as opposed to O.SOOa), giving it the advantage of a greater potential interaction with any one host atom. An octahedral hole, with its larger nearest-neighbor distance, can accommodate a larger impurity or alloying atom than can a tetrahedral hole, without straining the host lattice. 10.3. CsCl crystallizes in a cubic structure that has a C1- at each corner and a Cs+ at the center of the unit cell. Use the ionic radii listed in Table 10.1 to predict the lattice constant a and compare with the value of a calculated from the observed density of CsC1, 3.97g/cm3. Figure 10-5(a) shows a schematic view of the unit cell, where the open circles represent C1- and the filled circles Cs'. The circles are made small with respect to the unit cell length a in order to show more clearly the locations of the various ions. Figure 10-5(b) is a more realistic representation of the right triangle ABC, showing anion-cation-anion contact along the body diagonal A C. Let us assume that the closest CsS-to-C1- distance is the sum of the ionic radii of Cs' and Cl-, or CHAP. 101 SOLIDS AND LIQUIDS 167 Fig. 10-5 169 + 181 = 350 pm This distance is one-half the cube diagonal, or afi/2. Then -:= U f i (b) 2 350 pm or a= 2(350 pm) fi = 404 pm The density can be used to compute U if we count the number of ions of each type per unit cell. The number of assigned Cl- ions per unit cell is one-eighth of the number of corner C1 ions, or (8) = I . The only Cs+ in the unit cell is the center Cs', so that the assigned number of cesium ions is also 1 . (This type of assignment of ions or atoms in a compound must always agree with the empirical formula of the compound, as the 1 : 1 ratio in this problem does.) The assigned mass per unit cell is thus that of 1 formula unit of CsCI, 132.9 + 35.5 6.02 x 1023 Volume of unit cell = u3 = whence U - 2.797 x lop*' g x mass ~ density - 2.797 - g 3.97 g/cm3- = 70.4 x 10-24 cm3 = v70.4 x cm3 = 4.13 x 10-* cm = 413 pm This value, based on the experimental density, is to be considered the more reliable since it is based on a measured property of CsC1, while the ionic radii are based on averages over many different compounds. Unit cell dimensions can be measured accurately by X-ray diffraction, and from them the theoretical density can be calculated. The measured density is usually lower because most samples which are large enough to measure are not perfect single crystals and contain empty spaces in the form of grain boundaries and various crystalline iniperfections. The CsC1 structure is not described as body-centered, since the particle occupying the center is different from the particles occupying the corners of the unit cell. There are two ways of describing the structure. One way is to say that Cs' occupies the central holes of the Cl-. simple cubic lattice. Another way is to say that the structure is made up of two interpenetrating simple cubic lattices, one made up of C1- and one of Cs'. The Cs+ lattice is displaced from the C1- lattice along the direction of the unit cell diagonal by one-half the length of the unit cell diagonal. 10.4. The CsCl structure (Fig. 10-5) is observed in alkali halides only when the radius of the cation is sufficiently large to keep its eight nearest-neighbor anions from touching. What minimum value for the ratio of cation to anion radii, r + / r - , is needed to prevent this contact? 168 SOLIDS AND LIQUIDS [CHAP. 10 In the CsCl structure, the nearest cation-anion distance occurs along the diagonal of the unit cell cube, while the nearest anion-anion distance occurs along a unit cell edge. This relationship is shown in Fig. 10-5(b). In the figure, - A B = ~ BC=aJZ AC=ad3 If we assume anion-cation contact along AC, then #c = 2(r+ + Y-) = a d . In the limiting case, where anions touch along the edge of the unit cell, 2r- = a. Dividing the former equation by the latter, If the ratio were less than this critical value, anions would touch (increasing the repulsive forces) and cation and anion would be separated (decreasing the attractive forces). Both effects would tend to make the structure unstable. 10.5. Ice crystallizes in a hexagonal lattice. At the low temperature at which the structure was determined, the lattice constants were a = 453 pm and c = 741 pm (Fig. 10-2). How many H20 molecules are contained in a unit cell? The volume V of the unit cell in Fig. 10-2 is V = (area of rhombus base) x (height c ) = (a2 sin 60")c = (453 pm)'(O.866)(741 pm) = 132 x 106 pm3 = 132 x 10-24 cm3 Although the density of ice at the experimental temperature is not stated, it could not be very different from the value at O"C, 0.92g/cm3. Mass of unit cell = V x density = (132 x cm3)(0.92g/cm3)(6.02 x 1023u/g) = 73 U This is close to 4 times the molecular mass of H20; we conclude that there are four molecules of H20 per unit cell. The discrepancy between 73 U and the actual mass of four molecules, 72 U, is undoubtedly due to the uncertainty in the density at the experimental temperature. 10.6. BaTi03 crystallizes in the perovskite structure. This structure may be described as a bariumoxygen face-centered cubic lattice, with barium ions occupying the corners of the unit cell, oxide ions occupying the face centers, and titanium ions occupying the centers of the unit cells. (a) If titanium is described as occupying holes in the Ba-0 lattice, what type of hole does it occupy? (b) What fraction of the holes of this type does it occupy? (c) Can you suggest a reason why it occupies those holes of this type which it does but not the other holes of the same type? These are octahedral holes. The octahedral holes at the centers of unit cells constitute just one-fourth of all the octahedral holes in a face-centered cubic lattice. (See Problem 10.2.) An octahedral hole at the center of a unit cell, occupied by a titanium ion, has six nearest-neighbor oxide ions. The other octahedral holes, located at the centers of the edges of the unit cell, have six nearest neighbors each, as is the case with any octahedral hole, but two of the six neighbors are barium ions (at the unit-cell corners terminating the given edge) and four are oxide ions. The proximity of two cations, Ba2+ and Ti4+, would be electrostatically unfavorable. CRYSTAL FORCES 10.7. The melting point of quartz, one of the crystalline forms of Si02, is 1 610"C, and the sublimation point of CO2 is -79C. How similar do you expect the crystal strucures of these two substances to be? CHAP. 101 SOLIDS AND LIQUIDS 169 The big difference in melting points suggest a difference in type of crystal binding. The intermolecular forces in solid CO2 must be very low to be overcome by a low-temperature sublimation. CO2 is actually a molecular lattice held together only by the weak van der Waals forces between discrete CO2 molecules. Si02, on the other hand, is a covalent lattice, with a three-dimensional network of bonds; each silicon atom is bonded tetrahedrally to four oxygens and each oxygen is bonded to two silicons. 10.8. In the hexagonal ice structure (Fig. 10-6 showing the positions of the oxygen atoms), each oxygen is coordinated tetrahedrally with four other oxygens, with an intervening hydrogen between adjoining oxygens. A H of sublimation of ice at 0C is 51.0 kJ/mol H20. It has been estimated by comparison with non-hydrogen-bonded solids having intermolecular van der Waals forces similar to those in ice that the A H of sublimation would be only 15.5 kJ/mol H 2 0 if ice were not hydrogen-bonded. From these data estimate the strength of the hydrogen bond in ice. The excess of A H of sublimation above that of a non-hydrogen-bonded solid can be attributed to hydrogen bonds. AH(excess) = 51 .O - 15.5 = 35.5 kJ/mol H,O Each H 2 0 is hydrogen-bonded to four other H20 molecules, through 0-H---0 linkages (indicated in Fig. 10-6 only for the two interior molecules). Each such hydrogen-bonded linkage is shared by two H 2 0 molecules (to which the two oxygen atoms belong). Thus, on the average, each H20 can be assigned 4 halves, or 2 hydrogen bonds. AH(hydrogen bond) = Observe from Fig. 10-6 that 35.5 kJ/mol H 2 0 = 17.8 kJ/mol hydrogen bond 2 mol H-bonds/mol H,O Four H 2 0 molecules are uniquely assigned to the unit cell. This is in agreement with the result of Problem 10.5. 10.9. Which would have the higher melting point (a) V or Ca, (b) MgO or KCl? Explain. (a) V; vanadium would have a much greater charge density in the electron sea since it has three 3d electrons in addition to the two 4s electrons to contribute. Also the ionic cores, being smaller, will r a Fig. 10-6 170 SOLIDS AND LIQUIDS [CHAP. 10 be closer together. Thus the crystal forces will be greater. The actual melting points are V 1 890C, Ca 845C. The relative closeness of the cores is indicated by the densities: V 6.11 g/cm3,Ca 1.55 g/cm3 (b) MgO; the crystal forces will be greater on two accounts: (1) the smallness of both cation and anion in MgO allows closer approach, (2) both cation and anion carry twice as much charge. (Recall that electrostatic force is proportional to the product of the charges and inversely proportional to the square of the distance between them.) Actual melting points are MgO 2800"C, KCl 776C. FORCES IN LIQUIDS 10.10. In each case indicate which liquid will have the higher boiling point and explain why. (a) (c) C12 or Br2 (a) CO2 or SO2 (b) (CH3)2CHCH(CH3)2 CH3CH2CH2CH2CH2CH3 or ( d ) C2H5SHor C2H50H SO2. The molecule is bent and has a permanent dipole moment, in contrast to CO2 which is straight. (Note: CO2 is never liquid at 1 atm pressure but sublimes directly from the solid to the gas at this pressure.) The (b) CH3CH2CH2CH2CH2CH3. straight-chain isomer has a greater area of contact with its neighbors. (c) Br2. The higher atomic number leads to a greater London force of attraction between molecules. ( d ) C2H50H. Hydrogen bonding is present in the oxygen compound but is negligible in the sulfur compound. Supplementary Problems CRYSTAL DIMENSIONS 10.11. Potassium crystallizes in a body-centered cubic lattice, with a unit cell length a = 520pm. (a) What is the distance between nearest neighbors? (b) What is the distance between next-nearest neighbors? (c) How many nearest neighbors does each K atom have? ( d ) How many next-nearest neighbors does each K have? (e) What is the calculated density of crystalline K? Ans. (a) 450pm; (b) 520pm; (c) 8; (d) 6; (e) 0.924g/cm3 There is an atom at 10.12. The hexagonal close-packed lattice can be represented by Fig. 10-2, if c = a d = 1.633~. each corner of the unit cell and another atom which can be located by moving one-third the distance along the diagonal of the rhombus base, starting at the lower left-hand corner, and moving perpendicularly upward by c/2. Mg crystallizes in this lattice and has a density of 1.74g/cm3. (a) What is the volume of the unit cell? (b) What is a? (c) What is the distance between nearest neighbors? (d) How many nearest neighbors does each atom have? Ans. (a) 46.4 x 106pm3; (b) 320pm; (c) 320pm; (d) 12 10.13. The NaCl lattice has the cubic unit cell shown in Fig. 10-7. KBr crystallizes in this lattice. (a) How many Kf ions and how many Br- ions are in each unit cell? (b) Assuming the additivity of ionic radii, what is a? ( c ) Calculate the density of a perfect KBr crystal. (d) What minimum value of r + / r - is needed to prevent anion-anion contact in this structure? Ans. (a) 4 each; (b) 656pm; (c) 2.80g/cm3; ( d ) 0.414 CHAP. 101 SOLIDS AND LIQUIDS 171 Fig. 10-7. NaCl Unit Cell 10.14. MgS and CaS both crystallize in the NaC1-type lattice (Fig. 10-7). From ionic radii listed in Table 10.1, what conclusion can you draw about anion-cation contact in these crystals? Ans. Ca2+ and S2- can be in contact, but Mg" and S2- cannot. In MgS, if Mg2+ and S2- were in contact there would not be enough room for the sulfide ions along the diagonal of a square constituting one-quarter of a unit cell face. To state this another way, for MgS, Y + / Y _ is less than 0.414 [see problem 10.13(d)]. 10.15. Each rubidium halide crystallizing in the NaC1-type lattice has a unit cell length 30 pm greater than that for the corresponding potassium salt of the same halogen. What is the ionic radius of Rb+ computed from these data? Ans. 148pm 10.16. Iron crystallizes in several modifications. At about 910C, the body-centered cubic a-form undergoes a transition to the face-centered cubic y-form. Assuming that the distance between nearest neighbors is the same in the two forms at the transition temperature, calculate the ratio of the density of y-iron to that of airon at the transition temperature. Ans. 1.09 10.17. The ZnS zinc blende structure is cubic. The unit cell may be described as a face-centered sulfide ion sublattice with zinc ions in the centers of alternating minicubes made by partitioning the main cube into eight equal parts. (a) How many nearest neighbors does each Zn2+ have? (b) How many nearest neighbors does each S2- have'? (c) What angle is made by the lines connecting any Zn" to any two of its nearest neighbors? ( ) What minimum Y + / Y - ratio is needed to avoid anion-anion contact, if closest cationd anion pairs are assumed to touch? Ans. (a) 4; (b) 4; (c) 109'28'; ( d ) 0.225 10.18. Why does ZnS not crystallize in the NaCl structure? (Hint: Refer to Problem 10.13.) Ans. The T+/E ratio is 0.402, too low to avoid anion-anion contact in the NaCl structure. 10.19. Compute the packing factor for spheres occupying ( a ) a body-centered cubic, and structure, where closest neighbors in both cases are in contact. (b) a simple cubic Ans. (a) 0.680; (b) 0.524 172 SOLIDS AND LIQUIDS [CHAP. 10 10.20 Many oxide minerals can be visualized as a face-centered oxide ion lattice, with cations distributed within the tetrahedral and octahedral holes. Calculate the lattice constant, a, for a face-centered 02-lattice. If cations occupy all the octahedral holes in MgO and CaO, calculate a for these minerals. Use data in Table 10-1. Ans. For an oxide lattice a = 396 pm. With Mg2+ and Ca2+ in the octahedral holes anion-anion contact is broken and a expands to 410pm and 478 pm, respectively. (Actual values are a little greater.) 10.21. Lithium iodide crystallizes in the NaCl lattice despite the fact that r+/r_ is less than 0.414. Its density is 3.49g/cm3. Calculate from these data the ionic radius of the iodide ion. Ans. 224pm (note: value in Table 10-1 is 216pm). 10.22. Thallium (I) bromide crystallizes in the CsCl lattice. Its density is 7557 kg/m' and its unit cell edge-length, a, is 397pm. From these data estimate Avogadro's number. Ans. 6.01 x 1023mole-' CRYSTAL FORCES 10.23. In each of the following cases, which of the two crystals has the higher melting point, and why? (a) Cs, Ba; ( b ) Si, P,; (c) Xe, Kr; ( d ) MgF2, CaC12. Ans. (a) Ba, denser electron sea; ( h ) Si, covalently bonded network versus molecular crystal; (c) Xe, higher atomic number gives stronger London forces; ( d ) MgF2, cations and anions are both smaller. 10.24. With reference to Fig. 10-8, how do you account for the differences in melting point between (a) and (b), between (c) and (d), and between these two differences? Ans. The crystal forces in ( b ) and (d) are largely van der Waals. Compounds (a) and (c), containing the polar -OH group, are capable of hydrogen bonding. In the case of (c), the hydrogen bonding is from the -OH of one molecule to the doubly bonded oxygen of the neighboring molecule; and the resulting intermolecular attraction leads to a very large increase in melting point as compared with (d), the non-hydrogen-bonded control. In the case of (a), the molecular structure allows intramolecular hydrogen bonding from the -- OH group of each molecule to the doubly bonded oxygen of the same molecule; in the absence of strong intermolecular hydrogen bonding the difference in melting point as compared with the reference substance (b) should be small, related perhaps to differences in the crystal structure or to the van der Waals forces, which should be slightly larger for (b) than for (a) on account of the extra CH3 group. 0 O \ o-Hydroxybenzaldehyde m.p. = 266 "C ( a1 o-Methoxybenzaldehyde m.p. = 309 "C p-Hydroxybenzaldehyde m.p. = 388 "C ( c1 p-Methoxybenzaldehyde m.p. = 273 "C C H3 (b) Fig. 10-8 (4 CHAP. 101 SOLIDS AND LIQUIDS 173 FORCES IN LIQUIDS 10.25. Which one of each of the following pairs of liquids has the higher boiling point? Why? (a) CH3CH2CH20H, HOCH2CH20H; (b) CH3CH2CH3, CH3,CHzF; (c) Xe, Kr; ( d ) H20, H2S; (e) CH3CH2CH2CH2CH2CH3, C H ~ C H Z C H ~ C H ~ . Ans. (a) HOCHZCH~OH, which has twice as many H-bonds per molecule; (b) CH3CH2F, which has a strong dipole moment; (c) Xe, which has a higher atomic number, and hence greater London forces; ( d ) H20, which forms strong H-bonds; (e) CH3CH2CH2CH2CH2CH3, which has a greater area of intermolecular contact, because it is a longer molecule. 10.26. Which of the following pairs of liquids is (are) miscible? Why, or why not? (a) butane (C4HI0),pentane (C5HI2); (b) butane, water; (c) 1-butanol (C4HgOH), water. Ans. (a) Miscible; forces of attraction between like and unlike molecules are about the same. (b) Not miscible; mixing would disrupt the strong H-bonds in water while there is no especially strong attraction between unlike molecules to compensate. Mixing would thus be energetically unfavorable. (c) Miscible; both components have hydrogen bonding. Breaking of H-bonds in water is compensated for by formation of H-bonds between unlike molecules. 10.27. Explain why UF6 (molecular mass 352) is more volatile than SbC15 (molecular mass 299). Ans. The intermolecular forces are greater for SbC15because the outer atoms have higher 2, hence greater London forces, than in UF6. 10.28. Explain why water dissolves in acetone (CH3COCH3), but not in hexane (C6H14). Ans. When water dissolves much energy is required to break its hydrogen bonds. In acetone this is compensated for by hydrogen-bonding between water and the oxygen atom of CH3COCH3, but there is no compensating strong interaction in hexane. This page intentionally left blank Chapter 11 Oxidation-Reduction OXIDATION-REDUCTION REACTIONS Oxidation is a chemical change in which electrons are lost by an atom or group of atoms, and reduction is a chemical change in which electrons are gained by an atom or group of atoms. These definitions can be applied most simply in the case of elementary substances and their ions. A transformation that converts a neutral atom to a positive ion must be accompanied by the loss of electrons and must, therefore, be an oxidation. Consider the following example: Fe -+ Fe2+ + 2eElectrons (symbol e-) are written explicitly on the right side and provide equality of total charge on the two sides of the equation. Similarly, the transformation of a neutral element to an anion must be accompanied by electron gain and is classified as a reduction, as in the following case: C12 + 2e3 2C1- Oxidation and reduction always occur simultaneously, and the total number of electrons lost in the oxidation must equal the number of electrons gained in the reduction. OXIDATION STATE It is not immediately apparent from the ionic charge alone whether a compound substance is undergoing oxidation or reduction. For example, Mn02 reacts with hydrochloric acid to produce, among other things, chlorine gas and the Mn2+ ion. It is obvious that neutral chlorine is produced by oxidation of Cl-. We infer, therefore, that Mn02 is undergoing reduction, in spite of the fact that a neutral substance, Mn 0 2 is converted partly to a cationic substance, Mn2+. In another example, arsenious acid, H3As03, reacts with 12 to form, among other things, the arsenate ion, HASO:-, and the iodide ion, I-. Since the iodine is reduced (neutral halogen to anion), the arsenious acid must be oxidized, in spite of the conversion of neutral H3As03 to anionic HASO:-. Oxidation state is a concept that is helpful in diagnosing quickly the state of oxidation or reduction of particular atoms in such compound species as Mn02, H3As03, and HASO:-. Oxidation state of an atom in some chemical combination is the electrical charge assigned to that atom according to a prescribed set of rules. Other expressions often used to refer to oxidation state are oxidation number or valence state. It should be made clear that oxidation state is not the same as formal charge (Chapter 9). Formal charge is based on an attempted mapping out of the real charge distribution of a molecule or ion among the constituent atoms in accordance with detailed knowledge of structure and electronic binding. Oxidation state, however, is a simpler assignment that does not require information about such electronic variables as single or multiple bonding and octet versus nonoctet structures. Oxidation state is computed directly from the compositional formula itself, such as Mn02 or H3As03. The two basic rules for oxidation state assignment follow. 1. In ionic binary compounds the oxidation state is the charge per atom. EXAMPLE 1 CdC12 is an ionic compound and may be designated Cd2+(C1-)2 to show its ionic character. The cadmium possesses a true charge of + 2 hence an oxidation state of +2. Each chloride ion possesses a true charge of - 1 hence an oxidation state of - 1. In Hg2C12, each mercury in Hg:+ has an average charge of -t- 1 and thus the oxidation state is + 1. The chlorine in C1- is again -1. 175 176 OXIDATION-REDUCTION [CHAP. 11 2. In covalent or nonionic compounds, the electrons involved in bond formation are not completely transferred from one element to the other, but are shared more or less equally by the bonding atoms. For purposes of oxidation state computation, however, it is conventional to assign each bonding electron to some particular atom. If the atoms are of the same kind, half of the bonding electrons are assigned to each of the two atoms. If the atoms are different, all electrons in the bond are assigned to that atom which has the greater electronegativity (Chapter 9). Rules 1 and 2 have several important corollaries. (a) The oxidation state of a free and uncombined element is zero. EXAMPLE 2 Hg in Hg, H in H2, 0 in 0 2 , S in Sg, etc. In HZ, of the two electrons in the molecule, one is assigned to each hydrogen atom. A hydrogen with one electron is the same as a neutral free hydrogen atom. Thus the oxidation state is zero. (b) The oxidation state of hydrogen in compounds is usually + I , except in the case of the metallic hydrides, where it is -1. EXAMPLE 3 In NH3, the nitrogen atom is bonded directly to each of the three hydrogen atoms. Since nitrogen is more electronegative than hydrogen, all the bonding electrons are assigned to nitrogen. Each hydrogen is thus left with zero assigned electrons, one less than a free hydrogen atom. Thus the hydrogen has an apparent charge, or an oxidation state of + l . The arbitrary nature of this charge designation is apparent in NH3, in which the true charge separation between the nitrogen and the hydrogens is very slight. NH3 never ionizes in water, for example, to produce hydrogen ions. In CaH2, on the contrary, each hydrogen, being more electronegative than calcium, is assigned two electrons, one more than a free hydrogen atom. The oxidation state of hydrogen is therefore - 1. (c) The oxidation state of oxygen in compounds is usually -2, except in peroxides, where it is - 1, or in fluorine compounds, where it may be positive. ( d ) The algebraic sum of the positive and negative oxidation states of all atoms in a neutral molecule is equal to zero. (e) The algebraic sum of the positive and negative oxidation states of all atoms in an ion is equal to the charge of the ion. EXAMPLE 4 In H2Si03, the oxidation state of Si is +4, in accordance with corollary ( d ) . 2( + 1) + 4 In PO:-, +5 + 3( - 2) = 0 the oxidation state of P is +5, in accordance with corollary (e): + 4( - 2) = -3 Only the phosphate ion as a whole has a true charge. Each atom within this complex ion has only an arbitrarily assigned charge, its oxidation state. The definitions of oxidation and reduction may now be generalized as follows: The increase in oxidation state is oxidation and the decrease in oxidation state is reduction. In the reduction of Mn02 to Mn2+,for example, the oxidation state of manganese is changed from +4 to +2; in the oxidation of H3As03 to HASO:-, the oxidation state of arsenic is changed from +3 to +5. IONIC NOTATION FOR EQUATIONS In writing oxidation-reduction equations, care should be taken to write formulas only for compounds or ions that have true chemical existence, like Mn02, H3As03,HASO:-. Mn (+4), As ( + 3), and As ( + 5) are not true species and should not be written as such in equations. CHAP. 111 OXIDATION-REDUCTION 177 When the ionic convention is to be used for writing formulas, the following rules are customarily observed: 1. Ionic substances are written in the ionic form only if the ions are separated from each other in the reaction medium. Thus, NaCl would be the conventional notation for reactions involving solid salt, because the ions in the solid are bound together in the crystal. Salt reactions in solution, however, would be indicated by Na' and Cl-, or by either of these ions alone if only the sodium or the chlorine undergoes a change in oxidation state. Fairly insoluble salts, like CuS, are always written in the neutral form. 2. Partially ionized substances are written in the ionic form only if the extent of ionization is appreciable (about 20 percent or more). Thus water, which is ionized to the extent of less than one part in a hundred million, is written as H20. Strong acids, like hydrochloric acid and nitric acid, may be written in the ionized form; but weak acids, like nitrous, acetic, and sulfurous acids, are always written in the molecular form. Ammonia, a weak base, is written NH3. Sodium hydroxide, a strong base, is written in the ionized form Na' and OH-, for water solutions. 3. Some complex ions are so stable that one or more of the groups from which they are formed do not exist in appreciable amounts outside the complex. In such a case the formula for the entire complex is written. Thus, the ferricyanide ion is always written as [Fe(CN),I3-, never as separate ferric and cyanide ions. [ C U ( N H ~ ) ~is ~ + notation for the comon blue complex ] the ion formed by cupric salts in ammonia solutions. 4. A mixed convention will be employed in this chapter as an aid to indicate whether a given compound may be written in the ionized form. Thus, Ba2+(N0y)2should indicate to the student that barium nitrate has the overall composition of Ba(N03)2, that it is soluble and ionized, and that either the barium ion, Ba2+, or the nitrate ion, NOT, may be written separately if desired. The absence of an ionic notation, as in AS&, means that only the neutral formula of the whole compound should be used. This mixed convention will not be uniformly used in subsequent chapters. Experienced chemists may write Ba(N03)2 in a reaction in aqueous solution, recognizing the ionic character of this salt even without an explicit ionic notation. BALANCING OXIDATION-REDUCTION EQUATIONS The principles of oxidation-reduction provide the basis of two simple systematic methods for balancing these equations. If all the products of reaction are known, the balancing may be done either by the ion-electron method or by the oxidation-state method. (The two methods are compared following Problem 11.8.) After students have acquired more experience, they will be able to predict some or all of the principal products if they keep in mind such facts as the following: If a free halogen is reduced, the product of reduction must be the halogenide ion (charge= - 1). If a metal that has only one positive oxidation state is oxidized, the oxidation state of the product is obvious. Reductions of concentrated nitric acid lead to N02, whereas the reduction of dilute nitric acid may lead to NO, N2, NH:, or other products, depending on the nature of the reducing agent and the extent of the dilution. Permanganate ion, Mn07, is reduced to Mn2+ in distinctly acid solution, as is Mn02. The reduction product of permanganate in neutral or alkaline solution may be MnO(OH), Mn02, or MnOi-. If a peroxide is reduced, the product of reduction must contain oxygen in the -2 oxidation state, as in H20 or OH-. If a peroxide is oxidized, molecular oxygen is formed. (A superoxide is possible.) Dichromate, Cr20;-, is reduced in acid solution to Cr3+. 178 OXIDATION-REDUCTION [CHAP. 11 Ion-electron method 1. Write a skeleton equation that includes those reactants and products that contain the elements undergoing a change in oxidation state. 2. Write a partial skeleton equation for the oxidizing agent, with the element undergoing a reduction in oxidation state on each side of the equation. The element should not be written as a free atom or ion unless it really exists as such. It should be written as part of a real molecular or ionic species. 3. Write another partial skeleton equation for the reducing agent with the element undergoing an increase in oxidation state on each side. 4. Balance each partial equation as to number of atoms of each element. In neutral or acid solution, H 2 0 and H+ may be added for balancing oxygen and hydrogen atoms. The oxygen atoms are balanced first. For each excess oxygen atom on one side of an equation, balance is secured by adding one H20 to the other side. Then H+ is used to balance hydrogens. Note that 0 2 and H2 are not used to balance the oxygen and hydrogen atoms unless they are known to be principal participants in the reaction. If the solution is alkaline, OH- may be used. For each excess oxygen on one side of an equation, balance is secured by adding one H20 to the same side and 20H- to the other side. If hydrogen is still unbalanced after this is done, balance is secured by adding one OH- for each excess hydrogen on the same side as the excess and one H 2 0 on the other side. If both oxygen and hydrogen are in excess on the same side of the skeleton equation, an OH- can be written on the other side for each paired excess of H and 0. 5. If an element undergoing a change in oxidation state is complexed in one of its states with some other element, balance the complexing groups with a species of that element in the same oxidation state occurring in the complex. 6 . Balance each partial equation as to the number of charges by adding electrons to either the left or the right side of the equation. If the preceding steps have been followed carefully, it will be found that electrons must be added to the left in the partial equation for the oxidizing agent and to the right in the partial equation for the reducing agent. 7. Multiply each partial equation by a number chosen so that the total number of electrons lost by the reducing agent equals the number of electrons gained by the oxidizing agent. 8. Add the two partial equations resulting from the multiplications. In the sum equation, cancel any terms common to both sides. All electrons should cancel. 9. For an understanding of the nature of the reaction, step 8 can be considered the last step. For calculations involving the masses of reactants or products, transform the ionic equation of step 8 into a molecular equation. This is done by adding to each side of the equation equal numbers of the ions which do not undergo electron transfer but which are present together with the reactive components in neutral chemical substances. Proper pairs of ions may be combined to give a molecular formula. 10. Check the final equation by counting the number of atoms of each element on each side of the equation. If step 9 was omitted, also check the net charge on each side. Oxidation-state method 1. Write a skeleton equation that includes, as principal formulas, those reactants and products that contain the elements undergoing a change in oxidation state. 2. Determine the change in oxidation state which some element in the oxidizing agent undergoes. The number of electrons gained is equal to this change times the number of atoms undergoing the change. 3. Determine the same for some element in the reducing agent. 4. In the skeleton equation, multiply the principal formulas by such numbers as to make the total number of electrons lost by the reducing agent equal to the number of electrons gained by the oxidizing agent. CHAP. 111 OXIDATION-REDUCTION 179 5. By inspection supply the proper coefficients for the rest of the equation. 6. Check the final equation by counting the number of atoms of each element on both sides of the equation. Solved Problems FORMULAS AND OXIDATION STATE 11.1. Given that the oxidation state of hydrogen is +1, of oxygen -2, and of fluorine -1, determine the oxidation states of the other elements in (a) PH3, (b) H2S, (c) CrF3, ( d ) H2S04, (e) H2S03, (f) A1203. H3 represents an oxidation-state sum of +3( 1 for each of the three hydrogens). Then the oxidation state of P must be -3, since the algebraic sum of the oxidation states of all atoms in a compound must equal zero. (6) Oxidation-state sum of H2 is +2; then oxidation state of S is -2. Oxidation-state sum of F3 is -3; then oxidation state of Cr is +3. (c) ( d ) Oxidation-state sum of H2 is +2 and of 0 4 is -8, or a total of -6; then oxidation state of S is +6. The sulfate ion, SO:-, has an ionic charge of -2, since the positive ionic charge of H2 (i.e., 2H') is f2. (e) Oxidation-state sum of H2 is +2 and of 0 3 is -6, or a total of -4; then oxidation state of S is +4. The sulfite ion, SO:-, has an ionic charge of -2, since the ionic charge of H2 is +2. Oxidation-state sum of O3 is -6; then oxidation-state sum of A12 is +6, and oxidation state of A1 is (f) 1 2 ( 6) = +3. (a) + + BALANCING OXIDATION-REDUCTION EQUATIONS 11.2. Balance the oxidation-reduction equation HCNOT + HZS --+ NO + S + H20 Ion-Electron Method. (Steps are numbered to correspond to the rules on the preceding page.) 1. The skeleton equation is given above. (H20 is not necessary to the skeleton equation.) The oxidizing agent is nitrate ion, NO,, since it contains the element, N, which undergoes a decrease in oxidation state. The reducing agent is HZS, since it contains the element, S, which undergoes an increase in oxidation state. (S2- might have been selected as the reducing agent, but H2S is preferable because of its very slight degree of ionization in acid solution.) 2. The partial skeleton equation for the oxidizing agent is NO, -+ NO 3. The partial skeleton equation for the reducing agent is H2S -+ S 4. (a) In the first partial equation, 2H20 must be added to the right side in order to balance the oxygen atoms. Then 4H' must be added to the left side to balance the H atoms. 4H' + NO, -+ NO + 2H20 (4 (b) (b) The second partial equation can be balanced by adding 2H' to the right side. H2S -+ S + 2Ht 180 OXIDATION-REDUCTION [CHAP. 11 5. Not applicable in this example. 6. (a) In equation 4(a) above, the net charge on the left is +4 3 electrons must be added to the left side. 4Hf - 1 = +3, and on the right it is 0. Therefore + NO, + 3e- -+ NO + 2H20 (4 (b) In equation 4(b), the net charge on the left is 0, and on the right is +2. Hence 2 electrons must be added to the right side. H2S + S + 2H+ + 2e7. Equation 6(a) must be multiplied by 2, and equation 6(b) by 3. 8H' (b) + 2N03 + 6e- -+ 2 N 0 + 4H20 3H2S -+ 3s + 6H+ + 6e- (b) 8. Addition of equations 7(a) and 7(b) results in 8H' + 2NOT + 3H2S + 6e2H' -+ 2 N 0 + 4 H 2 0 + 3s + 6H' 2 N 0 + 4H20 + 3s + 6e- Since 6H' and 6e- are common to both sides, they may be canceled. + 2N03 + 3H2S -+ This form of the equation shows all the reactant ions and compounds in the proper form. 9. If we want to know how much HN03 is required, we merely combine H+ with NOT. 2H+N03 + 3H2S -+ 2 N 0 + 4H20 + 3s 10. Each side of the equation has 8 H atoms, 2 N atoms, 6 0 atoms, and 3 S atoms. Oxidation-State Method. 1. 2. 3. 4. Given. Note that the oxidation state of N changes from +5 in NOT to +2 in NO. The oxidation state of S changes from -2 in H2S to 0 in S. Electron balance diagrams can be written as follows. (These are not equations.) N( + 5 ) + 3e- + N( + 2) (1) (2) S( - 2) + S(0)+2e- In order that the number of electrons lost shall equal the number gained, we must multiply ( I ) by 2, and (2) by 3. 2N( + 5)+6e- -+ 2N( + 2) 3S( - 2) -+ 3S(O) + 6eHence the coefficient of HfNOT and of NO is 2, and of H2S and S is 3. Part of the skeleton equation can now be filled in. 2H fN 03 + 3H2S -+ 2 N 0 + 3s 5. The 8 atoms of H on the left (2 from H'N03 plus 6 from HZS) must form 4H20 on the right. The final and complete equation is 2H+N05 + 3H2S + 2 N 0 + 3s + 4H20 6. Note that the oxygen atoms were balanced automatically, without special attention. Each side has 8 H, 2 N, 6 0, and 3 S atoms. The first partly filled skeleton equation could have been written in terms of NO, rather than H+NOT. For variety in the subsequent solved problems, the neutral-compound notation will be used in the oxidationstate method, and the ionic notation in the ion-electron method. CHAP. 111 OXIDATION-REDUCTION 181 11.3. Balance the following oxidation-reduction equation: K'Mn0; + K+Cl- + (H+),SO:- -+ Mn2+SO:- + (K+)2S0i- + H 2 0 + C12 Ion- Electron Method. The skeleton partial equations may be written MnO, -+ Mn2+ c - + c2 1 1 8H' (4 Partial (a) requires 4H20 on the right to balance the oxygen atoms; then 8H" on the left to balance the hydrogen. Partial (b) balances in routine fashion. + MnO, 2c1- -+ Mn2+ + 4H20 c2 1 (4 (b) --f The net charge on the left of partial (a) is +8 - 1 = +7, and on the right it is +2; therefore 5 electrons must be added to the left. In partial (b)the net charge on the left is -2, and on the right it is 0; therefore 2 electrons must be added to the right. 8H+ + Mn04+5e-+ Mn2+ + 4 H 2 0 (4 The multiplying factors are seen to be 2 and 5, respectively. 16H+ + 2MnO;+lOe- -+ 2Mn2+ + 8 H 2 0 1 c - --+ 5Cl2 + 10eo1 16H' + 2Mn04 + 1OC1- --+ 2Mn2+ + 8 H 2 0+ 5Cl2 Finally, check charges on each side: +4 on the left (+16 - 2 - lO), +4 on the right ( + 2 x 2). Since MnO, was added as KMn04, 2Mn04 introduces 2Kf to the left side of the equation; and since K+ does not react, the same number will appear on the right side. Since C1- was added as KCl, the lOC1introduces 10K' to each side of the equation. Since H+ was added as H2SO4, 16Hf introduces 8SO:- to each side of the equation. Then 16H' + 8SO;- + 2K+ + 2Mn04 + 10K+ + 1OC1- --f 2Mn2+ + 8 H 2 0+ 5Cl2 + 12K+ + 8SO:- Pairs of ions may be grouped on the left side to show that the chemical reagents were H2SO4, KMn04, and KC1. Pairs of ions may be grouped on the right to show that evaporation of the solution, after reaction has taken place, would cause the crystallization of MnS04 and K2SO4. 8(H'),SO;- + 2K'MnO; + 10K'ClMn( + 7) + 5e- + 2Mn2+SOi- + 6(K+),SO:- + 5C12 + 8 H 2 0 Mn undergoes a change in oxidation state from +7 in MnO, to +2 in Mn2+.C1 undergoes a change in oxidation state from -1 in C1- to 0 in Cl2. The electron balance diagrams are -+ Oxidation-State Method. Mn( + 2) (1) 2C1( - 1) -+ 2C1(0)+2e- (2) Diagram (2) was written in terms of two C1 atoms because these atoms occur in pairs in the product Clz. The multiplying factors are 2 and 5, just as in the previous method. 2Mn( + 7)+10eHence the coefficient of K'Mn04 2KfMn0i -+ 2Mn( + 2) 1OCl( - 1) -+ lOCl(O)+lOeand of Mn2+SO:- is 2, of K'Clis 10, of C12 is 5, ( f x 10). (incomplete) + 10K+C1- -+ 2Mn2+SO:- + 5c12 182 OXIDATION-REDUCTION [CHAP. 11 So far no provision has been made for the H20, (H+)zSOi-, and (K+)2S0i-. The 8 atoms of oxygen from 2K+MnOT form 8H20. For 8H20 we need 16 atoms of hydrogen, which can be furnished by 8(H+)2SOi-. The 12 atoms of K(lOK'C1- + 2KtMn0;) yields 6(K+),SOf. Note that all the oxygen in the oxidizing agent is converted to water. The sulfate ion retains its identity throughout the reaction. 2K+Mn0; + 10K+C1- + 8(H+)2SOi- -+ 2Mn2+SO;- + 5Cl2 + 6(K+),sOi- + 8 H 2 0 11.4. Balance the following oxidation-reduction equation: (K+)2Cr20;Ion-Electron Method. + H+Cl- --+ K'Cl- + Cr3'(C1-), + H 2 0+ C12 The balancing of the partial equation for the oxidizing agent proceeds as follows: cr20:cr20:14H+ + Cr20:-+6e14H' -+ cr3+ 2cr3+ 2cr3++ 7 H 2 0 2Cr3+ + 7 H 2 0 c~,o;-+ + Cr20:c1- -+ 4 2Cr3+ + 7 H 2 0 (balanced) The balancing of the partial equation for the reducing agent is as follows. 4 c2 1 2c12C1The overall equation is -+ c2 1 + C12 + 2e- (balanced) 1 x [14Hf + Cr2O:-+ 6e2C13x1 14Hf -+ 2Cr3+ 7H20] + C12+2e] + + + Cr20;- + 6C1- 2Cr3+ + 7H20 + 3c12 + 3Cl2 + 7H20 + 2K4C1- + 3Cl2 + 7 H 2 0 The 14H+ was added as 14H+Cl-, and 6 of the 14 chloride ions were oxidized. To each side of the equation 8 more C1- can be added to represent those C1- which were not oxidized. Similarly, 2K' may be added to each side to show that Cr20:- came from (K+)2CrO:-. 14Hf + 6C1- + 8C1- + 2K+ + Cr20:14H'Cl- + (K+),Cr20:- -+ + 2Cr3+ 2K+ + 8Cl2Cr3'(C1-), + The electron balance diagrams are written in terms of 2 atoms of Cr and 2 atoms of C1 because of the appearance of pairs of atoms of these kinds in (Kt)2Cr20:- and C12. For the oxidizing agent: 2Cr( +6) 2Cr( +6)+6eFor the reducing agent: 2C1( - 1) 2C1(0) 2C1( - 1) -+ 2C1(0)+2e---f Oxidation-Sta te Met hod. ---f 4 2Cr( +3) 2Cr( +3) Multiplying and adding: 1 x [2Cr( +6) 3x [ 2Cr( +6) + 6C1( - I ) + 6e- 2Cr( +3) ] 2C1( - 1) -+ 2C1(0) + 2e-] + -+ 2Cr( +3) + 6C1(0) CHAP. 111 OXIDATION-REDUCTION 183 Hence (Kt),Cr2O;- + 6H'Cl- -+ 2Cr3+(C1-), + 3c12 (incomplete) The equation is still unbalanced, as no provision has been made for K+Cl-, H20, or the H+Cl- that acts as an acid (as opposed to the H'Cl- that acts as reducing agent). By inspection, the 7 atoms of oxygen in (K+),Cr20?- form 7H20. For 7H20 we need 14 atoms of H, which can be furnished by 14H'C1-. Since 6 of the chloride ions were oxidized to Cl2, the remaining 8, (14 - 6), should appear on the right as K+Cl- or Cr3+(Cl-)3. Moreover, one (K+)2Cr20:- yields 2K+C1-. Thus the coefficient of H'C1- is 14, of H20 is 7, and of K'C1- is 2. (K+)2Cr20:- + 14H'Cl- + 2Cr3+(C1-), + 3Cl2 + 7H20 + 2K'Cl- (balanced) Note that here again all the oxygen in the oxidizing agent is converted to water. 11.5. Balance the following oxidation-reduction equation by the oxidation-state method. FeS2 + O2 -+ Fe203 SOz + The two special features of this problem are that both the ion and sulfur in FeS2 undergo a change in oxidation state and that the reduction product of oxygen gas occurs in combination with both iron and sulfur. The electron balance diagram for oxidation must contain Fe and S atoms in the ratio of 1 to 2, since this is the ratio in which they are oxidized. The two diagrams, with their multiplying factors, are 4 x [Fe( +2) 11 x [ 4Fe( +2) Hence + 2S( - 1) -+ Fe( +3) + 2S( +4)+1 le-] + 20( 20(0)+4e- -2) + 8S( - 1) + 220(0) -+ 4Fe( +3) + 8S( +4) + 220( -2) 4FeS2 + I 102 -+ 2Fe203+ 8S02 (balanced) I In problems like the above, involving unusual oxidation states, the student may ask, "What will happen if I choose the wrong oxidation numbers?" The answer is that the same result will be obtained so long as the method is followed consistently. If one chooses to assign +4 to the Fe and -2 to the S in the above example the total oxidation state increase is still 1 1 . An advantage of the ion-electron method is that such decisions are avoided, since the oxidation state is not used to determine the electron change. 11.6. Balance the following oxidation-reduction equation by the ion-electron method: Zn + Na'NO, + Na'OHNO, -+ (Na+),ZnO:- + NH3 + H 2 0 The skeleton partial for the oxidizing agent is + NH3 In alkaline solutions, each excess oxygen atom is balanced by adding 1H20 to the same side of the equation and 20H- to the opposite side. Each excess hydrogen atom is balanced by adding 10H- to the same side and 1H20 to the opposite side. In this example we must add 3H20 to the left and 60H- to the right to balance the excess oxygen of the NO,. Also, we must add 30H- to the right and 3H20 to the left to balance the excess hydrogen of the NH3. In all, 90H- must be added to the right and 6 H 2 0 to the left. NO, NO, + 6 H 2 0 NH3 + 90H+ 6 H 2 0+ 8e- -+ NH3 + 90H-+ (balanced) The balancing of the partial equation for the reducing agent is as follows: 40H40H- + Zn + Zn Zn -+ ZnO$Zn0;- -+ -+ + 2H20 ZnO$- + 2 H 2 0 + 2e- (balanced) 184 OXIDATION-REDUCTION [CHAP. 11 The overall equation is NO, Canceling, + 6 H 2 0+ 8e- + NH3 + 90HI 40H- + Zn Zn0;- + 2H20+2e-] 4 4 + 6 H 2 0+ 4Zn + 160H- NH3 + 90H- + 4Zn0:- + 8 H 2 0 NO; + 4Zn + 70HNH3 + 4Zn0;- + 2H20 1 x [NO, -+ t -+ Adding ions and combining, Na'NO, + 4Zn + 7Na'OH- -+NH3 + 4(Na'),ZnO:- + 2H20 11.7. Balance the following equation by the ion-electron method: HgS + H'Cl- + H'N03 4H' --+ (H+),HgCl;- + NO + S + H20 The partial equation for the oxidizing agent is balanced as in Problem 11.2. + N0,+3e- -+ NO + 2H20 The skeleton for the reducing agent is HgS + S The above skeleton contains both reduced and oxidized forms of sulfur, the only element undergoing a change in oxidation state. The unbalance is not one of hydrogen or oxygen atoms but of mercury. According to the overall equations, the form in which mercury exists among the products is the ion HgCli-. If such an ion is added to the right to balance the mercury, then chloride ions must be added to the left to balance the chlorine. In general, it is always allowable to add ions necessary for complex formation when such an addition does not require the introduction of new oxidation states (rule 5, page 164). HgS The overall equation is 2 x [4H+ + NO, 3x [ 8H' + 2N0, Adding ions and combining, 3HgS + 4C1- +S + HgCl$-+2e--t (balanced) + 3e- NO + 2H20 ] HgS + 4C1- -+ S + HgCl:-+ 2e-] + 3HgS + 12C1- 2 N 0 + 4H20 + 3s + 3HgC1:----t --+ + 2H'NOy + 12H'Cl- 3s + (H'),HgCli- + 2 N 0 + 4 H 2 0 11.8. Complete and balance the following skeleton equation for a reaction in acid solution: H202+Mn04+ --+ The products must be deduced from chemical experience. MnO, contains manganese in its highest oxidation state. Hence, if reaction occurs at all, MnO, is reduced. Since the solution is acid the reaction product will be Mn2+.The H202 must therefore act as a reducing agent in this reaction, and its only possible oxidation product is 02. The usual procedure may be followed from this point, leading to the following solution: 2 x [MnOz + 8H' 5x1 After canceling, 2Mn04 + 5eH202 -+Mn2' -+ + 16H' + 5H202 + +4H20 ] 0 2 + 2H' + 2e-] 2Mn2+ + 8 H 2 0 + 5 0 2 + lOH+ + 2Mn04 + 6H' + 5H202 2Mn2' + 8H20 + 502 CHAP. 111 OXIDATION-REDUCTION 185 The above equation is as definite a statement as can be made within the given overall skeleton. If we are to write an equation in terms of neutral substances, we are free to decide which salt of MnO, and which acid to use. If we use K'Mn04 and (HT)SOi-, we obtain 2K'MnO; + 3(H+)2soi- + 5H202 -+ 2Mn2+SO$- + 502 + (K+)2S024-+ 8 H 2 0 COMPARISON OF THE TWO METHODS OF BALANCING Both methods lead to the correct form of the balanced equation. The ion-eletron method has two advantages: 1. It differentiates those components of a system which react from those which do not. In Problem 11.8, any permanganate and any strong acid could have been used, not necessarily K'Mn04 and (H+)2SOi--.The use of complete formulas for neutral substances is helpful only for the calculation of mass relations. Since permanganate ion cannot be weighed on a balance, it is necessary to choose one particular permanganate salt, like K'MnO;, and weigh an amount of K'MnO;, that will give the correct weight of Mn0;. 2. The half-reactions given by the partial equations of the ion-electron method cafi actually be made to occur independently. Many oxidation-reduction reactions can be carried out as galvanic cell processes for producing an electrical potential. This can be done by placing the reducing agent and oxidizing agent in separate vessels and making electrical connections between the two. It has been found that the reaction taking place in each beaker corresponds exactly to a partial equation written according to the rules for the ion-electron method. (This will be treated in Chapter 19.) Some chemists prefer to use the ion-electron method for oxidation-reduction reactions carried out in dilute aqueous solutions, where free ions have a more or less independent existence, and to use the oxidation-state method for oxidation-reduction reactions in other liquid media, in solids, or in gases. STOICHIOMETRY IN OXIDATION-REDUCTION 11.9. In Problem 11.3, how much C12 could be produced by the reaction of lOOg KMn04? This mass problem is solved like all other mass problems if the balanced molecular equation is used; it shows that 2 mol KMn04 can produce 5 rnol C12. The problem can also be solved directly from the balanced ionic equation showing that 2Mn04 ions yield 5 molecules of Cl2. We need only the chemical equivalence of 2Mn04 with 2KMn04. In either case, the solution is x g CI2 = (100 g KMn04) I mol KMn04 158 g KMn04 )( 5 rnol CI2 (70.9 g (21,) = 112 g Cl2 2 rnol KMn04 1 rnol C12 ) Supplementary Problems 186 OXIDATION-REDUCTION [CHAP. 11 Write balanced ionic and molecular equations for the following: 11.11. Na'Br11.12. 11.13. + C12 -+ Na'Cl- + Br2 Sn + O2 + H'C1- -+ Sn"(Cl-), + H 2 0 CuS + H'NO; (dilute) C U ~ + ( N O Y ) ~ + H 2 0 + NO +S 4 11.14. Fe2+(C1-), 11.16. Cu 11.17. 11.18. 11.19. 11.20. 11.21. 11.22. 11.23. + H 2 0 2+ H'Cl(conc.) + + Fe3+(CI-)i +H20 11.15. As2SS+ H'NO; + + NO2 (dilute) CU~'(NOY)~ H20 + NO + (dilute) Zn2+(NO;)2 + H 2 0 + NH:NO: (Na+),C20;- + K'Mn04 + (Hf)2SO$- (K+),SO;- + (Na'),SO$- + H 2 0 + Mn2+SO$- + CO2 K'C107 + H'ClKfC1- + Cl2 O2 + H+I- + H 2 0 + H'(13)MnO + Pb02 + H+NO; H'MnO, + Pb2'(NOy), + H 2 0 Cr3+(1-)3 + K'OH- + C12 (K')2CrOi- + K'IOZ + K'CI- + H 2 0 CU~'(NO;)~ H20 ---f + H'NO: Cu + H'NO: Zn + H'NO: (conc.) -+ H3As04+ H'HSO; + H 2 0+ NO2 -+ + -+ --$ --+ (Note that both the chromium and the iodide are oxidized in this reaction.) + 11.24. (Na+),HAsO:11.25. 11.26. 11.27. 11.28. 11.29. 11.30. 11.31. 11.32. 11.33. 11.34. 11.35. 11.36. 11.37. 11.38. + K'Br0: + H'Cl~ Na'Cl- + K'Br- + H3As04 (Na+),TeO:- + Na+I- + H+C1Na'C1- + Te + H 2 0 + I2 U4'(SO:-)2 + K+MnO; + H 2 0 (Ht)2S0$- + (K'),SO$- + Mn2+SO$- + UO;+SO:12 + (Naf),S20:(Na'),S40;- + Na+ICa2t(OC1-)2 + K'I- + H'Cl1, + Ca2+(C1--)2 H 2 0 + K'CI+ Bi2O3 + Na+OH- + Na'OCINa'BiO, + Na'CI- + H 2 0 (Ki)4Fe(CN)2--+ (K+)2CrO:- + H 2 0 (K')3Fe(CN)i- + Cr2O3 + K+OHMn2+SOf + (NH:)2S20i- + H 2 0 Mn02 + (H')2SO$- + (NHi)2SOiC O ~ + ( C I -+~ ) (Na+),O:- + Na'OH- + H 2 0 Co(OH), + Na'C1C U ( N H ~ ) T ( C ~+ ) ~ - K+CN- + H20 -+ NH3 + NH,fCl- + (K+)2Cu(CN):- + K'CNO- + K+ClHSb(OH), + K'CI- + ICI Sb203 + K'IOY + H+CI- + H20 Ag + K'CN- + 0 2 + H 2 0 K'Ag(CN), + K'OHWO3 + Sn2+(C1-)2+ H'CI- + W308+ (H')2SnCli- + H 2 0 (K'),Co(NO2)i- + NO + K+C2H30, + K+Cl- + H 2 0 Co2+(C1)2+ K'NO? + H'C2H30, V(OH),fCI- + Fe2'(C1-)2 + H+CI- -+ V02t(CI-)2 + Fe3'(C1-), + H 2 0 -+ + --f ---f -+ -+ -+ ---f -+ -+ 4 Balance the following equations by the oxidation-state method: 11.39. NH3 + 0 11.41. 11.42. 11.43. 11.44. 2 --+ NO + H20 + N2 11.40. CUO+ NH3 + H20 + CU PbO2 + HI -+ PbI2 + I2 + H 2 0 Ag2S04 + AsH3 + H20 Ag + As203+ H2SO4 NaN3 Na3N + N2 KC103 + H2SO4 KHS04 + O2 + C102 + 2 H 2 0 -+ 4 -+ CHAP. 111 OXIDATION-REDUCTION 187 11.45. Sn 11.46. 11.47. + HN03 I2+ HN03 KI + H2SO4 -+ + + + NO2 + H20 HI03 + NO2 + H 2 0 K2SO4 + I2 + H2S + H20 SnO2 11.48. KBr + H2SO4 + -+ 11.49. Cr2O3 11.50. P2H4 + NazC03 + KN03 PH3 K2SO4 + Br2 + SO2 H20 --+ Na2Cr04 + 11.51. Ca3(P04)2 Si02 + + P4H2 + CO2 + KNO2 +C 3 CaSi03 + P4 + CO (acid solution) (aqueous solution) (alkaline solution) (alkaline solution) (acid solution) (acid solution) (alkaline solution) (alkaline solution) Complete and balance the following skeleton equations by the ion-electron method: 11.52. I11.53. + NO; I2 + NO AU+ CN- + 0 2 + Au(CN), -+ 11.54. MnO, 11.55. P 11.56. Zn 11.57. 11.58. 11.59. t --+ MnOiAsH3 ClO, 10, + As203 Zn + ReO, C102 + 0;C12 + 10, PH3 + H2PO; + O2 + -+Re--t + 11.60. V -+ HV,O:I + H2 (alkaline solution) 11.61. In Problem 11.8, what volume of O2 at S.T.P. is produced for each gram of H202 consumed? Ans. 0.66L 11.62. How much KMn04 is needed to oxidize 100 g Na2C204? See the equation for the reaction in Problem 11.19. Ans. 52g 11.63 Automotive air bags can be inflated by the reaction of Problem 11.43, which proceeds very rapidly once initiated. How many grams of NaN3(s) are required to produce 69.5L (at S.T.P.) of N2(g)? Ans. 151.3g This page intentionally left blank Chapter 12 Concentrations of Solutions SOLUTEANDSOLVENT In a solution of one substance in another substance, the dissolved substance is called the solute. The substance in which the solute is dissolved is called the solvent. When the relative amount of one substance in a solution is much greater than that of the other, the substance present in greater amount is generally regarded as the solvent. When the relative amounts of the two substances are of the same order of magnitude, it becomes difficult, in fact arbitrary, to specify which substance is the solvent. CONCENTRATIONS EXPRESSED IN PHYSICAL UNITS When physical units are employed, the concentrations of solutions are generally expressed in one of the following ways: 1. By the mass of solute per unit volume of solution (e.g., 20g of KCl per liter of solution) 2. By the percentage composition, or the number of mass units of solute per l00mass units of solution. EXAMPLE 1 A 10% aqueous NaCl solution contains 10 g of NaCl per 100 g of solution. Ten grams of NaCl is mixed with 90 g of water to form 100 g of solution. 3. By the mass of solute per mass of solvent (e.g., 5.2g of NaCl in lOOg of water) 4. In some fields of study, such as air and water pollution, there is an interest in very small concentrations of solute. These are frequently specified as parts-per-million (ppm), or partsper-billion (ppb), based on mass. CONCENTRATIONS EXPRESSED IN CHEMICAL UNITS Molar concentration The molar concentration (M)is the number of moles of the solute contained in one liter of solution. EXAMPLE 2 A 0.500 molar (0.500M) solution of H2S04 contains 49.04g of H2SO4 per liter of solution, since 49.04 is half of the molar mass of H2S04, 98.08. A one molar (1.00 M) solution contains 98.08 g H2S04 per liter of solution. Note that M is the symbol for a quantity, the molar concentration, and M is the symbol for a unit, mol/L. (The term molurity is often used to refer to molar concentration; it will not be used in this book, in order to avoid confusion with rnolality, defined below.) Normality The normality of a solution ( N ) is the number of equivalents of the solute contained in one liter of solution. The equivalent mass is that fraction of the molar mass which corresponds to one defined unit of chemical reaction, and an equivalent (eq) is this same fraction of a mole. Equivalent masses are determined as follows: 1. The defined unit of reaction for acids and bases is the neutralization reaction 189 190 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 HS+ OH- --+ H20 The equivalent mass of an acid is that fraction of the molar mass which contains or can supply for reaction one mole of acid H'. (In other words, the equivalent mass is the molar mass divided by the number of HS furnished per molecule.) EXAMPLE 3 The equivalent masses of HCI and HC2H302 are the same as their molar masses, since each contains one acidic hydrogen per molecule. An eq of each of these molecules is the same as 1 mol. The equivalent mass of H2SO4 is usually half the molar mass and an eq is f mol, since both hydrogens are replaceable in most reactions of dilute sulfuric acid. An eq of H3P04 may be 1 mol, f mol, or f mol, depending on whether one, two, or three hydrogen atoms per molecule are replaced in a particular reaction. An eq of H3B03 is always Imol, since only one hydrogen is replaceable in neutralization reactions. The equivalent mass of SO3 is one-half the molar mass, since SO3 can react with water to give two H+. SO3 + H 2 0 t 2H' + SO:- There are no simple rules for predicting how many hydrogens of an acid will be replaced in a given neutralization. 2. The equivalent mass of a base is that fraction of the molar mass which contains or can supply one mole of OH-, or can react with one mole of HS. EXAMPLE 4 The equivalent masses of NaOH, NH3 (which can react with water to give NH: +OH-), Mg(OH),, and Al(OH)3, are equal to f , f , f , and of their molar masses, respectively. 4 3. The equivalent mass of an oxidizing or reducing agent for a particular reaction is equal to its molar mass divided by the total number of moles of electrons gained or lost when the reaction of one mole occurs. A given oxidizing or reducing agent may have more than one equivalent mass, depending on the reaction for which it is used. Equivalent masses are so defined that equal numbers of equivalents of two substances react exactly with each other. This is true for neutralization because one H+ neutralizes one OH-, and for oxidationreduction because the number of electrons lost by the reducing agent equals the number of electrons gained by the oxidation agent. EXAMPLE 5 I mol HCI, rnol H2SO4, and mol K2Cr207 (as oxidizing agent), each in 1 L of solution, give normal (I N) solutions of these substances. A normal (1 N) solution of HC1 is also a molar ( 1 M) solution. A normal (1 N) solution of H2SO4 is also a one-half molar (0.5 M) solution. Note that N is the symbol for a quantity, the normality, and N is the symbol for a unit, eq/L. 4. Reagents are frequently used for precipitation reactions and the defined unit of reaction is the amount which provides N A positive (or negative) charges. For example, for AgN03 one equivalent is the same as one mole, but for BaC12 one equivalent is one-half mole. Molality The molality of a solution is the number of moles of the solute per kilogram of solvent contained in a solution. The molality (m) cannot be computed from the molar concentration (M> unless the density of the solution is known (see Problem 12.58). EXAMPLE 6 A solution made up of 98.08 g of pure H2SO4 and 1 000 g of water would be a 1.OOO molal solution of H2SO4. (Some books use the symbol rn to designate both the quantity, the molality, and the unit, mol/kg. In this book, rn is used to refer to the quantity only and not to the unit.) Mole fraction The mole fraction (x) of any component in a solution is defined as the number of moles ( n ) of that component, divided by the total number of moles of all components in the solution. The sum of the mole CHAP. 121 CONCENTRATIONS OF SOLUTIONS 191 fractions of all components of a solution is 1. In a two-component solution, x(so1ute) = n(so1ute) n(so1ute) + n(so1vent) x(so1vent) = n(so1vent) n(so1ute) + n(so1vent) Expressed as a percentage, mole fraction is called mole percent. COMPARISON OF THE CONCENTRATION SCALES The molar concentration and normality scales are useful for volumetric experiments, in which the amount of solute in a given portion of solution is related to the measured volume of solution. As will be seen in subsequent chapters, the normality scale is very convenient for comparing the relative volumes required for two solutions to react chemically with each other. A limitation of the normality scale is that a given solution may have more than one normality, depending on the reaction for which it is used. The molar concentration of a solution, on the other hand, is a fixed number because the molar mass of a substance does not depend on the reaction for which the substance is used, as may the equivalent mass. The molality scale is useful for experiments in which physical measurements (as of freezing point, boiling point, vapor pressure, etc.) are made over a wide range of temperatures. The molality of a given solution, which is determined solely by the masses of the solution components, is independent of temperature. The molar concentration or the normality of a solution, on the other hand, being defined in terms of the volume, may vary appreciably as the temperature is changed, because of the temperature dependence of the volume. In dilute aqueous solutions ( < 0.1 M) the molality is very close numerically to the molar concentration. The mole fraction scale finds use in theoretical work because many physical properties of solutions (Chapter 14) are expressed most clearly in terms of the relative numbers of solvent and solute molecules. (The number of moles of a substance is proportional to the number of molecules.) SUMMARY OF CONCENTRATION UNITS 4 Molar concentration of a solution = A = number of moles of solute number of liters of solution number of millimoles of solute number of cubic centimeters (mL) of solution number of equivalents of solute Normality of a solution = N = number of liters of solution number of milliequivalents of solute number of cubic centimeters (mL) of solution number of moles of solute Molality of a solution = m = number of kilograms of solvent number of moles of that component Mole fraction of any component = x = total number of moles of all components DILUTION PROBLEMS The volumetric scales of concentration are those, like molar concentration and normality, in which the concentration is expressed as the amount of solute per fixed volume of solution. When the concentration is expressed on a volumetric scale, the amount of solute contained in a given volume of solution is equal to the product of the volume and the concentration: Amount of solute = volume x concentration 192 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 If a solution is diluted, the volume is increased and the concentration is decreased, but the total amount of solute is constant. Hence, two solutions of different concentrations but containing the same amounts of solute will be related to each other as follows: Volume1 x concentrationl = volume2 x concentration2 where the subscripts 1 and 2 refer to the conditions "before dilution" and "after dilution", respectively. If any three terms in the above equation are known, the fourth can be calculated. The quantities on both sides of the equation must be expressed in the same units. Solved Problems CONCENTRATIONS EXPRESSED IN PHYSICAL UNITS 1 . . Explain how you would prepare 60cm3 of an aqueous solution of AgN03 of strength 0.030g 21 AgN03 per cubic centimeter. Since each cubic centimeter of solution is to contain 0.030 g AgN03, 60 cm3 of solution should contain (0.030g/cm3)(60 cm3) = 1.8 g AgN03 Thus, dissolve 1.8 g AgN03 in about 50 cm3 of water. Then add sufficient water to make the volume exactly 60cm3. Stir thoroughly to insure uniformity. (Note that 60cm3 is to be the volume of the final solution, not of the water used to make up the solution.) 1 . . How many grams of a 5.0% by weight NaCl solution are necessary to yield 3.2g NaCl? 22 A 5.0% NaCl solution contains 5.0g NaCl in lOOg of solution. Then: 1 g NaCl is contained in and 3.2 g NaCl is contained in (3.2) - g solution -g 100 5.0 solution ::( Or, by proportion, letting x = required number of grams of solution, 1 = 64 g solution 5.0 g NaCl 100 g solution - 3.2 g NaCl X whence x = 64 g solution Or, by the use of a quantitative factor, x g solution = (3.2 g NaC1) 100 g solution 5.0 g NaCl = 64 g solution 1 . . How much NaN03 must be weighed to out make 50 cm3 of an aqueous solution containing 70 mg 23 Na+ per cm3? Mass of Na' in 50 cm3 of solution = (50 cm3)(70 mg/cm3) = 3 500 mg = 3.5 g Na+ Molar mass of NaN03 is 85; of Na is 23. Then: CHAP. 121 CONCENTRATIONS OF SOLUTIONS 193 23 g Na+ is contained in 85 g NaNO, 1 g Na' is contained in g NaN03 and 3.5 g Na' Or, by direct use of quantitative factors, x g NaN03 = (50 cm3 solution) is contained in (3.5)($ g) = 12.9 g NaN03 70 mg Na+ ( 8 5 g NaNO3) (1 cm3 solution 23 g Na+ ) (1000gmg) 1 = 12.9 NaNO3 12.4. A 500mL sample of the effluent from a water softener required six drops of standard soap solution to produce a permanent lather. The soap solution had been calibrated against an artificial hard water solution containing 0.136 g of CaC12 per liter. On the average it required 28 drops of standard soap solution to lather 500mL of the artificial solution. Calculate the "hardness" of the effluent sample expressed as ppm of CaC03. (CaC03 is very insoluble and does not exist as such in hard water. The measure of hardness is really the amount of CaC03 that would be formed if all the Ca2+ were precipitated as CaC03.) The hardness of the effluent sample is clearly 6/28 of that of the CaC12 solution. It remains to express this in the standard units, assuming that each mole of CaC12 is equivalent to a mole of CaC03 in natural hard water. In a liter of the artificial hard water, we have: ((0.136 g CaC12)/(11 1 g CaC12/mol)} x (100 g CaC03/mol) = 0.123 g CaC03 One liter of this nearly pure water weighs lOOOg. 0.123 g CaC03 1000 g H 2 0 123 i 6 = 1 000 000 = 123 ppm 66 1000 Hardness of the effluent is (123)(6/28) = 26 ppm Comment: The above value is lower than that of most natural waters, but poor for treated water. The watersoftening cartridge should be replaced or recharged. 12.5. Describe how you would prepare 50 g of a 12.0% BaC12 solution, starting with BaC12 . 2H20 and pure water. A 12.0% BaCI2 solution contains 12.0g BaC12 per 1OOg of solution, or 6.00g BaC12 in 50.0g of solution. Molar mass of BaC12 is 208; of BaC1, . 2H20, 244. Therefore, 208 g BaC12 is contained in 244 g BaC1, a2H20 1 g BaC1, 244 is contained in - g BaC12 . 2 H 2 0 208 and 6.00 g BaC12 is contained in (6.00) - g = 7.04 g BaC12 a2H20 50 g solution - 7.04 g of salt = 43 g of water needed (Z ) The desired solution is prepared by dissolving 7.0 g BaC12 . 2H20 in 43 g (43 cm3) of water. Note: A small part of the solvent water came with the hydrated salt. 1 . . Calculate the mass of anhydrous HCl in 5.00cm3 of concentrated hydrochloric acid (density 26 1.19 g/cm3) containing 37.23% HCl by weight. 194 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 The mass of 5.00 cm3 of solution is (5.00 cm3)(l.19g/cm3)= 5.95 g. The solution contains 37.23% HC1 by weight. Hence the mass of HC1 in 5.95g solution is (0.372 3)(5.95 g ) = 2.22 g anhydrous HC1 12.7, Calculate the volume of concentrated sulfuric acid (density 1.84g/cm3), containing 98% H2SO4 by weight, that would contain 40.0g of pure H2SO4. One cm3 of solution has a mass of 1.84g and contains (0.98)(1.84g) = 1.80g of pure H2SO4. Then 40.0 g H2SO4 is contained in (=)(I Or, by direct use of conversion factors, (40*0g H2S04)( cm3 solution) = 22.2 cm3 solution) = 22.2 cm3 solution 12.8. Exactly 4.00g of a solution of sulfuric acid was diluted with water, and an excess of BaC12 was added. The washed and dried precipitate of BaS04 weighed 4.08 g. Find the percent H2SO4 in the original acid solution. First determine the mass of H2SO4 required to precipitate 4.08 g BaS04 by the reaction H2S04+ BaCI2 t 2HC1+ BaS04 The equation shows that 1 mol BaS04 (233.48) requires 1 mol H2SO4 (98.08g). Therefore 4.08g BaS04 requires ( ~ ~ ~(98.08 ~ g H2S04) = 1.72 g H2S04 ~ ~ a ~ 4 ) and Fraction H2S04 by weight = mass of H2SO4 mass of solution - ___ 1.72 g = 0.430 = 43.0% 4.00 g CONCENTRATIONS EXPRESSED IN CHEMICAL UNITS 12.9. How many grams of solute are required to prepare 1 L of 1 M Pb(N03)2? What is the molar concentration of the solution with respect to each of the ions? A molar solution contains 1 mol of solute in 1 L of solution. The molar mass of Pb(N03)2 is 331.2; hence 33 1.2g of Pb(NO& is needed. A 1 M solution of Pb(N03)2 is 1 M with respect to Pb2+ and 2 M with respect to NOT. 12.10. What is the molar concentration of a solution containing 16.0g CH30H in 200cm3 of solution? Molar mass of CH30H is 32.0. 16.0 g n (solute) M = molar concentration = - 32.0 g/mol = 2.50 mol/L = 2.50 M volume of solution in L - 0.200 L . . CHAP. 121 CONCENTRATIONS OF SOLUTIONS 195 12.11. Determine the molar concentration of each of the following solutions: (a) 18.0g AgN03 per liter of solution, (b) 12.00g A1Cl3 - 6H20 per liter of solution. Molar mass of AgN03 is 169.9; of AlC13.6H20, 241.4. 18*0g/L = 0.106 mol/L = 0.106 M 169.9 g/mol 12'oo g/L = 0.049 7 mol/L = 0.049 7 M 241.4 g/mol 12.12. How much (NH&SO4 is required to prepare 400cm3 of M/4 solution? (The notation M/4 is sometimes used in place of $ M.) Molar mass of (NH4)2S04 is 132.1. One liter of M/4 solution contains (132.1 g) = 33.02 g (NH4)2S04. Then 400cm3 of M/4 solution requires (0.400 L)(33.02 g/L) = 13.21 g (NH4)2S04 Another Method. Mass = (molar concentration) x (molar mass) x (volume of solution) = (i mol/L)( 132.1 g/mo1)(0.400 L) = 13.21 g (NH4)2S04 i 12.13. What is the molality of a solution which contains 20.0 g of cane sugar, C12H2201 dissolved in 125g of water? Molar mass of C12H22011is 342. m = molality = n(so1ute) - (20.0 g)/(342 g/mol) = 0.468 mol/kg mass of solvent in kg 0.125 kg 12.14. The molality of a solution of ethyl alcohol, C~HSOH, water is 1.54mol/kg. How many grams in of alcohol are dissolved in 2.5 kg of water? Molar mass of C2H50H is 46.1. Since the molality is 1.54, 1 kg water dissolves 1.54mol alcohol. Then 2.5 kg water dissolves (2.5)( 1.54)= 3.85mol alcohol, and Mass of alcohol = (3.85 mo1)(46.1 g/mol) = 177 g alcohol 12.15. Calculate the (a) molar concentration and (b) molality of a sulfuric acid solution of density 1.198 g/cm3, containing 27.0% H2S04 by weight. It is recommended, when quantities of substances are not stated, to select an arbitrary quantity as the basis of the calculation. In this case let the basis be 1.OOOL of solution. (a) One liter has a mass of 1 198 g and contains (0.270)(1 198) = 323 g of H2S04 of molar mass 98.1. Hence: (b) From (a), there is 323 g, or 3.29 mol, of solute per liter of solution. The amount of water in 1 L of solution is 1 198 g - 323 g = 875 g H20. Hence m= n(so1ute) mass of solvent in kg - 3.29 mol H2SO4 = 3.76 mol/kg 0.875 kg H 2 0 196 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 12.16. Determine the mole fractions of both substances in a solution containing 36.0 g of water and 46 g of glycerin, C3H5(OH)3. Molar mass of C ~ H S ( O His ~ of H20, 18.0. ) 92; n(g1ycerin) = ~ 46 g = 0.50 mol 92 g/mol n(water) = = 0.50 36.0 g = 2.00 mol 18.0 g/mol Total number of moles + 2.00 = 2.50 mol x(g1ycerin) = mole fraction of glycerin = x(water) = mole fraction of water = Check: Sum of mole fractions = 0.20 + 0.80 = 1.OO. n(g1ycerin) 0.50 - -= 0.20 total numbr of moles 2.50 n(water) 2 00 - -= 0.80 total number of moles 2.50 12.17. How many equivalents of solute are contained in (a) 1 L of 2 N solution. solution, (c) 0.5 L of 0.2 N solution? (b) 1 L of 0.5 N A normal solution contains 1 eq of solute in 1 L of solution. Amount of solute = (volume) x (concentration). (a) 1 L of 2 N contains 2 eq of solute. (b) 1 L of 0.5N contains 0.5eq of solute. (c) 0.5 L of 0.2 N contains (0.5 L)(0.2 eq/L) = 0.1 eq of solute. 12.18. How many (a) equivalents and (b) milliequivalents (meq) of solute are present in 60cm3 of 4.0 N solution? (a) Number of equivalents = (number of liters) x (normality) = (0.060 L)(4.0 eq/L) = 0.24 eq (b) (0.24 eq)( 1 000 meq/eq) = 240 meq Another Method. Number of meq = (number of cm3) x (normality) = (60 cm3)(4.0 meq/cm3) = 240 meq 12.19. How many grams of solute are required to prepare 1 L of 1 N solution of (a) LiOH, (as oxidizing agent), (c) H3P04 (for a reaction in which three H are replaceable)? (b) Br2 A normal solution contains 1 eq of solute in 1 L of solution. Molar mass of LiOH is 23.95; of Br2, 159.8; of H3P04, 98.00. (a) One liter of 1 N LiOH requires (23.95/1)g = 23.95 g LiOH. (b) Note from the partial equation Br2 + 2e- + 2Br- that two electrons react per Br2. Thus, the equivalent mass of Br2 is half its molar mass, and 1 L of 1 N Br2 requires (159.8/2)g = 79.9 g Br2. (c) One liter of 1 N H3P04 requires (98.00/3) g = 32.67 g H3P04. 12.20. Calculate the normality of each of the following solutions: (a) 7.88g of H N 0 3 per liter of solution, (b) 26.5g of Na2C03 per liter of solution (if neutralized to form CO2). (a) Equivalent mass of H N 0 3 (when used as an acid, not as an oxidizing agent) = molar mass = 63.02 CHAP. 121 CONCENTRATIONS OF SOLUTIONS 197 (b) The reaction is CO:- + 2H+ -, CO2 + H 2 0 . Equivalent mass of Na2C03 = x (molar mass) = (i) (i)( = 53.0 106.0) 12.21. How many cubic centimeters of 2.00 M Pb(NO& contain 600 mg Pb2+? One liter of 1 M Pb(N03)2 contains 1 mol Pb2+, or 207 g. Then a 2 M solution contains 2 mol Pb2+ or 414g Pb2+ per liter, or 414mg Pb2+ per cubic centimeter. Hence 600mg Pb2+ is contained in 600 mg = 1.45 cm3 of 2.00 M Pb(N03), 414 mg/cm3 Another Method. Mass = A x volume x molar mass 4 Volume = mass/(M x molar mass) Volume = 600 mg -= 1.45 mL (2 mmol/mL) (207 mg/mmol) 12.22. Given the unbalanced equation K'Mn04 (a) + K+I- + (H'),SO;- -+ (K+),SO;-- + Mn2+SO;- + I2 + H20 How many grams of KMn04 are needed to make 500cm3 of 0.250N solution? (b) How many grams of KI are needed to make 25 cm3 of 0.36N solution? In this oxidation-reduction reaction, the oxidation state of Mn changes from +7 in MnO, to +2 in Mn2+, a gain of 5 electrons. Hence Equivalent mass of KMnO, = molar mass 158 - -= 31.6 oxidation state change 5 (a) Then 0.500 L of 0.250 N requires (0.500 L)(0.250 eq/L)(31.6 g/eq) = 3.95 g KMn04 If the KMn04 made above were to be used in the reaction: 2Mn04 + 3H202 + 2H+ + 2Mn02 + 302 + 4H20 the label 0.250 N would no longer be appropriate because the oxidation number change in this case is 3, not 5. The appropriate normality would be f F ) ( 3 ) = 0.150 N (b) The oxidation state of I changes from -1 in I- to 0 in 12. Hence Equivalent mass of KI = molar mass = 166 Then 0.025 L of 0.36N solution requires (0.025 L)(0.36 eq/L)(l66 g/eq) == 1.49 g KI 198 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 12.23. Calculate the molar concentration of an NaBr solution which has a density of 1.167 g/cm3 and a molality of 2.28 mol/kg. The molar mass of NaBr is 102.9. For each kilogram of water there is (2.28)(102.9)=235g of NaBr, or a total mass of 1000+235=1235g. The mass of solution occupies a volume of 1235g/ (1.167 g/cm3) = 1058 cm3 or 1.058L. The molar concentration then is 2.28 mo1/1.058 L = 2.16 mol/L or 2.16 M 12.24. A solution of a certain organic halide in benzene has a mole fraction of halide of 0.082 1. Express its concentration in terms of molality. The formula for benzene is C6H6 and its molar mass is 78.1. A total of 0.082 1 mol of halide is mixed with 1 - 0.082 1 = 0.9179 mol of benzene, which has a mass of (0.917 9 mo1)(78.1 g/mol) = 71.7g = 0.071 7 kg. Hence Molality = 0.082 1 rnol = 1 145 mol/kg 0.071 7 kg DILUTION PROBLEMS 12.25. To what extent must a solution of concentration 40 mg AgN03 per cubic centimeter be diluted to yield one of concentration 16 mg AgN03 per cubic centimeter? Let V be the volume to which 1 cm3 of the original solution must be diluted to yield a solution of concentration 16mg AgN03 per cubic centimeter. Because the amount of solute does not change with dilution, Volume1 x concentrationl = volume2 x concentration2 1 cm3 x 40 mg/cm3 = V x 16 mg/cm3 Solving, V = 2.5 cm3. Each cubic centimeter of the original solution must be diluted to a volume of 2.5 cm3. Note that 2.5 is not the number of cubic centimeters of water to be added, but the final volume of the solution after water has been added to 1 cm3 of the original solution. The dilution formula always gives answers in terms of the total volume of solution. If we can assume that there is no volume shrinkage or expansion on dilution, the amount of water to be added in this problem is 1.5cm3 per cubic centimeter of original solution. Unless this assumption is made, the answers in the subsequent problems will be left in terms of the total volumes of the solutions. (The assumption of volume additivity is usually good for dilute solutions, e.g., < 0.1 M). 12.26. To what extent must a 0.50M BaC12 solution be diluted to yield one of concentration 20.0mg Ba2+ per cubic centimeter? The original solution contains 0.50 mol of BaC12 or of Ba2' per liter. The mass of Ba2+ in 0.50 rnol is (0.50 mo1)(137.3 g/mol) = 68.7 g Ba2+ Thus 0.50 M BaC12 contains 68.7 g Ba2+ per liter, or 68.7 mg Ba2+ per cubic centimeter. The problem now is to find the extent to which a solution of 68.7 mg Ba2+ per cubic centimeter must be diluted to yield one of concentration 20.0 mg Ba2+ per cubic centimeter. Volume1 x concentrationl = volume2 x concentration2 1 cm3 x 68.7 mg/cm' = V x 20.0 mg/cm3 Solving, V 3.43 cm3. = 3.43 cm3. Each cubic centimeter of 0.50 M BaC12 must be diluted with water to a volume of CHAP. 121 CONCENTRATIONS OF SOLUTIONS 199 12.27. A procedure calls for 100 cm3 of 20% H2S04, density 1.14 g/cm3. How much of the concentrated acid, of density 1.84g/cm3 and containing 98% H2S04 by weight, must be diluted with water to prepare 100cm3 of acid of the required concentration? The concentrations must first be changed from a mass basis to a volumetric basis, so that the dilution equation will apply. Mass of H2SO4 per cm3 of 20% acid = (0.20)(1.14 g/cm3) = 0.228 g/cm3 Mass of H2SO4 per cm3 of 98% acid = (0.98)(I .84 g/cm3) = 1.80 g/cm3 Let V be the volume of 98% acid required for 100cm3 of 20% acid. Volume1 x concentration, = volume2 x concentration2 100 cm3 x 0.228 g/cm3 = Solving V = 12.7cm3 of the concentrated acid. V x 1.80 g/cm3 12.28. What volumes of N/2 and of N/lO HCl must be mixed to give 2 L of N/5 HCl? Let x = volume of N/2 required; 2L - x = volume of N/10 required. Number of eq of N/5 = (number of eq of N/2)+(number of eq of N/10) (2 L ) ( f N ) = x ( i N ) + ( 2 L - x ) ( k N ) Solving x = 0.5 L. Thus 0.5 L of N/2 and 1.5 L of N/lO are required. Note that the assumption was made, implicitly, that no shrinkage (or the reverse) has occurred. 12.29. How many cubic centimeters of concentrated sulfuric acid, of density 1.84 g/cm3 and containing 98% H2SO4 by weight, should be taken to make (a) 1 L, of normal solution, (b) 1 L of 3.00N solution, (c) 200 cm3 of 0.500 N solution? Equivalent mass of H2S04 = $(molar mass) = f (98.1) = 49.0 The H2SO4 content of I L of the concentrated acid is (0.98)(1 000 cm3)(I .84 g/cm3)= 1 800 g H2SO4. The normality of the concentrated acid is 8oo g H2S04/L = 36.7 eq/L 49.0 g H2S04/eq The dilution formula, Vconc, Neon,, = x Vdil, x Ndil,, can now be applied to each case. (4 Vconc. = L)(l *O0 N, = 0.027 2 L = 27.2 cm3 conc. acid 36.7 N Vconc. = ( I L)(3*00 36.7 N = 0.081 7 L = 81.7 cm3 conc. acid (4 1 Vconc. = (200 cm3)(0.500 N) = 2.72 cm3 conc. acid 36.7 N 200 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 Supplementary Problems CONCENTRATIONS EXPRESSED IN PHYSICAL UNITS 12.30. How much NH4C1 is required to prepare 100cm3 of a solution of concentration 70mg NH4C1 per cubic centimeter? Ans. 7.0g 12.31. How many grams of concentrated hydrochloric acid, containing 37.9% HCl by weight, will give 5.0 g HCl? Ans. 13.2g 12.32. It is required to prepare 100 g of a 19.7% by weight solution of NaOH. How many grams each of NaOH and H 2 0 are required? Ans. 19.7g NaOH, 80.3 g H 2 0 12.33. How much CrC13 . 6 H 2 0 is needed to prepare 1 L of solution containing 20mg Cr3+ per cubic centimeter? Ans. 102g 12.34. What volume of dilute nitric acid, of density 1.11 g/cm3 and 19% H N 0 3 by weight, contains l o g HN03? ~ n s . 47cm3 12.35. How many cubic centimeters of a solution containing 40 g CaCI2 per liter are needed to react with 0.642 g of pure Na2C02? CaC03 is formed in the reaction. Ans. 16.8cm3 12.36. Ammonia gas is passed into water, yielding a solution of density of 0.93 g/cm3 and containing 18.6% NH3 by weight. What is the mass of NH3 per cubic centimeter of solution? Ans. 173mg/cm3 12.37. Given 100cm3 of pure water at 4"C, what volume of a solution of hydrochloric acid, density 1.175g/cm3 and containing 34.4% HCl by weight, could be prepared by absorption of the appropriate amount of HCl(g)? Ans. 130cm3 12.38. A liter of milk weighs 1 032 g. The butterfat which it contains to the extent of 4.0% by volume has a density of 0.865g/cm3. What is the density of the fat-free "skimmed" milk? Ans. 1.039g/cm3 12.39. To make a benzene-soluble cement, melt 49g of rosin in an iron pan and add 28g each of shellac and beeswax. How much of each component should be taken to make 75 kg of cement? Ans. 35 kg rosin, 20 kg shellac, 20 kg beeswax 12.40. How much CaCI2 . 6 H 2 0 and water must be weighed out to make lOOg of a solution that is 5.0% CaC12? Ans. 9.9 g CaC12. 6H20, 90.1 g water 12.41. How much BaC12 would be needed to make 250cm3 of a solution having the same concentration of C1- as one containing 3.78 g NaCl per 100cm3? Ans. 16.8g BaC12 CHAP. 121 CONCENTRATIONS OF SOLUTIONS 20 1 12.42. To determine the sulfate content of drinking water a 6.00liter sample was first concentrated to a small volume. Treatment of this with barium chloride solution resulted in the precipitation of 0.096 5 g of BaS04. Express the concentration of sulfate ion in the water in ppm. Ans. 6.62ppm 12.43. Analysis by mass spectrometer of an air sample from a chemical plant revealed the presence of 1.2 x 10-* moles of benzene (C6H6) per mole of air. Express the concentration by weight of benzene in the air in ppb. (Assume the average molar mass of air is 29). Ans. 32ppb CONCENTRATIONS EXPRESSED IN CHEMICAL UNITS 12.44. What is the molar concentration of a solution containing 37.5 g Ba(Mn04)2per liter, and what is the molar concentration with respect to each type of ion? Ans. 0.100 M Ba(Mn04)2, 0.100 M Ba2+, 0.200 M MnO, 12.45. A solution is labeled "0.100M Ba(Mn04)2."With what normality should it be labeled if used (a) as an oxidizing agent in strong acid (end-product Mn2+), (b) as an oxidizing agent in slightly acid solution (endproduct Mn02), or (c) as a precipitant for BaS04? Ans. (a) 0.500N; (b) 0.300N; ( c ) 0.200N 12.46. How many grams of solute are required to prepare 1 L of 1 M CaC12 . 6H20? Ans. 219.1 g 12.47. How many grams of anhydrous CaC12 are required to prepare (a) one liter of one molar CaC12, (b) 2.50 L of 0.200 M CaC12 2H20, (c) 650 cm3 of 0.600 M (3aC12? Ans. (a) 111 g; (b) 55.5g; (c) 43.3g 12.48. A 6.00 g sample of a polymer was dissolved in 280 mL of a salvent. Osmotic pressure measurements revealed that its concentration was 2 . 1 2 ~ M. Calculate the molar mass of the polymer. 10-4 ~ n s . A = 1.01 x 105g/mol. 12.49. Exactly lOOg of NaCl is dissolved in sufficient water to give 1 500cm3 of solution. What is the molar concentration? Ans. 1.14M (b) 45 g 12.50. Calculate the molality of a solution containing (a) 0.65 mol glucose, C6H1206, in 250 g water, glucose in 1 kg water, (c) 18g glucose in 200 g water. Ans. (a) 2.6 molal; (b) 0.25molal; 12.51. How many grams of CaC12 should be added to 300mL of water to make up a 2.46molal solution. Ans. 82g 12.52. A solution contains 57.5 cm3 of ethyl alcohol (C2H50H) and 600cm3 of benzene (C6H6). How many grams of alcohol are in 1000 g benzene? What is the molality of the solution? Density of C2H50His 0.80 g/cm3; of C6H6, 0.90 &In3. Ans. 85 g, 1.85mol/kg + (c) 0.50molal. 202 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 12.53. A solution contains 1O.Og of acetic acid, CH3COOH, in 125g of water. What is the concentration of the solution expressed as (a) mole fractions of CH3COOH and H20, (b) molality? Ans. (a) x(acid) = 0.024, x(water) = 0.976; (b) 1.33mol/kg 12.54. A solution contains 116 g acetone (CH3COCH3), 138 g ethyl alcohol (C2H~OH) 126 g water. Determine and the mole fraction of each. Ans. x(acetone) = 0.167, x(alcoho1) = 0.250, x(water) = 0.583 12.55. What is the mole fraction of the solute in a 1.00molal aqueous solution? Ans. 0.0177 12.56. An aqueous solution labeled 35.0% HC104 had a density 1.251 g/cm3.What are the molar concentration and molality of the solution? Ans. 4.36 M, 5.36 mol/kg 12.57. A sucrose solution was prepared by dissolving 13.5g C12H22011 enough water to make exactly 100cm3of in solution, which was then found to have a density of 1.050g/cm3. Compute the molar concentration and molality of the solution. Ans. 0.395 M, 0.431 mol/kg 12.58. For a solute of molar mass A, show that the molar concentration M and molality rn of the solution are related by M(&+-!-) =d where d is the solution density in g/cm3. (Hint: Show that each cubic centimeter of solution contains MA/lOOO grams of solute and M/rn grams of solvent.) Use this relation to check the answers to Problems 12.56 and 12.57. 12.59. What volume of a 0.232 N solution contains (a) 3.17 meq of solute, (b) 6.5 eq of solute? Ans. (a) 13.7cm3; (b) 28.0L 12.60. Determine the molar concentration of each of the following solutions: (a) 166g KI per liter of solution, (b) 33.0 g (NH4)2S04 in 200 cm3 of solution, (c) 12.5 g CuSO4.5H20 in 100 cm3 of solution, ( d ) 10.0mg A13+ per cubic centimeter of solution. Ans. (a) l.00M; (b) 1.25M; (c) 0.500M; ( d ) 0.370M 12.61. What volume of 0.200 M Ni(N03)2.6H20 contains 500 mg Ni2+? Ans. 42.6cm3 12.62. Compute the volume of concentrated H2SO4 (density 1.835g/cm3, 93.2% H2S04 by weight) required to make up 500 cm3 of 3.00 N acid. Ans. 43.0cm3 12.63. Compute the volume of concentrated HCl (density 1.19g/cm3, 38% HCl by weight) required to make up 18 L of N/50acid. Ans. 29cm3 CHAP. 121 CONCENTRATIONS OF SOLUTIONS 203 12.64. Determine the mass of KMn04 required to make 80 cm3 of N/8 K M n 0 4 when the latter acts as an oxidizing agent in acid solution and Mn2+ is a product of the reaction. Ans. 0.316g 12.65. Given the unbalanced equation Cr20;- + Fe2+ + H+ + Cr3' + Fe3+ + H 2 0 (a) What is the normality of a K2Cr207solution 35.0 cm3 of which contains 3.87 g of K2Cr207? (b)What is the normality of an FeS04 solution 750cm3 of which contains 96.3 g of FeS04? Ans. (a) 2.25N; (b) 0.845N 12.66. What mass of Na2S203-5H20 needed to make up 500cm3 of 0.200N solution for the following reaction? is 2s20:Ans. 24.8g + I2 + s& + 21- 12.67. A 4.51 g sample of an unknown compound was dissolved in 98.0 g of solvent. By close observation of its freezing point, the solution was found to be 0.388 rn. Calculate the molar mass of the unknown. Am. 119 g/mol 12.68. An analyst planned to use barium chloride to make up 60.0 mL of 0.500 M Ba2+ solution but his only source of the chloride in stock was 2.66g of BaC12.2H20. Knowing that NO, would not interfere with his procedure, he decided to make up the difference using Ba(N03)2.How much Ba(N03)2does he need? Ans. 4.99g DILUTION PROBLEMS 12.69. A solution contains 75 mg NaCl per cubic centimeter. To what extent must it be diluted to give a solution of concentration 15mg NaCl per cubic centimeter of solution? A m . Each cubic centimeter of original solution is diluted with water to a volume of 5cm3 12.70. How many cubic centimeters of solution of concentration l00mg Co2+ per cubic centimeter are needed to prepare 1.5 L of solution of concentration 20 mg Co2+ per cubic centimeter? Ans. 300cm3 12.71. Calculate the approximate volume of water that must be added to 250cm3 of 1.25 N solution to make it 0.500 N (neglecting volume changes). Ans. 375cm3 12.72. Determine the volume of dilute nitric acid (density 1.1 1 g/cm3, 1!J.O% H N 0 3 by weight) that can be prepared by diluting with water 50cm3 of the concentrated acid (density 1.42g/cm3, 69.8% H N 0 3 by weight). Calculate the molar concentrations and molalities of the concentrated and dilute acids. Ans. 234cm3; molar concentrations, 15.7 and 3.35; molalities, 36.7 and 3.72 12.73. What volume of 95.0% alcohol by weight (density 0.809 g/cm3) must be used to prepare 150cm3 of 30.0% alcohol by weight (density 0.957 g/cm3)? Ans. 56.0cm3 204 CONCENTRATIONS OF SOLUTIONS [CHAP. 12 12.74. What volumes of 12 N and 3 N HCl must be mixed to give 1 L of 6 N HCl? liter 12N, liter 3 N 12.75. What is the molar concentration of a solution made by mixing 300 mL of 0.0200 M H2S04with 200 mL of 3 0.0300 M H2S04? Ans. 0.024M 12.76. A chemist boiled down 500 mL of 0.0865 M NaCl until only 127 mL remained. Calculate its molar concen- tration. Ans. 0.341 M Chapter 13 Reactions Involving Standard Solutions ADVANTAGES OF VOLUMETRIC STANDARD SOLUTIONS Solutions of specified molar concentrations, as defined in Chapter 12, are called standard solutions and can be used conveniently for reactions involved in quantitative procedures. In a procedure called titration a standard solution is added slowly from a calibrated container to a reaction vessel until a first indication appears that the reactant in the vessel is completely consumed. Measured volumes of these solutions contain precisely determined amounts of solutes, according to the basic relation Number of moles = (number of liters) x (molar concentration) or Number of millimoles = (number of milliliters) :< (molar concentration) SOLUTION STOICHIOMETRY Stoichiometric calculations involving solutions of specified normalities are even simpler. By the definition of equivalent mass (Chapter 12), two solutions will react exactly with each other if they contain the same number of equivalents; that is, if Volume, x normality1 = volume2 x normality, In this relation the two volumes must be expressed in the same unit; the units may be chosen arbitrarily. Solutions of given normalities are useful even when only one of the reactants is dissolved. In this case, the number of equivalents (or milliequivalents) of the nondissolved reactant is computed in the usual way, by dividing the mass of the sample in grams (or milligrams) by the equivalent mass. The number of eq (or meq) of one reactant must still equal the number of eq (or meq) of the other. Solved Problems 13.1. What volume of 1.40 M H2S04 solution is needed to react exactly with 100 g Al? The balanced molecular equation for the reaction is 2A1+ 3H2S04+ A12(S04), + 3H2 Mole Method. 100 g Number of moles of A1 in 100 g A1 = = 3.70 mol 2'7.0 g/mol Number of moles of H2S04 required for 3.70 rnol A1 = f(3.70) 5.55 mol = Volume of 1.40 M H2SO4 containing 5.55 mol = 5.55 mol = 3.96 L 1.40 mol/L Factor- Label Me thod. 3 mol H2SO4 2 mol A1 205 ) () 1 L solution - = 3.96 L solution 206 REACTIONS INVOLVING STANDARD SOLUTIONS [CHAP. 13 1 . . In standardizing a solution of AgN03 it was found that 40.0mL was required to precipitate all 32 the chloride ions contained in 36.0mL of 0.520M NaCl. How many grams of Ag could be obtained from l00mL of the AgN03 solution? In the precipitation of AgCI, equimolar amounts of Ag+ and Cl- are needed; therefore equal numbers of moles of AgN03 and NaCl must have been used. n(AgN03) = n(NaC1) = (0.0360 L)(0.520 mol/L) = 0.018 72 mol Then 40.0 mL of the AgN03 solution contains 0.01 8 72 rnol AgN03, or 0.018 72 rnol Ag, and so 100 mL of solution contains (-)(0.01872 mol Ag)(107.9 g Ag/mol Ag) = 5.05 g Ag 1 . . Exactly 40.0mL of 0.225 M AgN03 was required to react exactly with 25.0 mL of a solution of 33 NaCN, according to the following equation: Ag' + 2CN- + Ag(CN), What is the molar concentration of the NaCN solution? n(AgN03) = (0.040 0 L)(0.225 mol/L) = 0.009 00 mol n(NaCN) = 2 x n(AgN03) = 0.018 0 mol Then 25.0 mL of the NaCN solution contains 0.018 0 rnol NaCN, so that M= 0.018 0 rnol = 0.72 M 0.025 L 1 . . How many milliliters of 6.0 N NaOH are required to neutralize 30 mL of 4.0 N HCl? 34 (Volume HC1) x (normality HCI) = (volume NaOH) x (normality NaOH) , (30 mL)(4.0 N) = (volume NaOH)(6.0 N) 1 . . Determine the normality of an H3P04 solution, 40mL of which neutralized 120mL of 0.531 N 35 NaOH. The solutions react exactly with each other. Therefore (Volume H3P04) x (normality H3P04)= (volume NaOH) x (normality NaOH) (40 mL)(normality H3P04)= (120 mL)(0.531 N) Note: In this problem we do not have to know whether one, two, or three hydrogens of H3P04 are replaceable. The normality was determined by reaction of the acid with a base of known concentration. Therefore the acid will have the same normality, 1.59N, in reactions with any strong base under similar conditions. In order to know the molar concentration of the acid, however, it would be necessary to know the number of replaceable hydrogens in the reaction. In a case like this, where a substance can have several equivalent masses, the normality determined by one type of reaction is not necessarily the normality in other types of reaction. If a weak base like NH3 were used instead of a strong base for neutralizing the phosphoric acid, or if the method of detecting the point of neutralization were changed, the equivalent mass of phosphoric acid, and hence the normality of the above CHAP. 131 REACTIONS INVOLVING STANDARD SOLUTIONS 207 solution, might be different. In order to predict the number of replaceable hydrogens of the acid in each case, detailed information about the chemistry of the acid and its neutralization must be known. 13.6. (a) What volume of 5.00 N H2SO4 is required to neutralize a solution containing 2.50 g NaOH? (b) How many grams of pure H2SO4 are required? (a) One equivalent H2S04 reacts completely with one equivalent NaOH. The equivalent mass of NaOH is the molar mass, 40.0. 2 50 g Number of equivalents in 2.50 g NaOH = -L-.I= 0.062 5 eq NaOH 40.0 g eq Number of liters x N = number of` eq Number of liters = Number of eq - 0.0625 N 5.00 = 0.0125 L or 12.5 mL of 5.00 N solution (b) Equivalent mass of H2S04= x (molar mass) = i(98.08) = 49.04 Mass of H2S04 required = (0.0625 eq)(49.04 g/eq) = 3.07 g 13.7. A 0.250-g sample of solid acid was dissolved in water and exactly neutralized by 40.0mL of 0.125 N base. What is the equivalent mass of the acid? Number of milliequivalents of base = (40.0 mL)(O.125 meq/mL) = 5.00 meq Number of milliequivalents of acid = number of meq of base = 5.00 meq 250 mg Equivalent mass of acid = ______ = 50 mg/meq = 50 g/eq 5.00 meq 13.8. Exactly 48.4mL of HCl solution is required to neutralize completely 1.24Og of pure CaC03. Calculate the normality of the acid. Each CO:- ion requires two Hf for neutralization: CO$- -t2H+ -+ CO2 + HzO. Thus, the equivalent mass of CaC03 is one-half the molar mass, or 50.05. Number of equivalents in 1.240 g CaCO - 1..240g = 0.0248 eq CaC03 - XQm Hence, 48.4 mL of acid solution contains 0.024 8 eq HC1. N= number of eq 0.0248 eq -= 0.512 N number of liters 0.0484 L 13.9. When 50.0mL of a certain Na2C03 solution was titrated with 0.102 M HCl, 56.3mL were required for complete neutralization, according to the equation: CO:- + 2H+ --+ CO2 + H 2 0 . Calculate the number of grams of CaC03 that would be precipitated if an excess of CaC12 were added to a separate 50.0mL portion of the Na2C03 solution. Factor-Label Method. Amount of CaC03 = = 0.287 g CaC03 13.10. A 10.0-g sample of "gas liquor" is boiled with an excess of NaOH and the resulting ammonia is passed into 60 mL of 0.90 N H2S04.Exactly 10.0mL of 0.40 N NaOH is required to neutralize the 208 REACTIONS INVOLVING STANDARD SOLUTIONS [CHAP. 13 excess sulfuric acid (not neutralized by the NH3). Determine the percent ammonia in the "gas liquor" examined. Number of meq NH3 in 10.0 g of gas liquor = (number of meq H2SO4) - (number of meq NaOH) = (60 = 50 mL)(0.90 meq/mL) - (10.0 mL)(0.40 meq/mL) meq NH3 In neutralization experiments, the equivalent mass of NH3 is the same as the molar mass, 17.0,in accord with the equation NH3 + H+ + NH: Then the mass of NH3 in the sample is (50 meq)(l7.0 mg/meq) = 850 mg = 0.85 g, and Fraction NH3 in sample = 085 g = 0.085 = 8.5% 10.0 g 13.11. A 40.8 mL sample of an acid is equivalent to 50.0 mL of an Na2C03solution, 25.0 mL of which is equivalent to 23.8 mL of 0.102 N HC1. What is the normality of the first acid? The volume of HCI that would have been required for 50.0mL of Na2C03solution is (%)(23.8 mL) = 47.6 mL Volume1 x normality1 = volume2 x normality2 (40.8 mL)NI = (47.6 mL)(O.102 N) NI = 0.119 N 13.12. Calculate the number of grams of FeS04 that will be oxidized by 24.0 mL of 0.250 N KMn04 in a solution acidified with sulfuric acid. The unbalanced equation for the reaction is MnO, + Fe2+ + H+ --+ Fe3++ Mn2+ + H 2 0 and the normality of the KMn04 is with respect to this reaction. It is not necessary to balance the complete equation. All that need be known is that Fe changes in oxidation state from +2 in Fe2+ to +3 in Fe3+.Then Equivalent mass of FeSO, = molar mass - 152 - -= oxidation state change 1 152 The same result may be found from the balanced partial equation for the Fe2+; Fe2+ -+ Fe3+ + e - . Equivalent mass of FeS04 = Let x = required mass of FeS04. Number of eq KMn04 = number of eq FeS0, (Volume KMn04) x (normality KMn04) = (0.0240 L)(0.250 eq/L) = Solving x = 0.912g FeS0,. mass of FeS04 equivalent mass of FeS04 X ~ molar mass number of electrons transferred 152 - -= - 1 152 152 g/eq CHAP. 131 REACTIONS INVOLVING STANDARD SOLUTIONS 209 13.13. What volume of 0.100 0 N FeS04 is required to reduce 4.000 g KMn04 in a solution acidified with sulfuric acid? The normality of the FeS04 is with respect to the oxidation-reduction reaction given in Problem 13.12. In this reaction the Mn changes in oxidation state from +7 in MnO, to +2 in Mn2+.The net change is 5. Or, from the balanced partial equation MnO, + 8H++5e- -+Mn2+ + 4H20 it can be seen that the electron transfer is 5 for each Mn0;. The equivalent mass of K M n 0 4 in this reaction is then ix (molar mass) = i(158.0) = 31.6 Number of eq FeS04 = nurnber of eq KMnO, 4.000 g (Volume FeS04) x (0.100 0 eq/L) = -~ 31.6 g/eq Volume FeS0, = 1.266 L 13.14. A sample known to contain some As203 was brought into solution by a process which converted the arsenic to H3As03. This was titrated with a standard 12 solution according to the equation H ~ A s O ~2 + H20 --+ H ~ A s O ~ +1 + 21- + 2HS Exactly 40.27mL of standard 12 solution was required to reach the point of completion as indicated by the persistence of a faint I2 color. The standard solution had been prepared by mixing 0.419 2 g of pure KI03 with excess KI and acid and diluting to exactly 250.0 mL. The I2 is formed in the reaction: 10, + 51- + 6H+ + 312 + 3H20 Calculate the mass of As203 in the sample. First it is necessary to calculate the molar concentration of the I2 solution. The factor-labelmethod is used. M- 0.4192 g KIO, mol ~ ~ ~ ~ (214.0 g KI03/mol) (1 mol KI20 3) ( ' ~ ~ 0.023 51 .mol/L 0.023 51 M ) I = = Then, by the same method: (0.040 27 L)(0.023 51 mol/L) 1 mol H ~ A s ) (~~ ~ ~ ~ 2 , 1 ) ( 1 9 As203/mol) = 0.093 63 g of AS203 O g7 . 8 Supplementary Problems 13.15. How many milliliters of 0.25 M AgN03 are required to precipitate all the chromate ion from 20 mL of a solution containing lOOg of Na2Cr04 per liter? The reaction is: 2Ag' Ans. 99mL + CrOi- --+ Ag2Cr04(s) 210 REACTIONS INVOLVING STANDARD SOLUTIONS [CHAP. 13 1 . 6 A 50.0mL sample of Na2S04 solution is treated with an excess of BaC12.If the precipitated BaS04 is 1.756g, 31. what is the molar concentration of the Na2S04 solution? Ans. 0.1505M 13.17. What was the thorium content of a sample that required 35.0 mL of 0.020 0 M H2C2O4 to precipitate all the thorium as Th(C204)2? Ans. 81mg 1 . 8 What is the molar concentration of a &Fe(CN), solution if 40.0mL was required to titrate 150.0mg Zn 31. (dissolved) by forming K2Zn3[Fe(CN)&? Ans. 0.0382M 13.19. A 50.0mL sample of NaOH solution requires 27.8 mL of 0.100 N acid in a titration. What is its normality? How many milligrams NaOH are in each cubic centimeter? Ans. 0.055 6 N, 2.22 mg/cm3 1 . 0 In standardizing HC1, 22.5mL was required to neutralize 25.0mL of 0.050OM Na2C03 solution. What is 32. the molarity and the normality of the HC1 solution? How much water must be added to 200 mL of it to make it 0.100N? Ans. 0.111 M, 0.111 N, 22mL 1 . 1 Exactly 21 mL of 0.80 N acid was required to neutralize completely 1.12g of an impure sample of calcium 32. oxide. What is the purity of the CaO? Ans. 42% 1 . 2 By the Kjelduhl method, the nitrogen contained in a foodstuff is converted into ammonia. If the ammonia 32. ' from 5.0g of a foodstuff is just sufficient to neutralize 20mL of 0.100 M nitric acid, calculate the percentage of nitrogen in the foodstuff. Ans. 0.56% 1 . 3 What is the purity of concentrated H2SO4 (density 1.800g/cm3) if 5.00cm3 after diluting in water, is 32. neutralized by 84.6 mL of 2.000 M NaOH? Ans. 92.2% 1 . 4 A 10.0mL portion of (NH4)2S04solution was treated with an excess of NaOH. The NH3 gas evolved was 32. absorbed in 50.0mL of 0.100HCl. To neutralize the remaining HC1, 21.5mL of 0.098M NaOH was required. What is the molar concentration of the (NH4)2S04?How many grams of (NH4)2S04are in a liter of solution? Ans. 0.145 M, 19.1 g/L 1 . 5 Exactly 400mL of an acid solution, when acted upon by an excess of zinc, evolved 2.430 L of H2 gas 32. measured over water at 21C and 747.5 torr. What is the normality of the acid? Vapor pressure of water at 21C is 18.6 torr. Ans. 0.483N 1 . 6 How many grams of copper will be replaced from 2.0 L of 0.150 M CuS04 solution by 2.7 g of aluminum? 32. Ans. 9.5g CHAP. 131 REACTIONS INVOLVING STANDARD SOLUTIONS 21 1 13.27. What volume of 1.50M H2SO4 is needed to liberate 185 L of hydrogen gas at S.T.P. when treated with an excess of zinc? Ans. 5.51 I, 13.28. How many liters of hydrogen at S.T.P. would be replaced from 500 mL of 3.78 M HCl by 125 g of zinc? Ans. 21.21, 13.29. What is the equivalent mass of an acid 1.243g of which required 3 1.72mL of 0.192 3 N standard base for neutralization? Ans. 203.8 13.30. The molar mass of an organic acid was determined by the following study of its barium salt. 4.290g of the salt was converted to the free acid by reaction with 21.64 mL of 0.477 M H2S04. The barium salt was known to contain 2mol of water of hydration per mol Ba2+, and the acid was known to be monoprotic (monobasic). What is the molar mass of the anhydrous acid? Ans. 122.1 13.31. An iron(I1) sulfate solution was standardized by titration. A 25.OOmL portion of the solution required 42.08 mL of 0.080 0 N ceric sulfate for complete oxidation. What is the normality of the iron(I1) sulfate? Ans. 0.1347N 13.32. How many milliliters of 0.025 7 N KI03 would be needed to reach the end point in the oxidation of 34.2 mL of 0.041 6 N hydrazine in hydrochloric acid solution? Ans. 55.4mL 13.33. How many grams of FeC12 will be oxidized by 28 mL of 0.25 N K2Cr207in HC1 solution? The unbalanced equation is Fe2+ + Cr20:Ans. 0.89g + H+ -+ Fe3+ + Cr3++ H 2 0 13.34. In a standardization procedure 13.76mL of ion(I1) sulfate solution was required to reduce a 25.00mL portion of potassium dichromate solution, which was prepared by dissolving 1.692g of pure K2Cr207 in water and diluting to 500.0 mL. (See reaction in Problem 13.33.) Calculate the molarity and the normality of both the potassium dichromate and the iron(I1) sulfate solutions. Ans. K2Cr2070.011 50 My0.069 01 N; FeS04 0.125 4 M, 0.125 4 N 13.35. What mass of MnO2 is reduced by 35mL of 0.080M oxalic acid, H2C204, in sulfuric acid solution? The unbalanced equation is Mn02 + H+ + HZC2O4 --+ CO2 + H20 + Mn2+ Ans. 0.24g 13.36. How many grams of KMn04 are required to oxidize 2.40g of FeS04 in a solution acidified with sulfuric acid? What is the equivalent mass of KMn04 in this reaction? Ans. 0.500 g, 31.6 13.37. Find the equivalent mass of KMn04 in the reaction Mn2++ MnO, + H20 --+ Mn02 + H+ (unbalanced) 212 REACTIONS INVOLVING STANDARD SOLUTIONS [CHAP. 13 How many grams of MnS04 are oxidized by 1.25g KMn04? Ans. 52.7, 1.79g 13.38. (a) What volume of 0.066 7 M K2Cr207 is required to liberate the chlorine from 1.20g of NaCl in a solution acidified with H2SO4? Cr20? + C1- + Ht -+ Cr3+ C12 + +H20 (unbalanced) (b) How many grams of K2Cr207are required? Ans. (a) 51 mL; ( b ) 1.01 g; (c) 0.73g (c) How many grams of chlorine are liberated? 13.39. If 25.0 mL of an iodine solution is equivalent as an oxidizing agent to 0.125 g of K2Cr207,to what volume should 1000mL be diluted to make the solution 0.050OM? The half-reaction for the iodine is: I2 + 2eAns. + 21- 1020mL 13.40. How many grams of K M n 0 4 should be taken to make up 250mL of a solution of such concentration that 1 mL is equivalent to 5.00mg of iron in FeS04? Ans. 0.707g 13.41. How many grams of iodine are present in a solution which requires 40 mL of 0.1 12 M Na2S203 to react with it? S20:Ans. 0.57g + 12 -+ S40i- + I- (unbalanced) 13.42. All the manganese in a certain solution was converted to MnO, by contact with solid sodium bismuthate. A 25.00mL portion of 0.0200 M FeS04 was added, which was more than sufficient to completely reduce the MnO, to Mn2+in an acidified medium. The excess Fe2+ was then back-titrated in an acid solution, requiring 4.21 mL of 0.0106 M KMn04. How many milligrams of manganese were in the original solution? Ans. 3.04mg 13.43. Reducing sugars are sometimes characterized by a number RcU, which is defined as the number of milligrams of copper reduced by 1 gram of the sugar, in which the half-reaction for the copper is Cu2+ + OH-+ C u 2 0 + H 2 0 (unbalanced) It is sometimes more convenient to determine the reducing power of a carbohydrate by an indirect method. In this method 43.2 mg of the carbohydrate was oxidized by an excess of K3Fe(CN)6. The Fe(CN):- formed in this reaction required 5.29 mL of 0.034 5 N Ce(S04)2 for reoxidation to Fe(CN)i- (the normality of the ceric sulfate solution is given with respect to the reduction of Ce4+ to Ce3+).Determine the Rc,-value for the sample. (Hint: The number of milliequivalents of Cu in a direct oxidation is the same as the number of milliequivalents of Ce4+ in the indirect method.) Ans. 268 13.44. A volume of 12.53mL of 0.050 93 M selenium dioxide, Se02, reacted exactly with 25.52 mL of 0.100 0 M CrS04. In the reaction, Cr2+ was oxidized to Cr3+.To what oxidation state was the selenium converted by the reaction? Ans. 0 CHAP. 131 REACTIONS INVOLVING STANDARD SOLUTIONS 213 13.45. An acid solution of a KRe04 sample containing 26.83mg of combined rhenium was reduced by passage through a column of granulated zinc. The effluent solution, including the washings from the column, was then titrated with 0.1000N KMn04; 11.45mL of the standard permanganate was required for the reoxidation of all the rhenium to the perrhenate ion, Reoz. Assuming that rhenium was the only element reduced, what is the oxidation state to which rhenium was reduced by the zinc column? Ans. - 1 13.46. The iodide content of a solution was determined by titration with ceric sulfate in the presence of HCl, in which I- is converted to IC1 and Ce4' to Ce3+.A 250 mL sample of the solution required 20.0 mL of 0.050 M Ce4+ solution. What is the iodide concentration in the original solution, in mg/mL? Ans. 0.25 mg/mL 13.47. A 0.518 g sample of limestone is dissolved, and then the calcium is precipitated as calcium oxalate, CaC204. The precipitate is filtered, washed, and dissolved in sulfuric acid solution. It then requires 40.0mL of 0.050 0 M KMn04 solution to titrate it. What is the percent CaO in the limestone? The unbalanced equation for the titration is MnO, Ans. 54.2% 13.48. Assume the reaction between phosphate ions and calcium ions goes to completion to form Ca3(P04)2.If + CaC204+ (H+),SO:- --+ CaS04 + Mn2+ + CO2 + H 2 0 50.0mL of 0.400M Ca(NO& is mixed with 100.0mL of 0.300M Na3P04, calculate the final molar concentrations of Ca2', NOT, Na+, and PO:- in the resulting solution. Ans. no C:a2+, 0.267 M NOT, 0.600 M Na', and 0.111 M PO:13.49. How many milliliters of 0.087 6 M NaOH should be added to 50.0 mL of 0.091 6 M HC1 to yield a solution in which the H' concentration is 0.001 0 M? Ans. 51.1 mL 13.50. One problem with standard NaOH solutions is that they absorb CO2 from the air, which reacts with OH- to form CO:-, which obscures the end-point in titrations of acid. An enterprising young chemist had 975.0 mL of 0.3664 M NaOH, which she suspected had absorbed some CO2, so she added 10.0mL of 0.500 M Ba(OH)2, which precipitated all of the CO:- as BaC03(s). After the BaC03 was filtered off, the solution was restandardized and found to be 0.368 9 M in OH-. How many grams of solid BaC03 had been removed? Ans. 0.38g This page intentionally left blank Chapter 14 Properties of Solutions INTRODUCTION Just as dilute gases are characterized by more or less general adherence to a group of simple laws, so dilute solutions as a class have many properties that are determined by concentration alone, without reference to the particular nature of the dissolved materials. All solutions obey the laws described below when the concentration is sufficiently low. A few of the laws, specifically designated in the discussion below, are obeyed over the entire range of composition by certain pairs (or groups of more than two) of substances. Such pairs of substances are said to form ideal solutions. In an ideal solution the forces of interaction between solvent and solute molecules are the same as between the molecules in the separate components. In the formation of an ideal solution from the separate components, there are no volume changes and no enthalpy changes. Pairs of chemically similar substances, such as methanol (CH30H) and ethanol (C2H50H), or benzene (C6HS) and toluene (C7Hs), for ideal solutions. But dissimilar substances, such as C2H50H and C6H6, form nonideal solutions. VAPOR PRESSURE The vapor pressures of all solutions of nonvolatile solutes i n a solvent are less than that of the pure solvent. If we prepare solutions of different solutes in a given solvent by adding equal numbers o solute f molecules to a fixed amount of solvent, as we do in preparing solutions of equal molality, we find that the depression of the vapor pressure is the same in every case in dilute solutions of nonvolatile nonelectrolytes. Raoult 's law states that in dilute solutions of nonvolatile nonelectrolytes the depression is proportional to the mole fracture o the solute, or the solution vapor pressure is proportional to the mole fraction o f f the solvent. In equation form: Depression of solvent vapor pressure AP = (vapor pressure of pure solvent) - (vapor pressure of solution) = (vapor pressure of pure solvent) x (mole fraction of solute) = or Vapor pressure of solvent over solution = (vapor pressure of pure solvent) x (mole fraction of solvent) In the second form, the vapor pressure of the solution has been identified with the vapor pressure of the solvent over the solution, since the solute is assumed to be nonvolatile. In systems of liquids that mix with each other in all proportions to form ideal solutions, Raoult's law in the form of the second equation above applies to the partial presure of each volatile component separately. Partial pressure of any component over solution = (vapor pressure of that pure component) x (mole fraction of component) Raoult's law is explained by the hypothesis that solute molecules at the liquid surface interfere with the escape of solvent molecules into the vapor phase. Because of the vapor pressure lowering, the boiling point of the solution is raised and the freezing point is lowered, as compared with the pure solvent. FREEZING-POINT LOWERING, AT, When most dilute solutions are cooled, pure solvent begins to crystallize before any solute crystallizes. The temperature at which the first crystals are in equilibrium with the solution is called the freezing 215 216 PROPERTIES OF SOLUTIONS [CHAP. 14 point of the solution. The freezing point of such a solution is always lower that the freezing point of pure solvent. In dilute solutions, the lowering of the freezing point is directly proportional to the number of solute molecules (or moles) in a given mass of solvent. Thus, Lowering of freezing point = ATf = (freezing point of solvent) - (freezing point of solution) = Kfm where m is the molality (Chapter 12) of the solution. If this equation were valid up to a concentration of 1 molal, the lowering of the freezing point of a 1 molal solution of any nonelectrolyte dissolved in the solvent would be K f , which is thus called the molal freezing-point constant of the solvent. The numerical value of Kf is a property of the solvent alone. EXAMPLE 1 The molal freezing-point constant for water is 1.86 K . kg/mol or 1.86"C.kg/mol. Thus if 1 mol of cane sugar (342g sugar) is dissolved in 1 OOOg of water, the solution will freeze at -1.86"C. BOILING-POINT ELEVATION, A T b The temperature at which a solution boils is higher than that of the pure solvent if the solute is relatively nonvolatile. In dilute solutions, the elevation of the boiling point is directly proportional to the number of solute molecules (or moles) in a given mass of solvent. Again, the molality scale is usually used, and the equation is Elevation of boiling point = AT, = (boiling point of solution) - (boiling point of solvent) = Kbm Kb is called the molal boiling-point constant of the solvent. As with K f , the numerical value of Kb is a property of the solvent alone and is independent of the nature of the solute, within the general requirements of nonvolatility and nondissociation into ions. EXAMPLE 2 The molal boiling-point constant for water is 0.513"C kg/mol. Thus if 1 mol of cane sugar (342g sugar) is dissolved in 1 OOOg of water, the solution will boil at 100.513"C, assuming a pressure of 1 atm. (If pure water boils at a temperature slightly different from 100C because the air pressure is not exactly one standard atmosphere, the magnitude of ATb is still 0.513, and the boiling point of the solution is 0.513"C higher than the actual boiling point of the water.) If a solution contains fmol of sugar (171 g) and 1 OOOg water, it will boil at 100.256"C at 1 atm. 1 OSMOTIC PRESSURE If a solution is separated from a sample of pure solvent by a porous sheet that allows solvent, but not solute molecules to pass through, solvent will move into the solution in an attempt to equalize the concentrations on the two sides of the sheet. Such a dividing sheet is called a semipermeable membrane. If the membrane is placed vertically and the vessel holding the solution can extend indefinitely in a horizontal direction to accommodate the incoming material, solvent will continue to flow until it is all used up or until the solution becomes so dilute that there is no more driving force due to the difference in solvent concentrations on the two sides. If, however, the solution vessel is closed on all sides except for an extension tube on top as shown in Fig. 14-1, the incoming solvent will force some of the solution up the extension tube. The weight of this solution in the tube will exert a downward pressure that will tend to oppose the inward penetration of more solvent through the membrane. Eventually the two forces will just balance each other and no more solvent will enter. These two forces are the weight of the hydrostatic head of solution in the tube and the driving force tending to equalize the concentrations on the two sides of the membrane. The hydrostatic head at this balance point is called the osmotic pressure of the solution. It can be measured in usual units of pressure, such as Pa, atm, or torr. The osmotic pressure, 7r, of a dilute solution of a nonelectrolyte is given by an equation formally equivalent to the ideal gas law: 7r = MRT CHAP. 141 PROPERTIES OF SOLUTIONS 217 Fig. 14-1 With the molar concentration M in mol/L, the temperature T in K, and with R 7r = 0.082 1 L - atm K-' a . mol-' will be obtained in atm. DEVIATIONS FROM LAWS OF DILUTE SOLUTIONS The above laws are valid only for dilute solutions of nonelectrolytes. For electrolyte solutions each ion contributes independently to the effective molality or molar concentration. On account of the electrical interactions between ions, however, none of the effects is as large as would be predicted on the basis of simple counting of ions. EXAMPLE 3 A solution containing 0.100 mol of KCl per kilogram of water freezes at -0.345 "C. The observed lowering is slightly smaller than (1.86 "C . Kg/mol)(O.2 mol/kg) = 0.372 "C which would be predicted if each K+ and C1- were truly independently effective (0.1 molal K+ + 0.1 molal Cl-). A solution containing 0.100 mol of BaC12per kilogram of water freezes at -0.470 "C, showing a lowering somewhat less than the (1.86)(0.3) = 0.558 "C predicted from the simple additivity of molalities (0.1 molal Ba2+ + 0.2 molal Cl-). For any solution not too concentrated, whether electrolyte or nonelectrolyte, the deviations from any one of the laws of the dilute solution are equal to the deviations from any of the others, on a fractional or percentage basis. That is, in which a " labels the magnitude of the effect that would be predicted by the laws of the dilute solution. 218 PROPERTIES O F SOLUTIONS [CHAP. 14 SOLUTIONS OF GASES IN LIQUIDS At constant temperature, the concentration of a slightly soluble gas in a liquid (i.e., the mass or number of moles of gas dissolved in a given volume of liquid) is directly proportional to the partial pressure of the gas in the space above the liquid. This is known as Henry's law. When a mixture of two gases is in contact with a solvent, the amount of each gas that is dissolved is the same as if it were present alone at a pressure equal to its own partial pressure in the gas mixture. LAW OF DISTRIBUTION A solute distributes itself between two immiscible solvents so that the ratio of its concentrations in dilute solutions in the two solvents is constant, regardless of the actual concentration in either solvent. Here, both concentrations are assumed to be on the same volumetric basis (e.g., mol/L). EXAMPLE 4 For the distribution of iodine between ether and water at room temperature, the value of the constant is about 200. Thus Concentration of iodine in ether = K = 200 Concentration of iodine in water The value of this ratio of concentrations, called the distribution ratio or distribution coeficient, is equal to the ratio of the solubilities (per unit volume) in the two solvents if saturated solutions in these solvents are dilute enough for the law of distribution to apply. Solved Problems FREEZING-POINT LOWERING 14.1 The freezing point of pure camphor is 178.4 "C, and its molal freezing-point constant, Kf,is 40.0 "C - kg/mol. Find the freezing point of a solution containing 1.50 g of a compound of molar mass 125 in 35.0g of camphor. The first step is to find the molality (m) of the solution m= number of moles of solute number of kilograms of solvent - (1.50/125) mol solute = 0.343 mol/kg kg solvent (a) Lowering of freezing point = AT, Freezing point of solution = Kfm = (40.0 "C . kg/mo1)(0.343 mol/kg) = 13.7"C = (freezing point of pure solvent) - AT, = 178.4"C - 13.7"C= 164.7"C 14.2. A solution containing 4.50 g of a nonelectrolyte dissolved in 125 g of water freezes at -0.372 "C. Calculate the approximate molar mass of the solute. First Method. First compute the molality from the freezing-point equation. AT,. = Kfm 0.372 "C = (1.86 "C . kg/mol)m m= 0'372 1.86 "C . kg/mol = "' 0.200 mol/kg CHAP. 141 PROPERTIES O F SOLUTIONS 219 From the definition of molality compute the number of moles of solute in the sample. n(so1ute) = (0.200 mol solute/kg solvent)(O.l25 kg solvent) = 0.025 0 mol solute Then Molar mass = 4.50 g solute = 180 g/mol 0.025 0 mol solute Second Method. Putting the definition of molality into the freezing-point equation: ATf = Kfm = Kf x amt. of solute in grams/A of solute amt. of solvent in kilograms where A is the molar mass of the solute. Solving for A: amt. of solute in grams amt. of solvent in kilograms ) - (0.372)(0.125) (136)(4.50) = 180 g/mol BOILING-POINT ELEVATION 14.3. The molar mass of a nonvolatile solute is 58.0. Compute the boiling point of a solution containing 24.0 g of the solute and 600 g of water, when the barometric pressure is such that pure water boils at 99.725"C. Molality = m = (24.0/58.0) mol solute n(so1ute) = 0.690 mol/kg number of kg solvent 0.600 kg solvent Elevation of boiling point = AT, = Kbm = (0.513 "C - kg/mo1)(0.690 mol/kg) = 0.354"C Boiling point of solution = (boiling point of water) + AT, = 99.725 "C + 0.354 "C = 100.079 "C 14.4. A solution was made up by dissolving 3.75 g of a pure nonvolatile solute in 95 g of acetone. The boiling point of pure acetone was observed to be 55.95 "C, and of the solution, 56.50 "C. If the molal boiling-point constant of acetone is 1.7 1 "C - kg/mol, what is the approximate molar mass of the solute? First Method. Compute the molality (m) from the boiling-point equation AT, = Kbm 56.50 - 55.95 = 1.7l.m Solving, m = 0.322 mol solutelkg solvent. Now find the number of moles of solute in the weighed sample. mol solute (0.095 kg solvent) = 0.030 6 mol solute kg solvent Then Molar mass = 3.75 g solute - = 123 g/mol 0.030 6 mol solute Second Method. As in Problem 14.2, we can develop the expression A= amt. of solute in grams (1.71)(3.75) = 123 g/mol ( A T ) (amt. of solvent in kilograms) = (0.55)(0.095) 5 220 PROPERTIES OF SOLUTIONS [CHAP. 14 VAPOR PRESSURE 14.5. The vapor pressure of water at 28C is 28.35 torr. Compute the vapor pressure at 28C of a solution containing 68 g of cane sugar, C12H22OI1,in 1000 g of water. Number of moles of CI2H22Ol1 = Number of moles of H 2 0 = 68 = 0.20 mol C12H22011 342 g/mol Oo0 18.02 g/mol = 55.49 mol H 2 0 Total number of moles = 0.20 + 55.49 = 55.69 mol Mole fraction C12H22011 -= 0.003 6 = 0 20 55.69 Mole fraction H20 = -= 0.996 4 55 49 55.69 First Method. Vapor pressure of solution = (v.P. of pure solvent) x (mole fraction of solvent) = (28.35 torr)(0.996 4) = 28.25 torr Second Method. Vapor pressure depression = AP = (v.P. of pure solvent)x(mole fraction of solute) = (28.35 torr)(0.003 6) = 0.10 torr Vapor pressure of solution = (28.35 - 0.10) torr = 28.25 torr 14.6. At 30 "C, pure benzene (molar mass 78.1) has a vapor pressure of 121.8 torr. Dissolving 15.0g of a nonvolatile solute in 250 g of benzene produced a solution having a vapor pressure of 120.2 torr. Determine the approximate molar mass of the solute. Let A be the molar mass of the solute. Number of moles of benzene = 250 g = 3.20 mol benzene 78.1 g/mol A 15.0 Number of moles of solute = - mol solute Substituting in the relation v.p. solution = (v.P. pure solvent) x (mole fraction solvent), 120.2 torr = (121.8 torr) [ 3.20 mol (15.0/A) mol 3.20 mol + or 120.2 = (121.8)( 1 5 . ~ f ~ ~ o M ) 1 Solving A = 350. Note that the accuracy of the calculation is limited by the term 121.8 - 120.2 that appears in the expansion. The answer is significant only to 1 part in 16. 14.7. At 20 "C the vapor pressure of methyl alcohol (CH30H) is 94 torr and the vapor pressure of ethyl alcohol (C2HSOH) is 44 torr. Being closely related, these compounds form a two-component system which adheres quite closely to Raoult's law throughout the entire range of concentrations. If 20 g of CH30H is mixed with 100 g of C2HSOH, determine the partial pressure exerted by each and the total vapor pressure of the solution. Calculate the composition of the vapor above the solution by applying Dalton's law (Chapter 5). CHAP. 141 PROPERTIES OF SOLUTIONS 22 1 In an ideal solution of two liquids, there is no distinction between solute and solvent, and Raoult's law holds for each component of such solutions. Hence, when two liquids are mixed to give an ideal solution, the partial pressure of each liquid is equal to its vapor pressure (in the pure state) multiplied by its mole fraction in the solution. The molar masses of CH30H and C ~ H S O H 32 and 46, and so are Partial pressure of C H 3 0 H = (94 torr) = (94 8 mol C H 3 0 H +% mol C 2 H 5 0 H torr)(0.22) = 21 torr Partial pressure of C2H50H= (44 torr) $f' mol C 2 H 5 0 H CH30H +% mol C2H50H = (44 torr)(0.78) = 34 torr The total pressure of the gaseous mixture is the sum of the partial pressures of all the components (Dalton's law): total pressure of solution= 21 torr + 34 torr = 55 torr. Dalton's law also indicates that the mole fraction of any component of a gaseous mixture is equal to its pressure fraction, i.e., its partial pressure divided by the total pressure. Mole fraction of C H 3 0 H in vapor = Mole fraction of C2H50H in vapor = partial pressure of C H 3 0 H - 21 torr --...-. 55 torr total pressure - 0.38 ' partial presslire of C 2 H 5 0 H- 34 torr = 0.62 total pressure 55 torr Since the mole fraction of (ideal) gases is the same as the volume fraction, we may also say that the vapor consists of 38% CH30H by volume. Note that the vapor is relatively richer in the more volatile component, methyl alcohol (mole fraction 0.38), than is the liquid (mole fraction of CH30H, 0.22). OSMOTIC PRESSURE 14.8. What would be the osmotic pressure at 17 "C of an aqueous solution containing 1.75 g of sucrose (C12H2201 per 150 mL of solution? Molar concentration = M = number of moles of solute number of liters of solution - 1.75 g/(342 g/mol) = 0.034 1 mol/L 0.150 L = 0.812 Osmotic pressure = 7r = M R T = (0.034 1 mol/L)(0.082 1 L, . atm . K-' . mol-')(290 K) atm 14.9. The osmotic pressure of a solution of a synthetic polyisobutylene in benzene was determined at 25 "C.A sample containing 0.20 g of solute per 100 mL of solution developed a rise of 2.4 mm at osmotic equilibrium. The density of the solution was 0.88g/cm3. What is the molar mass of the polyisobut y lene? The osmotic pressure is equal to that of a column of the solution 2.4mm high. By the formula of Chapter 5. 7r = height x densityxg = (2.4 x 10-3 m)(0.88 x 103 kg/m3)(9.81 m/s2) = 20.7 Pa The molar concentration can now be determined from the osmotic pressure equation. 20.7 N/m2 = 8.3 x 10-3 mol/m3 = 8.3 x 10-6 mol/L (8.314 5 J K-' mol-')(298 K) M=-= IT RT The solution contained 0.20g solute per l00mL solution, or 2.0g per liter, and has been found to contain 8.3 x 10-6mol/L. Then 2.0 g = 2.4 x 105 g/mol Molar mass = 8.3 x 10-6 mol 222 PROPERTIES OF SOLUTIONS [CHAP. 14 14.10. An aqueous solution of urea had a freezing point of -0.52 "C. Predict the osmotic pressure of the same solution at 37C. Assume that the molar concentration and the molality are numerically equal. The concentration of the solution is not specified, but the effective molality may be inferred from the freezing-point lowering. m = -AT, = K,- 0.52 'C = 0.280 mol/kg 1.86 "C . kg/mol The assumption that the molality and molar concentration are equal is not very bad for dilute aqueous solutions. (The relation found in Problem 12.58 shows that M M rn when d M 1 g/cm3 and A << 1 000/m. Urea has a molar mass of 60.) Then 0.280mol/L may be used for the molar concentration in the osmotic pressure equation. 7r = M R T = (0.280 mol/L)(0.082 L . atm . K-' . mol-')(310 K) = 7.1 atm SOLUTIONS OF GASES IN LIQUIDS 14.11. At 20 "C and a total pressure of 760 torr, 1 L of water dissolves 0.043 g of pure oxygen or 0.019 g of pure nitrogen. Assuming that dry air is composed of 20% oxygen, and 80% nitrogen (by volume), determine the masses of oxygen and nitrogen dissolved by 1 L of water at 20 "C exposed to air at a total pressure of 760torr. The solubility of a gas, i.e., the concentration of the dissolved gas, may be expressed as Solubility of Y = kH(Y) x P ( Y ) In words, the solubility of a gas dissolved from a gaseous mixture (air in the present case) is directly proportional to the partial pressure of the gas, the proportionality constant, k H ,is called the Henry's law constant. (Some authors define the Henry's law constant as the reciprocal of the k H used here.) To evaluate kH from the data, note that when pure oxygen is equilibrated with water at a total pressure of 760torr, P ( 0 , ) = (760 torr) - (vapor pressure of water) Then, from the data, kH(02) = kH(N2) solubility of 0, - 0.043 g/L P(02) 760 torr - v.p. solubility of N, " P - = 0.019 g/L 760 torr - v.p. When water is exposed to air at a total pressure of 760torr P(0,) = (0.2@)(760 torr - v.P.) Hence P(N2) = (0.80)(760 torr - v.P.) ) Solubility of 0, from air = k H ( 0 2x P ( 0 , ) = (76:*::rg/L;p.) (0.20)(760 torr - v.P.) = 0.008 6 g/L Similarly, the solubility of N2 from air is (0.80)(0.019g/L) = 0.015 g/L. 14.12. A gaseous mixture of hydrogen and oxygen contains 70% hydrogen and 30% oxygen by volume. If the gas mixture at a pressure 2.5atm (excluding the vapor pressure of water) is allowed to saturate water at 20 "C,the water is found to contain 3 1.5 cm3 (S.T.P.) of hydrogen per liter. Find the solubility of hydrogen (reduced to S.T.P.) at 20C and 1 atm partial pressure of hydrogen. CHAP. 141 PROPERTIES OF SOLUTIONS 223 Since the volume of a gas at S.T.P. is proportional to its mass, the volume of the dissolved gas (reduced to S.T.P.) is proportional to the partial pressure of the gas. Partial pressure of hydrogen = (0.70)(2.5 atm:) = 1.75 atm Solubility of H2 at 20C and 1 atm = (i:;: ~ ::)(31.5 cm3/L) = 18.0 cm3 (S.T.P.)/L LAW OF DISTRIBUTION: EXTRACTION 14.13. (a) An aqueous solution of iodine, of volume 25 mL and containing 2 mg of iodine, is shaken with 5 mL of CC14, and the CC14 is allowed to separate. Given that the solubility of iodine per unit volume is 85 times greater in CC14 than in water at the temperature of the experiment and both saturated solutions may be considered to be "dilute," calculate the quantity of iodine remaining in the water layer. (b) If a second extraction is made of the water layer using another 5cm3 of CC14, calculate the quantity of iodine remaining after the second extraction. (a) Let x = number of milligrams of iodine in H 2 0 layer at equilibrium 2 - x = number of milligrams of iodine i n CCl, layer at equilibrium The concentration of iodine in the water will be x/25 (mg per mL of water), and the concentration of iodine in the CC14 layer will be (2 - x)/5 (mg per mL of CC14). Hence cone. I2 in CC14 - 85 conc. I2 in H 2 0 - 1 or _____ (2 - x)/S - 85 -x/25 1 or 2-x -X - 17 Solving, x = 0.11 mg iodine Note: Any volumetric concentration units could have been used in this problem, so long as the same units were used in both numerator and denominator. The choice of mg/mL was the most convenient for this particular case. (b) Let y = number of milligrams of iodine in H,O layer after second extraction 0.11 - y = number of milligrams of iodine in CCl, layer after second extraction The concentration of iodine in the water layer will be y/25, and the concentration in the CC14 layer will be (0.11 - y ) / 5 . Hence conc. I2 in CC14 - 85 cone. I2 in H 2 0 -- 1 Solving, y = 0.006 1 mg or (0.1 1 - y ) / 5 _- 85 -_ Y/25 1 or -= 0.11 - y Y 17 iodine. Supplementary Probl.ems 14.14. A solution containing 6.35 g of a nonelectrolyte dissolved in 500 g of water freezes at -0.465 "C. Determine the molar mass of the solute. Ans. 50.8 14.15. A solution containing 3.24g of a nonvolatile nonelectrolyte and 200 g of water boils at 100.130 "C at 1 atm. What is the molar mass of the solute? Ans. 63.9 224 PROPERTIES OF SOLUTIONS [CHAP. 14 14.16. Calculate the freezing point and the boiling point at 1atm of a solution containing 30.0 g cane sugar (molar mass 342) and 15Og water. Ans. - 1.09 "C, 100.300 "C 14.17. If glycerin, C ~ H S ( O H )and methyl alcohol, CH30H, sold at the same price per pound, which would be ~, cheaper for preparing an antifreeze solution for the radiator of an automobile? Ans. methyl alcohol. 14.18. How much ethyl alcohol, C2HSOH,must be added to 1 L of water so that the solution will not freeze above -4 O F ? Ans. 4958 14.19. A chemist wished to determine the molar mass of an unknown compound using a pure solvent which had a convenient freezing point of 30.16 "C. A solution of 0.617 g of pure paradichlorobenzene (C6H&12) in 1O.OOg of solvent had a freezing point of 27.81 "C, while 0.526g of the unknown in 1O.OOg of solvent lowered its freezing point to 26.47 "C. Calculate the molar mass of the unknown. Ans. 79.8 14.20. What is the freezing point of a 10% (by weight) solution of CH30H in water? Ans. -6.5"C 14.21. When 10.6 g of a nonvolatile substance is dissolved in 740 g of ether, its boiling point is raised 0.284 "C. What is the molar mass of the substance? Molal boiling-point constant for ether is 2.11 "C kg/mol. Am. 106 14.22. The freezing point of a sample of naphthalene was found to be 80.6"C. When 0.512g of a substance is dissolved in 7.03 g naphthalene, the solution has a freezing point of 75.2 "C. What is the molar mass of the solute? The molal freezing-point constant of naphthalene is 6.80 "C - kg/mol. Ans. 92 14.23. Pure benzene freezes at 5.45 "C. A solution containing 7.24 g of C2C14H2 in 115.3g of benzene was observed to freeze at 3.55 "C. From these data calculate the molal freezing-point constant of benzene. Ans. 5.08 "C - kg/mol. 14.24. What is the osmotic pressure at 0 "C of an aqueous solution containing 46.0 g of glycerin (C3H803) per liter? Ans. 11.2atm 14.25. A solution of crab hemocyanin, a pigmented protein extracted from crabs, was prepared by dissolving 0.750g in 125mL of an aqueous medium. At 4C an osmotic pressure rise of 2.6mm of the solution was observed. The solution had a density of 1.OO g/mL. Determine the molar mass of the protein. Ans. 5.4 x 10' 14.26. The osmotic pressure of blood is 7.65 atm at 37 "C. How much glucose, C6H12O6, should be used per liter for an intravenous injection that is to have the same osmotic pressure as blood? Ans. 54.2g/L CHAP. 141 PROPERTIES OF SOLUTIONS 225 14.27. The vapor pressure of pure water at 26C is 25.21 torr. What is the vapor pressure of a solution which contains 20.0g glucose, C6H1206,in 70g water? Ans. 24.51 torr 14.28. The vapor pressure of pure water at 25 "C is 23.76 torr. The vapor pressure of a solution containing 5.40 g of a nonvolatile substance in 90 g water is 23.32 torr. Compute the molar mass of the solute. Ans. 57 14.29. Ethylene bromide, C2H4Br2, and 1,2-dibromopropane, C3H6Br2, form a series of ideal solutions over the whole range of composition. At 85 "C the vapor pressures of these two pure liquids are 173torr and 127 torr, respectively. (a) If 10.0 g of ethylene bromide is dissolved in 80.0 g of 1,2-dibromopropane, calculate the partial pressure of each component and the total pressure of the solution at 85 "C. (b) Calculate the mole fraction of ethylene bromide in the vapor in equilibrium with the above solution. ( c ) What would be the mole fraction of ethylene bromide in a solution at 85 "C equilibrated with a 50:50 mole mixture in the vapor? Ans. (a) ethylene bromide, 20.5 torr; 1,2-dibromopropane, 112 torr; total, 132torr; (b) 0.155; (c) 0.42 14.30. An organic compound was found by combustion analysis to contain 38.7% carbon, 9.7% hydrogen, and the remainder oxygen. In order to determine its molecular formula a l.0Og sample was added to 1O.OOg of water, and the freezing point of the solution was found to be -2..94 "C. What is the molecular formula of the compound? Ans. C2H602 14.31. A 0.100molal solution of NaC103 freezes at -0.343 3 "C. What would you predict for the boiling point of this aqueous solution at l a t m pressure? At 0.001 molal coricentration of this same salt, the electrical interferences between the ions no longer exist, because the ions are, on the average, too far apart from each other. Predict the freezing point of this more dilute solution. Ans. 100.095"C, -0.003 7 "C 14.32. The molar mass of a newly synthesized organic compound was determined by the method of isothermal distillation. in this procedure two solutions, each in an open calibrated vial, are placed side by side in a closed chamber. One of the solutions contained 9.3mg of the new compound, the other 13.2mg of azobenzene (molar mass 182). Both were dissolved in portions of the same solvent. During a period of three days of equilibration, solvent distilled from one vial into the other until the same partial pressure of solvent was reached in the two vials. After this there was no further net distillation of solvent. Neither of the solutes distilled at all. The volumes of the two solutions at equilibrium were then read on the calibration marks of the vials. The solution containing the new compound occupied 1.72cm3 and the azobenzene solution occupied 1.02cm'. What is the molar mass of the new compound? The mass of solvent in solution may be assumed to be proportional to the volume of the solution. A m . 76. 14.33. Calculate the freezing point of a solution of 3.46 g of a compound, X, in 160 g of benzene. When a separate sample of X was vaporized its density was found to be 3.27 g/L at 116 "C and 773 torr. The freezing point of pure benzene is 5.45 "C and Kf is 5.12 "C - kg/mol Ans. 4.37"C 14.34. In a certain chemical process, it is necessary to monitor a stream of benzene, C6H6, which may be contaminated with toluene, C7HS. How finely calibrated should the thermometer scale be to detect 0.10% by weight of toluene in the benzene by measuring its freezing point? (See data in previous problem.) Arts. The freezing point is lowered 0.056"C. You would need calibrations every 0.1 "C at least; 0.01 "C would be better. Such thermometers are not ~ncommoii. 226 PROPERTIES OF SOLUTIONS [CHAP. 14 14.35. An alternative to the method of the previous problem might be to measure the vapor pressure of the benzene stream at 25C. Compare this method with that of the previous problem. Benzene and toluene form ideal solutions. The vapor pressure at 25 "C is approximately 95 torr for benzene, and 30 torr for toluene. Ans. The lowering of the total vapor pressure of the liquid by the addition of 0.10% toluene is about 0.06 torr, i.e., 0.06mm of Hg. Such an observation would require a microscope. Worse yet, since the vapor pressure of benzene increases about 3 torr/ "C, it would be extremely difficult to control the sample temperature adequately, i.e., to better than fO.O1 "C. 14.36. One method of producing potable water from saline water is reverse osmosis, in which a pressure just in excess of the osmotic pressure, applied to the solution, can reverse the flow of solvent. In principle, what pressure would be required to produce pure water from sea water at 25"C? Assume that sea water has a density of 1.021g/cm3 and is 3.00% by weight NaC1, which is 100% ionized. Express your answer in atm, kPa, and psi. Ans. 25.6atm, 2.59 x 103kPa, 376psi 14.37. At 20 "C and 1.OO atm partial pressure of hydrogen, 18 cm3 of hydrogen, measured at S.T.P., dissolves in 1 L of water. If water at 20 "C is exposed to a gaseous mixture having a total pressure of 1 400 torr (excluding the vapor pressure of water) and containing 68.5% H2 by volume, find the volume of H2, measured at S.T.P., which will dissolve in 1 L of water. Ans. 23cm3 14.38. A liter of CO2 gas at 15C and l.00atm dissolves in a liter of water at the same temperature when the pressure of CO2 is 1.OO atm. Compute the molar concentration of CO;! in a solution over which the partial pressure of CO2 is 150 torr at this temperature. Ans. 0.0083M 14.39. (a) The solubility of iodine per unit volume is 200 times greater in ether than in water at a particular temperature. If an aqueous solution of iodine, 30mL in volume and containing 2.0mg of iodine, is shaken with 30mL of ether and the ether is allowed to separate, what quantity of iodine remains in the water layer? (b) What quantity of iodine remains in the water layer if only 3 mL of ether is used? (c) How much iodine is left in the water layer if the extraction in (b) is followed by a second extraction, again using 3mL of ether? (6) Which method is more efficient, a single large washing or repeated small washings? Ans. (a) 0.010 mg; (6) 0.095 mg; (c) 0.004 5 mg; ( d ) repeated small washings. 14.40. The ratio of the solubility of stearic acid per unit volume of n-heptane to that in 97.5% acetic acid is 4.95. How many extractions of a 10mL solution of stearic acid in 97.5% acetic acid with successive 10mL portions of n-heptane are needed to reduce the residual stearic acid content of the acetic acid layer to less than 0.5% of its original value? Ans. 3 14.41. One method of purifying penicillin is by extraction. The distribution coefficient for penicillin G between isopropyl ether and an aqueous phosphate medium is 0.34 (lower solubility in the ether). The corresponding ratio for penicillin F is 0.68. A preparation of penicillin G has penicillin F as a 10.0% impurity. (a) If an aqueous phosphate solution of this preparation is extracted with an equal volume of isopropyl ether, what will be the % recovery of the initial G in the residual aqueous-phase product after one extraction, and what will be the YOimpurity in this product? (b) Compute the same two quantities for the product remaining in the aqueous phase after a second extraction with an equal volume of ether. Ans. (a) 75% recovery, 8.1% impurity; (6) 56% recovery, 6.6% impurity. Chapter 15 Organic Chemistry and Biochemistry INTRODUCTION Compounds of the element carbon, with a few exceptions, are called organic compounds; others are called inorganic. Originally, the distinction was based on the mistaken notion that living matter, organic, was chemically different from nonliving matter, inorganic. We now know that organic compounds, which constitute most of all known compounds, can be synthesized not only by the natural processes of living organisms but also in the laboratory by processes that do not require the intervention of biological events. The principles of chemistry as presented in this text apply equally of course to inorganic chemistry, organic chemistry, and biochemistry. This chapter is considered necessary, however, because the great variety of organic compounds demands, in addition to a consideration of their most important chemical reactions, a special nomenclature and special attention to the details of isomerism. Biochemistry will be introduced by way of examples. NOMENCLATURE Because neighboring carbon atoms readily form strong bonds with each other, organic compounds typically have a large number of atoms per molecule and their names must indicate not only the number of atoms of each kind but the pattern of connections. We begin with compounds of carbon and hydrogen, called hydrocarbons. If there are no multiple bonds these are termed alkanes, and if there are no rings they must all have the empirical formula C,H,+2 in order to satisfy the rules for bonding as in Chapter 9. The name of an organic compound always provides the number of carbon atoms explicitly. The root of the name indicates the number of carbon atoms, meth meaning one, eth two, prop three, but four, pent five, hex six, hept seven, and so forth. The number of atoms of other elements is often implied. For instance, in cyclic compounds each ring of carbon atoms reduces the number of hydrogens by 2; each double bond (in compounds called olefins or alkenes) likewise by 2; and each triple bond (in alkynes) by 4. In older less systematic nomenclature, still encountered, the root counted all the carbons. Thus "butane" meant C4HI0,the suffix ane indicating an alkane. To indicate that a structure was branched rather than straight the prefix is0 was used. Clearly such a scheme is inadequate for five or more carbons since various branched structures are possible. Modern organic nomenclature, as prescribed by the International Union of Pure and Applied Chemistry (IUPAC), avoids such ambiguities. For noncyclic alkanes the root gives the number of carbons in the longest chain. Each branch is described by a prefix indicating the number of carbons in the branch, and a numeral indicating the point of connection. (Numbering starts at the end of the chain which produces the smallest numerals.) EXAMPLE 1 Before the IUPAC rules were developed, both of the following would have been called "hexane." Give the IUPAC names. H H H H H H H H H H-C-C-C-C-C-H H I H-C-H I I I I I I H H H I l l H-C-C--CC-CC-~ I I I I I I H-C-H H 2-Methylpentane 227 I H-C---H H H 2,3-Dimethylbutane I I 228 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 The system is extended to include alkenes by changing the suffix from -ane to -ene, and to alkynes by the suffix -yne. A numeral preceding the root of the name gives the location of the multiple bond, which is always in the lower numbered end of the molecule. EXAMPLE 2 Give the IUPAC names of the hydrocarbons below. (Notice that the bonds to the individual H atoms have been omitted for simplicity.) CH,-CH-CECH CH3 3-Methyl- 1-butyne I CH,=CH--CH=CH, 1,3-Butadiene The prefix cyclo- appears before the roof of the name in ring compounds, except for those involving the benzene ring (see Problem 9.8), which form a special category of compounds called aromatic. The opposite of aromatic is aliphatic. Further modifications of the system when elements other than carbon and hydrogen are involved will be noted below when functional groups are discussed. ISOMERISM In keeping with the principles and definitions in Chapter 9, isomers are compounds with the same number of atoms of each kind per molecule, but that are different substances because of differences in molecular structure. There are three classes of isomers; (a) structural, (b) geometric, and (c) optical. In (a) structural isomerism, molecules differ in the sequence of atomic attachments, i.e., in the structural skeleton of the molecule. For example, the compounds in Example 1 are structural isomers. The IUPAC names clearly distinguish among structural isomers. Geometric isomers differ in the three-dimensional shape of the molecule, a significant part of which is rigidly fixed because of the presence of a double bond or a ring. In simple cases geometrical isomers can be distinguished in their names by the prefixes cis- or trans- according to whether two groups are close (same side of the double bond) or far apart (opposite sides of the double bond). Optical isomers are nonsuperimposable mirror images of one another. In most cases they arise from the presence in the molecule of a tetrahedral carbon atom with four different attachments. Such an atom is termed asymmetric or chiral. Two optical isomers can be distinguished in their names by the prefixes dextro- or Zevo- according to whether a solution of the compound rotates a beam of polarized light to the right or to the left. The abbreviations D- and L- are commonly used for dextro- and levo-, and in a newer system the abbreviations are R- (for rectus) and S- (for sinister). EXAMPLE 3 (a) Draw the geometric isomers of 2-butene. (b) Draw the optical isomers of 1-bromo l-chloro- Because of the double bond all atoms except the six end-hydrogens are constrained to one plane, which is chosen as the plane of the paper. cis trans If both carbons and the bromine are placed in the plane of the paper the chlorine and hydrogen atoms are on opposite sides of the plane and their bonds to carbon form approximately 109" angles to the other bonds of that carbon atom. The chlorine is shown above and the hydrogen below the plane of the paper. H CH, H3C\ (H C C/ ' Br' Cl ' C' l 'Br CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 229 FUNCTIONAL GROUPS Alkanes are comparatively unreactive and chemically uninteresting. But if a multiple bond or some other kind of atom is introduced, characteristic properties and reactions result. These properties depend rather little on the number or arrangement of the carbon atoms in the rest of the molecule. The group of atoms that determine the characteristic properties of the compound is called a functional group. Learning the properties and reactions of the functional groups greatly simplifies the study of organic chemistry. Table 15-1 lists some of the more common functional groups and illustrates how the IUPAC name is constructed, usually by using a characteristic suffix. In the case of acids the extra word "acid" is added in addition to the suffix "-oic." The word "ether" also appears separate from the rest of the name. Esters have two-part names indicating the structures of the two parts of the molecule joined by the ester group. In naming aldehydes, acids, and esters the roots for one and two carbons are usually "form-" and "acet-," as in "formaldehyde" and "acetic acid," these historical names being more commonly used than the strict IUPAC "methanal" and "ethanoic acid." Departures from IUPAC nomenclature often Table 15-1 Structure Group Name Alkene, double bond Alkyne, triple bond Halide Example CH,CH=CHCH, CH3-C-CH CH ,CHCH Name of Example 2-Butene 1-Propyne -c=c--c=c- -c-x ( X = F, C1, Br, I) I I F Alcohol, hydroxyl CH,OH Methanol I , 2-Fluoropropane -C-OH I I 0 -C-H I 1 Aldehyde Carboxylic acid CH,CHO CH,CHCOOH CH3 Ethanal 2-Methylpropanoic acid 0 -C-OH II I -c-c-c-c-0-cI I I I 0 II I I I 0 Ketone Ether CH,CH,CH,CCH, CH,OCH,CH, II 2-Pentanone Methyl ethyl ether I I I -C-N-H I 0 H Amine CH,CH,NH2 Ethylamine -C-NH, II Amide CH,CH,CONH, 0 Propanamide I II -c-c-0-cI 0 I I Ester CH,CH,CH,C--O-CH,CH, II Ethyl butanoate 230 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 occur for very common substances, and fortunately they rarely can be misunderstood, such as the use of "ethyl alcohol" for "ethanol." PROPERTIES AND REACTIONS Alkanes Alkanes are colorless, water-insoluble compounds the simplest of which have rather low boiling points and are commonly used as fuels. In combustion reactions the products are CO2 and H 2 0 along with some CO if the amount of oxygen is insufficient. The reaction of alkanes with halogens can be controlled, resulting in the substitution of a halogen for a hydrogen without disruption of the carbon skeleton. CH3CH2CH3 Alkenes Alkenes are physically similar to alkanes and are also easily combusted. However, they are much more likely to undergo addition reactions with halogens rather than substitution. Addition reactions occur with a wide variety of reagents. CH,CH=CH, + C12 --+ CH3CH2CH,Cl+ HC1 + C 1 , 3 CH,CHCH, I t c c 1 1 CN CH,CH=CH, + HCN -+ CH,CHCH, I A very important reaction, especially in the plastics industry, is addition polymerization, in which alkene molecules add to themselves to form long chains, for example, from CH3CH=CH2: etc. -CHCH,CHCH,CHCH,CHCH,CH, Halides Halogenated compounds are water-insoluble and have higher boiling points than the corresponding hydrocarbons. They find wide use as solvents, cleaning fluids, insecticides, and (when polymerized) plastics. They are also important starting materials for the synthesis of other compounds since the halogen can be replaced with OH (using NaOH) and other groups. Alcohols Alcohols are pleasant-smelling liquids, owing both their water solubility and high boiling point to extensive hydrogen bonding. Two very important condensution reactions are (a) with another alcohol to form an ether, and (b) with an acid to form an ester. A condensation reaction in general is one in which two molecules become joined along with the elimination of water or some other small molecule. The reaction of an ester with water to produce an alcohol and an acid, i.e., the reverse of esterification (the ester-forming condensation), is called hydrolysis. If the ester happens to be a natural fat, the process is called saponification since the sodium salt of the acid formed is a soap. The occurrence of particular organic reactions, including condensation and hydrolysis, is very dependent upon reaction conditions, such as acidity, temperature, or the presence of catalysts. The alcohol listed in Table 15-1, methanol, is also called methyl alcohol. The CH3 grouping is called the methyl group. Similarly other groups derived I CH, I CH, I CH, I etc. CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 23 1 from any alkane by the loss of one hydrogen atom are called alkyl groups. An alkyl group is not a compound by itself but retains its identity in most types of compounds, including halides, alcohols, and ethers. Aldehydes and ketones Aldehydes and ketones have much lower boiling points ancl are less water-soluble than the corresponding alcohols. In all cases, whether alcohols, aldehydes, ketones, or acids, water solubility gradually decreases as the number of carbon atoms increases. An alcohol can be viewed as the product of the first step in the oxidation of a hydrocarbon, in the sense that the overall result can be considered to be the insertion of one 0 atom between C and H. If another 0 atom is inserted on the same C, and H20 eliminated, the result is an aldehyde (if the C was at the end of chain) or a ketone (if in the middle). The third step, inserting the last possible 0 atom on the C of an aldehyde, results in an acid. Acids Short-chain carboxylic acids are quite soluble in water, in w:hich they ionize slightly (typically a few percent, depending on concentration), and undergo typical acid-base reactions. Besides the condensation reaction with alcohols, acids condense with ammonia or with amines to form amides. 0 CH,C-OH Acefic acid II O + CH3CH2CH2NH2 Propylamine ____* H 2 0 + ClH3C-N-CH2CH2CH3 n- Propylacetamide It H I Salts of carboxylic acids are fully ionized in the solid state and in aqueous solution. Salts of fatty acids (from the hydrolysis of fats) are soaps. Amines and amides Amines can be considered derivatives of ammonia, NH3, in which one or more H atoms is replaced by an organic group such as an alkyl group. The example in Table 15-1 is called a primary amine; replacement of two H atoms gives a secondary arnine, and a11 three a tertiary amine. Amines are somewhat soluble in water in which, like ammonia, they abstract a proton leaving the solution basic. They dissolve readily in strong acid to form salts analogous to ammonium salts. The condensation of a carboxylic acid with ammonia or a.n amine gives an amide. The amide in Table 15-1 resulted from the condensation of propanoic acid with ammonia. The example in the preceding section involved the condensation of acetic acid with a primary amine. A very important class of biochemical molecules are the amino acids which join to form protein molecules by the condensation of the amine group of one molecule to the acid group of the next. The amide linkage in this case has the special name "peptide." Ethers and Esters When an alcohol is heated under proper conditions with concentrated sulfuric acid, water is eliminated by the condensation of two molecules of alcohol and the two combining groups such as alkyl groups are joined through an oxygen atom to form an ether. The product formed from ethanol, diethyl ether, is the familar "ether" known for its use as a general anesthetic. Mixed ethers, such as methylethyl ether, may be formed by a condensation reaction of two different alcohols. Ethers are good organic solvents, only very slightly soluble in water. As discussed under alcohols, esters are the condensation products of alcohols and acids. They also are good organic solvents, insoluble in water. Many found in nature are flavorful and sweet smelling; amyl acetate is the fragrant component of "banana oil" ("amyl" is a traditional name for the 5-carbon 232 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 group). Natural fats and oils are esters of the trifunctional alcohol, glycerol, and long-chain acids called "fatty acids.'' Aromatic compounds Benzene and compounds containing the benzene ring are termed "aromatic." Aromatic compounds have somewhat different chemical properties than their aliphatic counterparts. For instance the hydroxyl group attached to a benzene ring is a weak acid (though much weaker than a carboxylic acid). The three alternating double bonds of the ring are not readily saturated. If benzene is treated with chlorine (in the dark) substitution rather than addition will occur. (o>+ Cl2 -+ 0 . 1 + HC1 Functional groups on aliphatic side chains attached to aromatic rings behave in a manner typical of alipha tic compounds. Multifunctional molecules Many molecules contain more than one functional group (either the same or different). In such cases condensation reactions involving two or more groups per molecule can lead to the formation of polymers, as illustrated by the formation of proteins from amino acids. Another example is the synthesis of polyester fiber. 0 HO-CH,CH,-OH 1,2-Ethanediol 0 0 0 + + H-H O -! O !Terephthalic acid H20 + 0 0 etc. -0-CH,CH,-0-C I'~!-O-CH,CH,-O-C Dacron polyester '*!etc. (The above equation is not balanced. Thousands of molecules of each reagent combine, eliminating a like number of water molecules, one for each ester linkage.) BIOCHEMISTRY Certain aspects of biochemistry which distinguish it from nonliving chemistry are listed below. Molecules may be very large, but unlike a synthetic polymer, a biopolymer usually has a fixed number and sequence of monomer units, and often a fixed three-dimensional shape as well. 2. Classes of molecules are based primarily on structure, but also on their function in the living cell. 3. Reactions occur at a modest temperature; extremely sensitive catalysts, called enzymes, are usually required; and the energy to drive one reaction may come from the energy released in some coupled reaction. 4. Throughout the animal and plant kingdom there is little or no difference in the structures of the molecules chosen by nature for a given biological function. 1. CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 233 Some of the principal classes of compounds are (a) proteins, (b) carbohydrates, (c) fats, and (6) nucleic acids. A brief account of their structures and functions follows. Proteins are polymers of amino acids joined by peptide linkages as discussed earlier. They form the bulk of living tissue. Enzymes, those very selective and powerful catalysts, are also proteins, which may contain a prosthetic group based on some metal atom. Many proteins serve special functions, such as hemoglobin and myoglobin which are oxygen carriers. The simplest carbohydrates are the sugars, relatively small molecules typically of four, five or six carbon atoms which have hydroxyl and aldehyde (or ketone) functional groups. They function as fuels or as building blocks for polymeric materials. Plants contain polymers of the sugars called starch and cellulose formed by ether linkages. They serve to store energy and to provide structural support. Fats are esters of glycerol and the fatty acids. In plants many of the fats are unsaturated (containing double bonds) and liquid, and thus are called oils. Fats also function as fuels, but for long-term energy storage, rather than the short-term delivery of energy provided by sugars. Fats are part of a class of substances called lipids based on their insolubility in water and solubility in ether. Among other lipids are cholesterol and some cell-membrane materials. Nucleic acids are very high polymers joined by ester linkages between phosphoric acid and sugar monomer units. These sugars carry nitrogenous side chains called purine or pyrimidine bases. The sequence of bases constitutes the genetic code of the individual living organism. The nucleic acid molecule directs the sequencing of amino acids in the synthesis of protein, as well as its own reproduction. Solved Problems NOMENCLATURE 15.1. Name each of the following compounds according to the IUPAC rules: ( a ) CH,CHCH,CHCH, CH, I CH2CH, I HC\ CH ' CH 0 0 ( c ) ClCH,CH,CCH,CH, (e) CH,NHCH,CH,CH, I1 ( d ) CH3CHCI-12-O-C-CH3 II CH3 I (f) CH,CHIC~H2-0-C-C,,H,, OH OH I I 0 II (a) Don't be deceived by the manner in which the structure is drawn. The longest chain of linearly connected carbon atoms is six carbon atoms and numbering must start at the left to minimize the numbers of the points of substitution: 2,4-dimethylhexane. 234 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 (b) The ring carbons are numbered so as to minimize the sum of the numerals: 1,3-cyclohexadiene. (c) The chloro group is treated as a prefix but the carbonyl oxygen is a suffix: I-chloro-3-pentanone. ( ) The compound is an ester. Contrary to the example in Table 15-1 the alocohol residue is to the left. In d either case the alcohol part is always named first: 2-methylpropyl ethanoate, or 2-methylpropyl acetate. (e) The compound is a secondary amine: methylpropylamine. (f) The fatty acids have special names. The saturated Clg acid is called "stearic", and the compound listed is 2,3-dihydroxypropyl stearate. The alcohol residue is glycerol. Since only one hydroxyl is esterified this type of ester is called a monoglyceride and the compound may be named monoglyceryl stearate. It is valuable in food processing as an emulsifier. ISOMERISM 15.2. Identify all the isomers with the formula C5H10,and give their IUPAC names. Consider first all the possible structural isomers. The cyclic structures will be saturated; the open-chain structures will have one double bond. (a) Cyclopentane CH2 / \ H2C-C,-CH3 (b) Methylcyclobutane (c) 1,2-Dimethylcyclopropane CH2 / \ H,C-CH-CH2-CH3 \ CH3 ( d ) 1,l-Dimethylcyclopropane CH2=CH--CH,-CH2--CH3 ( e ) Ethylcyclopropane CH3-CH=CH-CH2-CH3 (9) 2-Pentene (f) 1-Pentene CH2=C-CH2-CH3 CH3 I CH3-C=CH-CH, CH3 (i) 2-Methyl-2-butene I CH2=CH-CH-CH3 CH3 ( j ) 3-Methyl-1-butene I (h) 2-Methyl-1-butene Next consider cis-tram isomerism. The only case among the alkenes is (g).The two structures are shown below. cis tram However, structure ( c ) also shows cis-tram isomerism. The three carbons in the ring define a plane (the plane of the paper). Either both hydrogens of the CH groups are on the same side of this plane or one is above and the other below. Structure (c) is also the only one which has a chiral carbon atom (the carbons of the two CH groups). However the cis isomer has a plane of symmetry and thus cannot have an optical isomer. Only the tram isomer has an optical isomer. 15.3. Draw the structures of all the structural isomers with the formula C3H60.Point out any that have geometrical or optical isomers. (Not all structures satisfying the Lewis rules correspond to stable chemical substances.) CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 23 5 OH CH,-CH=CH-OH cis-tram forms CH2=CH-CH2-OH CH3-C=CH2 I 0 CH,-C-CH, CH2 H2C-CH-OH / \ II 0 CH3-CH2-C-H 0 H2C-CH-CH, / \ II CH2=CH-O-CH3 0-CH2 H2C-CH, I I CH carbon is chiral. FUNCTIONAL GROUPS AND REACTIONS 15.4. Draw the structure of the organic product resulting from each of the reactions below. 1-Propanol in dehydrating medium (b) 1-Propanol in mildly oxidizing medium (c) 2-Propanol in mildly oxidizing medium (6) 1-Propanol plus butanoic acid in dehydrating medium (e) 1-Propanol plus metallic sodium (a) (a) A condensation results in elimination of water and formation of an ether: CH3CH2CH2- 0 -CH2CH2CH3 When the alcohol is (b) One stage of oxidation of a hydroxyl group produces a carbonyl group (-C =0). at the end of the chain (primary alcohol) the product is an aldehyde: H CH,CH2-C=0 (c) I When the alcohol is not at the end (secondary alcohol in this case) the product is a ketone: 0 H,C-C-CH, II (d) A condensation results in elimination of water and forma.tion of an ester: 0 CH,CH,CH,C-O-CH,CH,CH, I1 (e) A possible side reaction, especially if the alcohol is in excess, is the formation of the corresponding ether as in part (a). Considering the alcohol to be a derivative of water, one expects the sodium to displace the hydrogen of the hydroxyl group. One product is H2; the other is the sodium salt of the alcohol: 15.5. Describe the products of the following reactions, using structural formulas: (a) vinyl chloride (alternate name: chloroethene) is treated with a polymerization catalyst. (b) The amino acid alanine (2-aminopropanoic acid) is treated with the appropriate polymerase enzyme and energy source molecules to cause it to polymerize. 236 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 (a) The product is a saturated high polymer, the double bonds having been used up in joining the monomer molecules together to form polyvinyl chloride: H H C=C I I H H H H H H --* H Cl I 1 etc.-C-C-C-C-C-C- H C1 H C1 H C1 I I I I I I I I I I I I etc. (b) Water is condensed out, H from the amino group of one molecule and OH from the acid of the next to form polyalanine: CH, 0 H-N-C-C-OH H I I1 --+ I H 1 I II I etc. -N-C-C-N-C-C-N-C-C-N-C-CI I I I H H H H CH, 0 CH, 0 I1 CH, 0 I II CH, 0 H I H 1 H I H I I II etc. Supplementary Problems NOMENCLATURE 15.6. Give the IUPAC name of each of the following compounds: CH,-CH,-CH-CH, CH3 CH H2Y/ \kH-CH, I H,C, ,CH, CH2 CH,=CH-CH,-CH-CH=CH, CH3 (a) 2-methylbutane; I CH,-CH,-CH-CH, H3C-CH-CH3 CH,-CH=CH-CH,-CH, I I (f1 H,C-CGC-CH, (b) 2,3-dimethylpentane; (c) methylcyclohexane; ( d ) 2-pentene; (e) 3-methyl-1,5-hexadiene; ( f ) 2-butyne 15.7. Name each of the following: (U) CH3-0-CH,-CH, 0 (b) CH3-CH-CH2-CH3 I c 1 I (c) H,C-CH-O-C-CH, /CH2 \ II ( d ) CH3-CH-CH2-CH2-OH CH,-CH, 0 (e) CH,-CH,-N-CH,-CH, CH2-CH3 I (1) CH3-C-CH2-CH2-CH, II CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 237 Ans. (a) methyl ethyl ether, (b) 2-chlorobutane, (c) cyclopropyl acetate (or ethanoate), ( d ) 3-methyl-1-pentanol, (e) triethylamine, (f)2-pentanone 15.8. Name each of the following: 0 (U) CH,-CH,-C-H II 0 ( b ) H-C-NH, II 0 (c) CH3-CH,-CH-CH,-C-OH CH,-CH, II O I ( d ) CH,-CH,-C-N-CH,-CH, II H I Ans. (a) propanal; (b) formamide (methanamide); ( c ) 3-ethylpentanoic acid; ( ) N-ethylpropanamide; ( e ) benzene: (f)1,2-dichlorobenzene d 15.9. Draw the structures of the following compounds: (a) 3-ethyl-4-methyl-2-hexanone, (b) 2-chloro-butyl acetate, ( c ) ethylbenzene, (d) 3-ethylcylohexene, ( e ) 2-meth y l-3-pentanol, (f)2-methylpropyl methyl ether. 0 Ans. (a) CH,-C-CH-CH-CH,-CH, II CH3 I CI ( b ) CH3-CH2-CH-CH2-O-C-CH, I 0 II CH,-CH, (c) I 0""2-cH3 CH, OH ( d ) H,?' CH 'kH H,C, ,CH CH CH,-CH, II I ( e ) CH,-CH-CH-CH,-CH, I 1 (f) CH3-CH-CH2-O-CH3 CH3 I 15.10. Explain what is wrong with each of the following names: (a) 3-methyl-2-propanol; (b) 3,3-dimethyl-2pentene; (c) 1,4-dichlorocyclobutane; ( d ) 2-propanal; ( e ) 2-methyl-1-butyne; ( f ) pentanoicacid. Ans. (a) The longest chain is four carbons. The correct name is 2-butanol. (b) Such a compound is impossible because it would require five bonds on the third carbon. ( c ) Positions 1 and 4 are equivalent to 1 and 2. The correct name is 1,2-dichlorocyclobutane. ( d ) Such a compound is impossible because the aldehyde carbon must be at the end of the chain. (e) Such a compound is The suffix "acid" should impossible because it would require five bonds on the second carbon. (f) be a separate word. The correct name is pentanoic acid.. ISOMERS 15.11. Draw the structures of all the structural isomers having the formulas: (a) C4H9Br, (6) C4Hs, (c) C2H402(omit any peroxides, -0-0-), ( d ) C6HI4, (e) C4H8C12. 238 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 Ans. ( a ) CH,-CH,-CH,-CH,-Br CH3-CH-CH, CH,Br Br CH,-C-CH, CH3 CH,-CH,-CH-CH, I Br I I I ( b ) CH3-CH2-CH=CH2 CHj-CH=CH-CH, CH,-CH2 CH,-C=CH, I CH3 0 I I CH,-CH, 0 CH,-CH-CH, /c\H2 (c) CH,-C-OH HO-CH=CH-OH II H-C-0-CH, H2C-0 0-CH, II 0 HO-CH,-C-H II I / 0 H,C-CH-OH / \ OH (Note: Not all the compounds above may be stable. Compounds like HO-C=CH2, groups on the same carbon, do not occur.) I with two hydroxyl ( d ) CH3-CH,-CH2-CH,-CH,-CH3 CH,-CH-CH,-CH,-CH, CH3 I CH3-CH2-CH-CH2-CH3 CH, CH3 I CH,-CH-CH-CH, CH, CH, I I CH3-C-CH2-CH3 CH3 I I c 1 (e) CH,-CH,-CH,-CH-CI I Cl-CH,-CH,-CH,-CH,CI CH3-CH-CHC12 CH3 I CH3-CH2-CH-CH,Cl c 1 CH,-CH-CH,-CH,CI C1 I CH,-CH-CH-CH, CH,-CH-CH,CI CH,CI CH,-C-CH,Cl I CH, I 1 c c 1 1 Cl I I CH,-CH,-C-CH, I c 1 I c 1 I 15.12. Examine all the answers to Problem 15.11 and select all that exhibit geometric isomerism, pointing out the two carbon atoms involved in each. Ans. (b) CH3-eH&H-CH3, (c) H O d H d H - O H CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 239 15.13. Examine all the answers to Problem 15.11 and select all that exhibit optical isomerism, pointing out the chiral carbon (or carbons) involved in each. Ans. ( a ) CH,-CH,-CH-CH, * Br (e) CH,-CH,-CH-CH,CI I * I (c) H,C-CH--OH / 0 \* ( e ) CH,-CH-CH * * ( e ) CH,-CH-CH-CH, I I c 1 c 1 * I ,-CH,CI c c 1 1 15.14. Draw the structure of the smallest noncyclic alkane that has a chiral carbon and mark it. Name the compound. Ans. H,C-CH-CH,-CH,-CH, CH2-CH, Also H,C-CH-CH-CH, CH,-CH, CH3 * I 3-methylhexane * I I 2,3-di1nethylpentane 15.15. Natural rubber is a polymer of isoprene and has the following structure: c 3 \ H /H ,CT2 CH,,C=C CH, etc., ,C=C\ CH, cH /H \3 CH, C=C CHZ CH2 / \ / \ / \ / CH, CH, C=C \ / \ H CH, H There is another natural product called gutta-percha which is a geometric isomer of rubber but is useless as an elastomer. Draw its structure. Which is cis and which is trans? Ans. Natural rubber is cis; gutta-percha is trans. etc. ,C=C\ CH, /cv / H ,Cp2 ,H CH, C=C CH, / \ /cF / H CH, CH, C=C CH, / \/'\ /H CH, C=C / \ CH, CH, 15.16. Which has more isomers, (a) dimethylbenzene or (b) dimethylcyclohexene? Explain. Ans. Because all corners of the benzene ring are equivalent ( a ) has only three isomers (structural) with the methyl groups in positions 1,2; 1,3; or 1,4. However (b) can have many more structural isomers; for example, 1,2 is different from 1,6 and 1,3 is different from 2,4. There are also many geometric isomers; for example, the 3,4 structural isomer must have cis and trans forms. Optical isomerism also occurs; for instance, both carbons 3 and 4 of the 3,4 isomer are chiral. 240 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 FUNCTIONAL GROUPS 15.17. Name the functional group in each of the following molecules: 0 ( a ) CH,CH,CH,OH (b) (CH,),N O=C-OH (c) CH,OCCH, I1 (d) CH,CHCH, c 1 I 0 (e) CH,CH,CH,CCH, II (f)CH,-CH-CH,-CH, I 0 (j) II Br (k) Br-CH,CH,CHCH, I C2H5CH (I) CH,CH,C=CCH, Ans. (a) alcohol, (hydroxyl); (b) amine, (tertiary amine); (c) ester; ( d ) chloride; (e) ketone; (f)acid, (carboxylic acid); ( g ) ester, (triglyceride); (h) amide; (i) ether; ( j ) aldehyde; (k) bromide, (dibromide); (I) alkyne, (triple bond). 15.18. What are the structures of the following compounds: (a) butanamide; (b) methylpropylamine; ( c ) diethyl ether; ( d ) 2,3-dimethyl-1-hexanol; (e) ethyl 2-methylpropionate; (f)3-iodo-2-pentanone. 0 Ans. ( a ) CH3CH,CH,CNH, II H (b) CH,-N-CH,CH,CH, ( d ) HO-CH2-CH-CH-CH,-CH2-CH3 I CH, CH, I I 0 (e) CH,-CH-C-0-C,H, II 0 CH3 I (f) CH,-C-CH-CH,-CH, I II I REACTIONS 15.19. Draw the structure of the polymer formed by addition polymerization of 2-methyl-1-propene. H Am. etc.-C-C-C-C-C-C-C-C-etc. I CH, 1 H CH, I I H I CH, I H CH, I I H CH, I 1 H CH, I I H I CH, I H CH, I 1 CHAP. 151 ORGANIC CHEMISTRY AND BIOCHEMISTRY 24 1 15.20. What are the main products of the following reactions? Give the structures of the organic products. 0 II C2H5C-NH2 C2H5OH dehydration + H2 hydrolysis 0 CH,C-OH II OH + CH3-CH-C2H, + C1, 0 --+ I dehydration (f) CH,CH,CH, OH + C1, -+ (first stage) CH,=CHCH, (first stage) (j) CH3CHCH2CH3 OH I II CH3CH2CH2C-H mild mild (j) CH,-CHCH,CH, I + K --+ 0 Am. ( a ) NH, and C,H,C-OH II (b) CO, and H,O 0 ( c ) H,Oand C,H,OC,H, 0 ( d ) H,O and CH,C-H ( f ) HCl plus CH,CH2CH2-Cl II ( e ) H,O and CH,C-0-CH-C,H, II CH3 and CH,CHCH, 0 (9) I c 1 II I CH2CHCH3 c c 1 1 I 1 (h) H 2 0 and CH,CCH,CH, 0 (i) CH,CH,CH,C-OH II 0-K+ ( j ) H, and CH,CHCH,CH, I 15.21. Draw a portion of the polymer formed by condensation of 1,Zethanediol and maleic acid (IUPAC name: cis-butenedioic acid). This, and similar polymers, are copolyrnerized with styrene to form the "polyester resins," which are widely used to make structural plastic moldings. 0 Ans. HO -- C2H4 -0 -CA I/ 0 C H I C -C -0- C2H4 -0 -- C -C =C -C -0 -C2H4 -0 -C - II 0 /I 0 I/ 0 I1 H / H I H I 15.22. Formaldehyde (methanal) can form addition polymers. Imagine adding a water molecule to a methanal molecule to form a diol, which then forms an ether linkage with a similar neighboring molecule (splitting out water) and so on. Draw a portion of the resultant product. H Ans. -0- I I I I I c -0- c -0 --c -0 -c -0 -cI H H H H H H I H I H H I 242 ORGANIC CHEMISTRY AND BIOCHEMISTRY [CHAP. 15 15.23. "Silicones" are polymers which find widespread uses from high-temperature hydraulic fluids to medical prostheses. Draw a portion of the polymer formed by condensing out water from dimethylsilanediol, (CH3)2Si(OH)2 CH3 Ans. 0 -Si -0 -Si -0 -Si -0 -Si CH3 I CH3 I CH3 I CH3 I I CH3 I CH3 I CH3 I 15.24. Two carboxylic acid groups can combine by condensing out a water molecule to form an acid anhydride. Draw the structures of (a) acetic anhydride, which involves two molecules, and (b) maleic anhydride (see problem 15.21), in which only one molecule is involved. H / C-C` \ H Chapter 16 Thermodynamics and Chemical Equilibrium THE FIRST LAW The principle of conservation of energy, referred to in Chapter 7, constitutes the first law of thermodynamics. The quantities E and H are properties of a system that along with others define the state of the system. Such properties are termed state functions. If any of these properties change, the system is said to have undergone a change in state. Any change in E must be equal to the amount of heat absorbed by the system plus the amount of work performed on the system. A formal statement of the first law is: AE=q+w (16-1) In this equation, w is defined as the work done on the system of interest. If a chemical reaction occurs within a system, work may be done upon it if gases are consumed and its volume is thereby decreased, or the system may perform work if gases are produced or if the system delivers an electric current to an external circuit. THE SECOND LAW The existence of an energy balance does not suffice to answer all questions about a chemical reaction. Does a given reaction take place at all? If so, to what extent does it proceed. Such questions require the introduction of some new thermodynamic functions, which like E and H are properties of the state of the system. These new functions are the entropy, S , and the Gibbs free energy, G . One mathematical statement of the second law of thermodynamics is A S 2 2 (16-2) T In words: when a system undergoes a change, the increase in entropy of the system is equal to or greater than the heat absorbed in the process divided by the temperature. The equality, which provides a definition of entropy increment, applies to any reversible process, whereas the inequality refers to a spontaneous, or irreversible, process, defined as one which cannot be made to go in the reverse direction by a very small change in one of the variables. Reversible and irreversible are illustrated in the following example. EXAMPLE 1 Consider a mixture of liquid and solid benzene at its normal freezing point, 5.45 "C. If the temperature is raised by the slightest amount, e.g., 0.01 "C, the solid portion will gradually melt. But if instead the temperature were lowered by that amount, the liquid would gradually crystallize. The freezing process (as well as the melting process) at 5.45 "C is reversible. It is possible to cool liquid benzene carefully to a temperature below its normal freezing point, say to 2.OO0C, without the occurrence of crystallization. The liquid is then said to be supercooled. If then a tiny crystal of solid benzene is added the entire mass will crytallize spontaneously and irreversibly. Raising the temperature by 0.01 "C (or even 1.OO "C) will not stop the crystallization. One would have to raise and maintain the temperature above 5.45 "C to restore the liquid state. The crystallization of liquid benzene at 2.00"C is an example of an irreversible process. The above statement of the second law (16-2) implies the differentiation between reactions that can occur spontaneously (for which A S > q / T ) and those that cannot (for which A S < q / t ) . To establish a succinct criterion of spontaneity for chemical processes, it is convenient to set up a state function, G, the free energy, which is defined as G=H-TS (16-3) Equation (16-2) leads, by a complex argument, to the following free-energy principle: 243 244 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 (16-4) In words equation (16-4) states that the change in free energy at constant temperature and pressure can be negative or zero; negative in the case of a spontaneous irreversible process, or zero in the case of a reversible process. This rule is restricted to processes in which the only form of work is expansion against the surroundings (or the reverse, contraction). Processes for which is positive can occur only with the application of work (or energy) from some external source (such as the process of electrical decomposition by the application of an electrical potential or a coupling with another chemical reaction for which A G T , p is negative). Another formulation of the free-energy principle is that the maximum amount of work which a system can perform at constant temperature and constant pressure in forms other than expansive or contractive work is equal to the decrease in free energy of the system. Entropy Via (16-2), entropy changes can be calculated from gross thermal measurements of reversible processes. However, they may also be related to the molecular properties of matter. Entropy is positively correlated with the number of different collections of molecular states consistent with the observed state of the bulk system. Some generally observed examples are given below. 1. Liquids have more entropy than their corresponding crystalline forms. (Every atom or molecule in a crystal is in a prescribed position in the lattice; in a liquid, positions of the particles may take on many different values consistent with the overall liquid nature.) 2. Gases have more entropy than their corresponding liquids. (Although the molecules in a liquid are free to occupy a variety of positions, they are constrained to be in close contact with their nearest neighbors; in a gas, the number of positional possibilities for the molecules is much greater because there is much more free space available per molecule.) 3. Gases at low pressure have higher entropy than at high pressure. (The argument is an extension of that given above in 2.) 4. A large molecule has a greater entropy than any of its submolecular fragments existing in the same phase of matter. (The internal vibrations and rotations of atoms within molecules give many possibilities for the distribution of intramolecular motions.) 5. The entropy of a substance always increases when its temperature is raised. [The temperature measures the average energy per molecule and, therefore, the total energy. The higher the temperature, the greater the total energy; and the greater the total energy, the greater the number of ways of apportioning this energy among the fixed number of molecules. Thus there are more different collections of molecular (energy) states corresponding to a higher temperature than to a lower temperaure.] 6. If a chemical reaction is accompanied by a change in the number of gas molecules, A S is positive in the reaction direction leading to an increase in the number of gas molecules. 7. When one substance dissolves in another, A S is positive. (The number of possible configurations of randomly arranged, unlike particles is greater than the number of configurations of separately packaged, like particles.) THE THIRD LAW Equation (16-2) allows the calculation of changes in the entropy of a substance, specifically by measuring the heat capacities at different temperatures, and enthalpies of phase changes. If the absolute value of the entropy were known at any one temperature, the measurements of changes in entropy in proceeding from that one temperature to any other temperature would allow the determination of the absolute value of the entropy at the other temperature. The third law of thermodynamics provides the basis for establishing absolute entropies; it states that the entropy of any perfect crystal is zero at the absolute zero of temperature. This is understandable in terms of the molecular interpretation of entropy. In a perfect crystal every atom is fixed in position; and at the absolute zero of temperature every form of CHAP. 161 THERMODYNAMICS AND CHEMICAL, EQUILIBRIUM 245 internal energy (such as atomic vibrations) has its lowest possible value, so that there is only one way of distributing the energy among the available atoms or molecules. STANDARD STATES AND REFERENCE TABLES Although enthalpies of substances are relatively independent of pressure (for gases) and of concentration (for dissolved species), their entropies, and thus the free energies as well, depend markedly on these variables. Tabulated values for S and G usually refer to the idealized state of 1 bar or 1 atm pressure for gases, 1 M concentration for solutes, and to the,pure substances for liquids and solids. Table 16-1 compiles some data for So, the molar entropy, and AG;, the free energy of formation from the elements, both at 25 "C and at standard states. AG; is defined, analogously to AH;, as the free-energy change associated with the reaction in which 1 mol of the substance in its standard state is formed from the stoichiometrically required numbers of moles of the constituent elements in their respective standard states. Note that the third law allows the determination of the entropy of a substance without reference to its constituent elements; note also that there are no zero values for the entropies of the elements at Table 16-1 Standard Entropies and Free Energies of Formation at 25 "C and 1 atm AG;/kJ. mol-' 121.3 152.23 245.35 5.74 126.8 282.6 197.56 213.68 222.96 266.10 130.57 69.95 188.72 191.50 239.95 304.18 205.03 217.1 31 1.7 52.3 95.6 51.5 44.1 -11.21 0 3.14 0 - 166.36 -162.00 -168.57 - 137.15 -394.37 0 97.9 0 -237.19 -228.59 0 51.30 97.82 0 -272.4 -267.8 -305.0 -369.0 -368.4 0 0.12 , 246 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 temperatures above 0 K as there are for AH; and AG;. Unless otherwise stated, calculations in this text will be referred to 1 atm standard state, still the common practice in the United States. However, use of the bar as standard state will be illustrated. In general, for the reaction aA + bB + cC + d D ~ the concentration dependence of the free-energy change may be expressed as A G = A G o + R T l n [CICIDld [AI"PIb ( 16-5) In this equation, [XI denotes the relative concentration of X, i.e., the ratio of the concentration to the concentration at the standard state. For a substance whose standard state is 1 mol/L, [XI is the numerical value of the molar concentration. Also, In is the symbol for the natural logarithm (see Appendix C ) ; AG is the free-energy change at the given concentrations. AGO is the free-energy change for the hypothetical reaction in which all reactants are in their standard states and all products are in their standard states; it is called the standard free energy change. Note that if the concentration of every species is 1, the logarithmic term in (16-5) becomes 0, and AG = AGO; this is as it should be, because unit concentration implies the standard state. In (16-5) if AG is to be expressed in joules, the universal gas constant must be taken as R = 8.314 5 J - K-' mol-' On the right side of (16-5) the term AGO lacks the unit mol-', which must be removed from the second term as well, having been absorbed into the logarithmic term by mathematical manipulation. The value of the second term applies to the number of moles indicated by the coefficients a, b, c, d. The above value for R must be used in all calculations in this chapter, except those directly involving the ideal gas law. The generalization of (16-5) to any number of reactants and products is AG = AGO + R T In Q where Q, the reaction quotient, is analogous to the argument of In in the last term of (16-5), but must include all reactants and products (not necessarily just two of each). CHEMICAL EQUILIBRIUM Theoretically, any chemical reaction could take place in the reverse direction to some extent. Often the driving force of a reaction favors one direction so greatly that the extent of the reverse reaction is infinitesimal and impossible to measure. The driving force of a chemical reaction is the change infree energy accompanying the reaction and it is an exact measure of the tendency of the reaction to go to completion. When the magnitude of AGO is very large and the sign negative, the reaction may go practically to completion in the forward direction; if AG" is only slightly negative, the reaction may proceed to a small extent until it reaches a point where AG (as distinct from AGO) for any further reaction would be zero, and the reaction could be reversed with a slight change in concentrations. In the latter case, the reaction is said to be thermodynamically reversible. Since many organic and metallurgical reactions are of this reversible type, it is necessary to learn how conditions should be altered to obtain economic yields, hasten desirable reactions, and minimize undesirable reactions. A chemical system which has reached a state of thermodynamic reversibility shows no further net reaction in either direction, since AG = 0. Such a system is said to be in a state of equilibrium, from which it could be displaced either forwards or backwards by an appropriate very small change in one of the variables. Although a system at equilibrium shows no net changes in the concentrations of any of the chemical species, the actual molecular or ionic rearrangements constituting both forward and reverse reactions may still be occurring, but the opposing forward and reverse reactions proceed at equal speeds. CHAP. 161 THERMODYNAMICS AND CHEMICAL, EQUILIBRIUM 247 Consequently the composition of the mixture remains constant and does not change with time. This condition of dynamic equilibrium, involving balanced reactions in both directions, contrasts with the static equilibrium of pulleys, levers, and springs. THE EQUILIBRIUM CONSTANT For a reversible reaction at equilibrium, AG = 0, and (16-6) becomes 0 = AGO + RT In Qeq or AGO = -RT In Qeq (16-7) Equation (16-7) is a remarkable statement. It implies that Qeq,the value of the reaction quotient under equilibrium conditions, depends only on thermodynamic quantities that are constant in the reaction (the temperature, and the standard free-energy change for the reaction at that temperature), and is independent of the actual starting concentrations of reactants or products. For this reason Qeq is usually denoted the equilibrium constant, K, and (16-7) is rewritten as AGO = -RT In K EXAMPLE 2 (16-8) Consider the reversible gaseous reaction system H2 + 12 + 2HI A reaction mixture could be made up by starting with H2 and I2 alone, 131 plus 12, HI plus H2, HI alone, or a mixture of all three substances. In each case a net reaction would occur in one direction or the other until the system comes to an eventual state of no further net change, i.e., equilibrium, which could be described by specifying the concentration of the three substances. Because of the variety of ways of making up the initial mixture, differing in the relative amounts of the various substances used, there are an infinite number of equilibrium states, each describable by a set of the concentrations of the three participating substances. Yet, for each of this infinite number of states, all at the same temperature, That is, the particular function of the three concentrations defined by Q is always the same, even though any individual concentration may vary by as much as 10 orders of magnitude. This unifying principle allows the calculation of conditions at equilibrium under as many sets of starting conditions as may be desired. Experimental measurements show that molecules in highly compressed gases or highly concentrated solutions, especially if electrically charged, abnormally affect each other. In such cases the true activity or eflective concentration may be greater or less than the measured concentration. Hence when the molecules involved in equilibrium are relatively close together, the concentration should be multiplied by an activity coeficient (determined experimentally). At moderate pressures and dilutions, the activity coefficient for nonionic compounds is close to unity. In any event, the activity coefficient correction will not be made in the problems in this book. The equilibrium constant K is a pure number whose magnitude depends not only on the temperature but also, generally, on the standard-state concentration to which all concentrations are referred. In this chapter, the standard state for dissolved substances will be taken as a 1 M concentration, unless a statement is made to the contrary. The magnitude of K is independent of the choice of standard concentration in the special case where the sum of the concentr;ation exponents in the numerator equals the sum of the exponents in the denominator. The concentration of a gas is proportional to its partial pressure ( n / V = P / R T ) . Hence, when all substances in the reversible reaction aA + bB + cC + d D 248 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 are gases, the equilibrium constant may be written as Kp can replace K in equation (26-8).Then if A O is obtained from a table based on the standard state of G 1 atm, KO will normally be correct only for P in atmospheres; and likewise for a standard state of 1 bar, Kp will normally apply only to P in bars. When the equation shows no change in the total number of moles of gas as the reaction proceeds (e.g., N2 + O2 + 2NO), Kp will be the same regardless of the pressure units used and will be identical with the K expressed in molar concentrations. Terms representing the concentrations of undissolved solid reactants or products are conventionally omitted from the expression for K because their concentrations cannot be varied. Such substances are always in their standard states and would merely contribute a [XI = 1 in the logarithmic terms of (26-5) through (16-8).Another exception occurs when the solvent is one of the products or reactants, as in the hydrolysis of urea in aqueous solution: So long as the solution is moderately dilute, the concentration of water cannot be much less than in pure water itself, 55.5 mol/L. It is therefore customary to remove [H20]from the expression for K by choosing 55.5mol/L as the standard state for water, while retaining 1 mol/L as the standard state for the other substances. LE CHATELIER'S PRINCIPLE If some stress (such as a change in temperature, pressure, or concentration) is brought to bear upon a system in equilibrium, a reaction occurs in the direction which tends to relieve the stress. This generalization is extremely useful in predicting the qualitative effects of changes in temperature, pressure, or concentration upon a system at equilibrium. Effect of changes in temperature The effect will first be discussed in qualitative terms. When the temperature of a system at equilibrium is raised, the reaction occurs in the direction which absorbs heat. EXAMPLE 3 In the thermochemical equation for the synthetic methanol process (all substances are in the gasCO + 2H2 + C H 3 0 H eous state) A H = -22 kcal the forward reaction liberates heat (exothermic), while the reverse reaction absorbs heat (endothermic). If the temperature of the system is raised, the reverse reaction (from right to left) occurs in an attempt to absorb the heat being supplied. Eventually a new state of equilibrium will be achieved at the higher temperature, containing more reactants (CO and H2) and less product than at the initial temperature. Conversely, the equilibrium yield of methanol is increased by lowering the temperature of this system. The same effect can be explained in quantitative terms. For any chemical process the change in free energy is: AG AH - TAS (26-9) If all substances are in their standard states, (26-9) becomes A O = AH" - TAS" G (16-20) CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 249 Combination of (16-8) and (16-10) gives AH" In K = --+RT AS" R (16-11) If AH" and A S o are fairly independent of temperature, as they are for most reactions, (16-11) shows that In K is a decreasing function of T for AH" < 0 (exothermic reaction). Then K itself is a decreasing function of T . A decrease in K means a shift of the equilibrium to favor the formation of the substances whose concentrations appear in the denominator of the K expression, namely the reactants, at the expense of the products of the reaction, whose concentrations appear in the numerator of the K expression. Conversely, if AH" > 0 (endothermic reaction), K increases with T , and the equilibrium shifts to favor the formation of the reaction products. Effect of changes in pressure When the pressure of a system in equilibrium is increased (for example, as shown in Fig. 16-1), the reaction occurs in the direction that lowers the pressure by reducing the volume of gas. CO+2H, 2 CH,OH Fig. 16-1 Cylinder EXAMPLE 4 In the synthetic methanol process (all substances are in the gaseous state) CO + 2H2 A c - CH30H 3 molecules of gas 3 volumes of gas 1 molecule of gas 1 volume of gas the forward reaction is accompanied by a decrease in volume. Then an increase in pressure will increase the equilibrium yield of CH30H. (This increase in yield occurs even though the value of K , depending only on temperature, does not change.) When the new equilibrium state is finally established the total pressure will exceed the original pressure, but will be much less than it would have been if no reaction had occurred. A pressure change will not affect the relative amounts of the substances at equilibrium in any gaseous system where the number of molecules reacted equals the number produced. For example, H2 + CO2 + CO -tH 2 0 (steam) and H2 + I2 + 2HI (all substances in the gaseous state). The effect of pressure on equilibrium systems involving gases and liquids or solids is usually due to a change in the number of gaseous molecules, since molar volumes of gases are so much larger than of liquids or solids. In the case of equilibrium reactions of solids or liquids, with no gases, the pressure effect is trivially small except at thousands of atmospheres. Effect of changes in amount of solvent For reactions that take place in solution, increasing the amount of solvent (dilution) will displace the equilibrium in the direction of forming the larger number of dissolved particles. This is analogous to decreasing the pressure in a gas reaction. 250 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 EXAMPLE 5 Consider the dimerization of acetic acid in benzene solutions. Let us imagine an equilibrium benzene solution of acetic acid. If the solution is suddenly diluted to twice its original volume, and if there were no change in the relative amounts of the two forms of acetic acid, the concentration of each form would be just one-half of what it had been before dilution. In the equilibrium constant equation, the 1 numerator would be its former value and the denominator would be its former value ( s uared). The ratio of 2 9 numerator to denominator would be 2 times its original value divided by i). this ratio must return to its But original value of K . It can do this if the numerator becomes smaller and the denominator larger. In other words, some of the dimer, (HC2H302)2, must react backwards to form 2HC2H302. (i Changes in the amount of solvent will not affect the equilibrium in any system where the number of dissolved particles reacted equals the number produced. For example, the esterification of methyl alcohol with formic acid in an inert solvent: CH30H + HC02H + HC02CH3 + H20 Effect of varying the concentration Increasing the concentration of any component of a system in equilibrium will promote the reaction which tends to consume some of the added substance. For example, in the reaction H2 + I2 + 2H1, the consumption of iodine is improved by adding excess hydrogen. This effect, too, can be explained in terms of the equilibrium constant. Consider a system at equilibrium with respect to the reaction A + B + C + D. Allow an increase in the concentration of A by introducing an additional amount of A. To bring the value of [C] [D]/[A] [B] back to K , further reaction takes place between A and B to form more of C and D, thus increasing the concentrations of C and D (the numerator) while reducing the resulting concentration of B (the denominator), until the fraction again attains the value of K . At this point equilibrium is again established. The resulting concentration of A is greater than before the supplementary addition of A but less than it would have been had there been no net chemical reaction of adjustment following the addition. Effect of catalysts Catalysts accelerate both forward and backward reaction rates equally. They speed up the approach to equilibrium, but do not alter the equilibrium concentrations. Solved Problems THERMODYNAMICS 16.1. Without consulting entropy tables, predict the sign of A S for each of the following processes: (a> 0 2 (g> 2 0 (g> (f) Desalination of seawater. (4 N2 (g>+ 3H2 (8) 2NH3 (g> (g) Devitrification of glass. ( h ) Hard-boiling of an egg. (4 c (4 H20 ( g > CO ( g >+ H2 (g> + (4 Br2 (4 Br2 (8) ( i ) C (s, graphite) C (s, diamond) (4 N2 ( g , 10 a w N2 ( g , 1 a t 4 + + + + + - + CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 25 1 Postive. There is an increase in the number of gas molecules. Negative. There is a decrease in the number of gas molecules. Positive. There is an increase in the number of gas molecules. Positive. S is always greater for a gas than for its corresponding liquid. Positive. Entropy increases on expansion. Negative. Desalination is the opposite of solution; a solute must be removed from a solution. Negative. Devitrification is the onset of crystallization in a supercooled liquid. Positive. The fundamental process in the "boiling" of an egg is not a literal boiling, in the sense of vaporization, but a denaturation of the egg protein. A protein is a large molecule which exists in a particular configuration in the so-called native state but may occupy a large number of almost random configurations in the denatured state, resulting from rotations around the bonds. The increase in the number of possible configurations is analogous to the melting process. Negative. Diamond, being a harder solid, would be expected to have more restricted atomic motions within the crystal. Diamond is denser and has less entropy on that account also. 16.2. Calculate A S for the following phase transitions: (a) melting of ice at 0 "C, (b) vaporization of water at 100C. Use data from Chapter 7. (a) A H of fusion of ice = 1.44 kcal/mol = 6.02 kJ/mol Since the melting of ice at 0 "C is a reversible process, (16-2)may be used with the equals sign. (Recall that q = A H at constant pressure.) 103 J/mol = 22,.0 J . K-' mol-' T 273.1 K A H of vaporization of water = 9.72 kcal/mol = 40.7 kJ/mol Since the vaporization at 100C is reversible, (16-2) may be used with the equals sign. x 104 J/mol = 10`3.1J . K-' As=&= 4.07 373.1 K T As __ &f __ 6.02 x (b) . mol-' 16.3. After comparing data in Table 7-1 and the answer to Problem 16.1(i), how do you account for the fact that A H and A S for the phase transition from diamond to graphite are not related by the same equation that applied in Problem 16.2? From `Table 7-1, the formation of diamond from graphite (the standard state of carbon) is accompanied by a positive A H of 1.88 kJ/mol at 25 "C. From Problem 16.1(i), A S for the same process is negative. Since 25C is not the transition temperature, the process is not a reversible one. In fact, it is not even a spontaneous irreversible process, and (16-2) does not apply with the inequality sign. On the contrary, the opposite process, the conversion of diamond to graphite at one atmosphere, is thermodynamically spontaneous, and A S for this process would obey (16-2) with the inequality sign. It may come as an unpleasant surprise to jewelers to learn that diamond is unstable. The term "spontaneous," however, implies nothing about the speed of a reaction, which in this case is interminably slow at normal temperatures in the absence of a catalyst. 16.4. Calculate AH; for C2HSOH(g) For the special process in which a substance in its standard state is formed from its elements in their standard states, (16-10) gives AG; = AH; - TAS; (16-12) A H ; can be computed from (16-12) with the use of data from Table 16-1. Write the balanced formation equation for C2H50H(g),and write the (nS")-value under each substance. 2 c ($1 (nS")/J . K-' : 2(5.74) + 3H2 ( g > 3( 130.57) + io, (g> !t f (205.03) C2H5OH (g> 282.6 252 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 For the process, then, ASo = 282.6 - 2(5.74) - 3(130.57) - f(205.03) = -223.1 J K-I whence AS" f' - -223.1 J . K-' = -223.1 J . K-' mol-' 1 mol C2H50H Now, from (16-12), AH; = AG; + TAS; = -168.57 kJ mol-' + (298.15 K)(-0.223 1 kJ . K-' . mol-I) = -235.09 kJ . mol-' Note again that AG; for a substance is listed as a single entry in Table 16-1, but AS: must be computed by taking the difference of the tabulated absolute entropies of the substance and its constituent elements. 16.5. (a) What is AGO at 25C for the following reaction? H2 ( g >+ CO2 (g> + H20 ( g >+ CO ( g > (b) What is AG at 25 "C under conditions where the partial pressures of H2, CO2, H20, and CO are 10, 20, 0.02, and 0.01 atm, respectively? First tabulate (nAG7) values under each substance (in this case, n = 1 mol for each substance). nAG:/kJ: (a) H2 (d 0 (-228.59 + CO2 (d -394.37 + H20 (4 -228.59 + CO ( 8 ) -137.1 5 The computation of AGO is analogous to that of AH" (see Problem 7.12). AG" = - 137.15) - (0 - 394.37) = 28.63 kJ (b) A G = AGO + RT In Q = (28.63 kJ) + (8.314 x 10-3 kJ/K)(298.15 K ) = (0*02)(0*01)] kJ ( 10)(20) = (28.63 - 34.25) kJ = -5.62 kJ [28.63 + 5.708 log 10e6]kJ = [28.63 - 6(5.708)] kJ Note that the reaction, although not possible under standard conditions, becomes possible (AG < 0) under this set of experimental conditions. Note the conversion from natural to common logarithms by use of the factor 2.302 6. 16.6. Calculate the absolute entropy of CH30H ( g ) at 25C Although there is no So-entry in Table 16-1 for CH30H(g), the AG_j--value this substance listed in for Table 16-1, the AH;-value listed in Table 7-1 and the So-values for the constituent elements may be combined to yield the desired value. From (26-22), - AG: (-200.7 + 162.0) kJ . mol-' = -129.8 J . K-' . mol-' T 298.2 K From the equation for the formation of 1 mol of CH30H(g) under standard conditions, AS; = AH; C(4 + 2H2 ( g ) + $ 0 2 (8) + CH30H ( g ) we can write - -129.8 J - K-' = (1 mol)[So(CH30H)] ( 1 mol)[S"(C)]- (2 mol)[S"(H2)]= (1 mol)[So(CH30H)] [5.7 - + 2( 130.6)+ f (205.0)] J . K-' (4 mol)[S0(02)] Solving, S"(CH,OH) = 239.6 J . K-' . mol-' , CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 253 16.7. Estimate the boiling point of PC13. The boiling point is the temperature at which AGO for the following reaction is zero: PC13( I ) + PC13(g) The reaction is not spontaneous at 25 "C, where, according to Table 16-1, AGO = (1 mol)[AG;(PC13, g)] - (1 mo1)[AG;(PC:l3, l ) ]= -267.8 + 272.4 = 4.6 kJ If we assume that AH" and AS" are both independent of temperature between 25 "C and the boiling point, then the temperature dependence of AGO is given by the factor T in (16-10). If AH" and AS" are known from the data at 25 "C, T can be calculated to satisfy the condition that AGO equals zero. AGO = AH" - TAS" = 0 Now AS" = (1 mol)[S" (g)]- ( 1 mol[S"(I)]= 311.7 - 217.1 = 94.6 J/K and, from Table 7-1, AH" = (1 mol)[AH? (g)]- ( 1 mol)[AH?(l)] = -287.0 Then AH" 32.7 x 103 J T(b.p.) = -= AS" 94.6 J/K= 346 - or AH" T(b.p.) = AS" (-319.7) = 32.7 kJ K or 73 "C. The observed boiling point, 75 "C, is very close to the estimated 73 "C. The validity of this type of estimate is no better than the assumption of constancy of AS" and A H . In general, the smaller the temperature range over which the extrapolation must be made, the greater the accuracy of the prediction. EQUILIBRIUM 16.8. For the gaseous reaction 1 pc15 ( g >=: pc13 (g> + c2 (g> explain the effect upon the equilibrium composition of (a) increased temperature, (b) increased pressure, (c) higher concentration of Cl;! ( d ) higher concentration of PCls, (e) presence of a catalyst. When the temperature of a system in equilibrium is raised, the equilibrium point is displaced in the direction which absorbs heat. Table 7-1 can be used to determine that theforward reaction as written is endothermic. A H = (1 mol)[AHf(PC13)] (1 mol)[AHf((312)]- (1 mol)[AHf(PC15)] 87.9 kJ 287.0 + 0 - (-374.9) Hence increasing the temperature will cause more PC15 to dissociate. When the pressure of a system in equilibrium is increased, the equilibrium point is displaced in the direction of the smaller volume. One volume each of PC13 and C12, a total of 2 gas volumes, forms 1 volume of PCls. Hence a pressure increase will promote: the reaction between PC13 and Cl2 to form more PC15. Increasing the concentration of any component will displace the equilibrium in the direction that tends to lower the concentration of the component added. Increasing the concentration of Clz will result in the consumption of more PC13 and the formation of more PCls, and this action will partially offset the increased concentration of' C12. Increasing the concentration of PC15 will result in the formation of more PC13 and Cl2. A catalyst accelerates both forward and backward reactions equally. It speeds up the approach to equilibrium, but does not favor reaction in either direction over the other. + 254 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 16.9. What conditions would you suggest for the manufacture of ammonia by the Haber process? N2 (g) + 3H2 (8) + 2NH3 (g) A H = -22 kcal From the sign of AH, we see that the forward reaction gives off heat (exothermic). Therefore, the equilibrium formation of NH3 is favored by as low a temperature as practicable. In a reaction like this, the choice of temperature requires a compromise between equilibrium considerations and rate considerations. The equilibrium yield of NH3 is higher, the lower the temperature. On the other hand, the rate at which the system will reach equilibrium is lower, the lower the temperature. The forward reaction is accompanied by a decrease in volume, since 4 volumes of initial gases yield 2 volumes of NH3. Therefore increased pressure will give a larger proportion of NH3 in the equilibrium mixture. A catalyst should be employed to speed up the approach to equilibrium. 16.10. Consider the gaseous reaction 2N02 + N2O4 (a) Calculate AGO and Kp for this reaction at 25C. (6) Calculate AGO and Kp for the reverse reaction: N2O4 (c) + 2N02 Calculate AGO and Kp for the forward reaction written with different balancing coefficients: NO2 + iN2O4 () Repeat calculation ( a ) for a standard state pressure of 1 bar. For NO2 ( g ) AGY based on d 1 bar is 5'4.32kJmol-', and for N204 ( g ) 97.89 kJ mol-'. = 97.82 - 2(51.30) = AGO = (1 mol)[AG?(N204)]- (2 mol>[AG?(N02)] -4.78 kJ = -RT In Kp log Kp = -AGO 4.78 x 103 J = 0.837 2.303RT - (2.303)(8.314 5 J/K)(298.2 K) Kp = 6.87 ~ (b) AGO = (2 mol)[AG?(NOz)]- (1 mol)[AG?(N2O4)] = 2(51.30) - 97.82 = 4.78 kJ -4.78 103 -AGO = -0.837 log Kp = ____ 2.303R T (2.303) (8.314 5) (298.2) Kp = 0.146 Parts (a) and (b)illustrate the general requirements that AG of a reverse reaction is the negative of AG for the forward reaction and K for a reverse reaction is the reciprocal of K (i.e., 1 divided by K) for the forward reaction. AGO = (f m o l ) [ A G ~ ( N 2 0- ) 1 mol)[AG~(NO2)] ~ (] = f (97.82) - 51.30 = -2.39 kJ -AGO 2.39 103 = 0.419 log Kp = ____ 2.303RT - (2.303) (8.314 5) (298.2) Kp = 2.62 Parts (a) and (c) illustrate the general result that AG for a reaction with halved coefficients is half the AG-value for the standard coefficients and that K for the reaction with halved coefficients is the one-half power (i.e., square root) of the K for the standard reaction. Note that the value of the ratio CHAP. 161 THERMODYNAMICS AND CHEMICAL, EQUILIBRIUM 255 at equilibrium must be independent of the way we write the balanced equation. Any equilibrium constant for the reaction must involve the two partial pressures in exactly this way. For part (a), K is equal to this ratio; for part (c), K is equal to the square-root of this ratio. AGO = (1 mol)[AG~(N204)] (2 mol)[AG;(N02)] = 97.89 - 2(51.32) = -4.75 kJ = -RT In Kp In Kp = 4.75 1035 = 1.916 (8.3145 J/K)(298.2 K) Kp = 6.79 Since 1 atm and 1 bar are close (1 atm = 1.013bar), the differences are small, but not insignificant. Kp (bar) can also be calculated from Kp (atm) by simply converting units. Kp(bar) = P(atm)(N204)x 1.013 - Kp(atm) -[P(atm)(N02)x 1.0131' 1.013 6.87 1.013 = 6'78 16.11. A quantity of PCls was heated in a 12-L vessel at 250C At equilibrium the vessel contains 0.21 mol PCls, 0.32 mol PC13, and 0.32 mol C12. (a) Compute the equilibrium constant Kp for the dissociation of PCls at 250 "C when pressures are referred to the standard state of 1 atm. (b) What is AGO for the reaction? ( c ) Estimate AGOfrom the data in Tables 7-1 and 16-1, assuming constancy of AH' and AS'. ( d ) Calculate Kp from the original data using SI units and a standard state of 1bar. (4 nRT (0.21 mol)[0.082 1 L - atm/(mol - K)](523 K) = 0.751 atm P(PC15) = -= 12 L V (0.32)(0.082 1)(523) = 1.145 atm P(C12) = P(PC13) = 12 (4 (c) A O = -RT In K = -(8.3145 J/K)(523 K) In 1.75 = -2.4 kJ G AG598 = -267.8 AH598 = -287.0 + 305.0 = 37.2 kJ + 374.9 = 87.9 kJ AT= AH" - AGO - (87.9 - 37.2)(1000) = 170.0 J/K -T 298.2 -1.0 kJ At 298.2 K, AGO = AH" - T AS" At 523 K, AG0523 = 87.9 - (523)(170'0) = 87.9 - 88.9 1000 The estimate is close to the experimental value based on the equilibrium measurement, despite the large temperature range over which AHo and ASo were assumed constant. (d) nRT (0.21 mol)[8.3145 m3 . Pa/(mol. K)](523 K)(1 bar/105 Pa) =0.761 bar P(PC15) = -= (12 L)(I m3/103 L) V P(C12) = P(PC13) = (0'32)(8'3145)(523)(103)= 1.160 bar 1 2 x 105 ( 1.160)( 1.160) = 1.77 Kp(bar) = 0.761 256 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 16.12. When lmol of pure ethyl alcohol is mixed with lmol of acetic acid at room temperature, the equilibrium mixture contains mol each of ester and water. (a) What is the equilibrium constant? (b) What is AGO for the reaction? (c) How many moles of ester are formed at equilibrium when 3 mol of alcohol is mixed with 1 mol of acid? All substances are liquids at the reaction temperature. 5 n at start: Change by reaction n at equilibrium: alcohol acid ester C2HjOH(I) + CH3COOH(I) *CH3COOC*Hs(I) 1 1 0 - -2 + H20(1) 0 2 2 3 water 1-2-1 3 3-3 -_2 3 1-2-1 3-3 +f 2 3 3 This tabular representation is a convenient way of doing the bookkeeping for equilibrium problems. Under each substance in the balanced equation an entry is made on three lines: (1) the amount of starting material; (2) the change (plus or minus) in the number of moles due to the attainment of equilibrium; and (3) the equilibrium amount, which is the algebraic sum of entries (1) and (2). The entries in line (2) must be in the same ratio to each other as the coefficients in the balanced chemical equation. The equilibrium constant can be found from the entries in line (3). Let v = number of liters of mixture and choose, as usual, 1 mol/L as the standard-state concentration. Then = [ester] [water] K= [alcohol] [acid] - (t) (t) (c) (c) Note that the concentration of water is not so large compared with the other reaction components that it remains constant under all reaction conditions; the concentration of water must therefore appear in the expression for K, along with the concentrations of the reactants and of the other product. (b) AGO = -RT In K = -(8.3145 J/K)(298.2 K)[(2.303)(log 4)] = -3.44 kJ (c) Because K is independent of the choice of standard-state concentration (the sum of the exponents in the numerator equals the sum of the exponents in the denominator), so also is AGO. Let x = number of moles of alcohol reacting. n at start Change by reaction: n at equilibrium: C2HSOH + CH3COOH 3 1 -x -X 3-x 1-x * CH3COOC2HS + H20 0 0 +X X +X X Then x2 = 4(3 - 4x + x2) or 3x2 - 16x + 12 = 0. Solving by the quadratic formula X = (G) (+) f d(16l2 - 4(3)(12) - (3 X2 3-4x+x2 -b f a +I6 2a - 2(3) --- 16 f 10.6 - 4.4 or 0.9 6 Of the two roots to the quadratic equation, only one has physical meaning. The applicable root can generally be selected very easily. In this problem, we started with 3 mol of alcohol and 1 mol of acid. We can see from the reaction equation that we cannot form more than 1 mol of ester, even if we use up all the acid. Therefore the correct root of the quadratic is 0.9. Consequently, 0.9 mol of ester is formed at equilibrium. In more complex problems, especially if the degree exceeds the quadratic, it may be best to find the equilibrium concentrations by a method of successive approximations. One selects a consistent set of concentration values near where one guesses the answer will be and calculates K , reiterating the process until a sufficiently precise result is obtained. Such a procedure was used to obtain the answer to problem 16.32. CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 257 It is wise to confirm the numerical result, especially if the calculation is complex, by substituting back in the expression for K . (0.9)(0.9) = 3.86 = 4 (within the limits of precision of the calculation) K= (2.1)(0.1) Note that the number of moles of ester formed is greater than the number of moles of ester in equilibrium in (a) (0.9 compared with 2/3). This result was to be expected because of the increased concentration of alcohol, one of the reactants. A further addition of alcohol would lead to an even greater yield of ester, but in no case could the amount of ester formed exceed 1 mol, since 1 mol would represent 100 percent conversion of the acid. In practice, the actual selection of the excess concentration to be used may depend on economic factors. If alcohol is cheap compared to acid and ester, a large excess of alcohol might be used to assure a high percentage conversion of the acid. If, on the other hand, alcohol costs more per mole than acid, it would be more sensible to use an excess of acid and aim for a high percentage conversion of the alcohol. 16.13. In a 10-L evacuated chamber, 0.5 mol H2 and 0.5 mol I2 are reacted at 448 "C. H2 (g>+ I2 (g>?= 2HI ( g > At the given temperature, and for a standard state of 1 mol/L, K = 50 for the reaction. (a) What is the total pressure in the chamber? (b) How many moles of the iodine remain unreacted at equilibrium? (c) What is the partial pressure of each component in the equilibrium mixture? (a) Before the reaction sets in, the total number of gas moles is 0.5 0.5 = 1. During the reaction, there is no change in the total number of moles. The total pressure can be computed from the ideal gas law: + n(tota1)RT - (1 mo1)(0.082 1 La atm. K-' . mol-')(721 K) = 5.9 atm V 10 L (b) Let x = number of moles of iodine reacting I2 +2HI H2 + n at start: 0.5 0.5 0 Change by reaction : -x -X 2x n at equilibrium: 0.5 - x 0.5 - x 2~ P(tota1) = Note the 2:l ratio of moles of HI formed to moles of H2 reacted. This ratio is demanded by the coefficientsin the balanced chemical equation. Regardless of how complete or incomplete the reaction may be, 2mol of HI is always formed for every 1 rnol of H2 that reacts. Since the number of moles of reactant gas equals the number of moles of product gas, moles may be used in place of concentration (as in Prob. 16-12), anti Kp = K . Kp = 50 = (0.5 - ~ ) ( 0 . - X) 5 (W2 or &=7.1 =- 2x 0.5 - x Then 2x = 7.1(0.5 - x). Solving, x = 0.39. Thus, 0.39mol I2 reacts, leaving 0.5 - 0.39 = 0.1 1 rnol I2 unreacted at equilibrium. Note that if the negative square root were chosen the value of x would have been 0.7 mol, an impossibly high result, because no more I2 can react than the amount initially added. Checking the result: K= (2 x 0.39)* = 50.3 = 50 (0.11)2 (within the limits of precision of the calculation) P(I2) = - (total pressure) = (?I) (5.9 atm) = 0.65 atm n(12) x n( total) P(H2) = P(I2)= 0.65 atm = P(H1) = (total pressure) - [P(H2)+ P(12)] 5.9 - 1.3 = 4.6 atm or P(H1) = I - (total pressure) = n(H1) x n( total) r$) (5.9 atm) = 4.6 atm 258 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 .6.14. Sulfide ion in alkaline solution reacts with solid sulfur to form polysulfide ions having formulas S;-, $-, Si-, and so on. The equilibrium constant for the formation of S:- is 12 and for the formation of Sz- is 130, both from S and S2-. What is the equilibrium constant for the formation of Sz- from S;- and S? To avoid confusion, let us designate the equilibrium constants for the various reactions by subscripts. Also, we note that only the ion concentrations appear in the equilibrium constant equations, because solid sulfur, S, is always in its standard state. 2s + S2- s + s2-+ s;- * S:s + s;- * s:- K1 = [$-]/[s2-] 12 = K 2 = [S:-]/[S2-] = 130 K3 = [s:-]/[s;-] The desired constant K3, expresses the equilibrium ratio of Si- to S:- concentrations in a solution equilibrated with solid sulfur. Such a solution must also contain sulfide ion, S2-, resulting from the dissociation of S;- by the reverse of the first equation. Since all four species, S, S2-, S;- and S:-, are present, all the equilibria represented above must be satisfied. The three equilibrium ratios are not all independent, because The result, K2 = K1K3, is a general one for any case where one chemical equation (the second in this case) can be written as the sum of two other equations (the first and third in this case). 16.15. At 27 "C and 1 atm, N2O4 is 20 percent dissociated into N02. Find (a) Kp and (b) the percent dissociation at 27 "C and a total pressure of 0.10 atm. ( c ) What is the extent of dissociation in a 69-g sample of N2O4 confined in a 20-L vessel at 27 "C? (a) When 1 mol of N2O4 gas dissociates completely, 2mol of NO2 gas is formed. Since this problem does not specify a particular size of reaction vessel or a particular weight of sample, we are free to choose 1 mol (92g) as the starting amount of N2O4. For the given total pressure of 1 atm, the tabular computation of the partial pressures proceeds as follows: N204 n at start: Change by reaction: n at equilibrium: k) + 2N02 (g) 0 f0.40 1 -0.20 1 - 0.20 = 0.80 0.80 Mole fraction: = 0.667 0.80 0.40 p.p. = (mole frac.) x (1 atm): 0.667 atm + 0.40 0.40 = 0.333 0.80 + 0.40 0.333 atm Then, in terms of the numerical values of the partial pressures, Kp = P(N02)2 - (0.333)2 ~ P(N2O4) - 0.667 ____ = 0.167 (b) Let a =fraction of N2O4 dissociated at equilibrium at 0.1 atm total pressure. N24 2 7 n at start Change by reaction n at equilibrium: Mole fraction: Partial pressure: : 1 -a 1-CY (& 0.1) atm (& 0.1) atm x x 1-CY l+CY 2N02 0 2a 2a 2a l+CY CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 259 From (a), Kp = 0.167. Thus, 0.167 = Kp = (S P(N204) 2 x 0.1) 0.4a2 1 -a2 . + a x 0.1 l I-a (4 or 0 . 4 = 0.167(1 - a 2 ) . ~ ~ Solving, a = 0.54 = 54% dissociated at 27 "C and 0.1 atm. Note that a larger fraction of the N2O4 is dissociated at 0.1 atm than at 1 atm. This is in agreement with Le Chatelier's principle: Decreasing the pressure shoiild favour the side with the greater volume (NO2). If the sample were all N204,it would contain ~ 69 g = 0.75 mol 92 g/mol 2N02 0 +2( 0 . 7 5 ~ ) 1.50a Let a be the fractional dissociation. The tabular analysis follows: N204 n at start : 0.75 Change by reaction: -0.75~~ n at equilibrium : 0.75(1 - a ) Because the total gas pressure is unknown, it is simplest to obtain the partial pressures directly from Dalton's law (Chapter 5). P(N204)= n(N204)RT __ [0.75(1 - a ) moll(0.082 L - atm . K-' . mol-')(300 K) v 20 L = 0.92(1 - a ) atm n(N02)RT - 1.50a [0.92(1 - a ) atm] = 1 . 8 4 ~ ~ atm P(N02) = V 0.75(1 - a ) (1.840)~ - - ~ 3.68~ 0.167 = Kp = P(N02)2 = P(N2O4) 0.92(1 -- a ) 1 - a ~ Then or Solving, a= 3.68~~~ 0.167a - 0.167 = 0 + 1.577 -0.167 f 4(0.167)2 + 4(0.167)(3.68) -0.1tj7 -= -0.24 or 2(3.68) 7.36 + 0.19 The negative root must obviously be discarded. The extent of dissociation is 19%. 16.16. Under what conditions will CuS04 - 5H20 be eflorescent at 25 "C? How good a drying agent is CuS04. 3H20 at the same temperature? For the reaction CuS04 - 5H20 (solid) F= CuS04 - 3H20 (solid) + 2H20 (gas) Kp at 25 "C is 1 . 0 8 6 lOV4;the vapor pressure of water at 25 "C is 23.8 torr. ~ An efflorescent salt is one that loses water to the atmosphere:. This will occur if the water vapor pressure in equilibrium with the salt is greater than the water vapor pressure in the atmosphere. The mechanism by which CuS04 . 5H20 could be efflorescent is that the salt would lose 2 molecules of H 2 0 and simultaneously form 1 formula unit of CuS04 - 3 H 2 0 for each unit of the original salt that dissociates. Then the above equilibrium equation would apply, since all three components would be present. Since CuS04 5H20 and CuS04 . 3 H 2 0 are both solids, Kp = P(H20)2= 1.086 x 10-4 260 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 where P(H20) is the partial pressure of water vapor (relative to the standard state pressure of 1 atm) in equilibrium with the two solids. Solving. P ( H 2 0 ) = 1.042 x 10-2 atm = (1.042 x 10-2 atm)(760 torr/atm) = 7.92 torr Since P(H20) is less than the vapor pressure of water at the same temperature, CuSO4 . 5H20 will not always effloresce. It will effloresce only on a dry day, when the partial pressure of moisture in the air is less than 7.92 torr. This will occur when the relative humidity is less than 7.92 -- torr - 0.333 = 33.3% 23.8 torr CuS04 3 H 2 0 could act as a drying agent by reacting with 2 molecules of H20 to form CuSO4 - 5H20. The same equilibrium would be set up as above, and the vapor pressure of water would be fixed at 7.92 torr. In other words, CuS04 . 3 H 2 0can reduce the moisture content of any confined volume of gas to 7.92 torr. It cannot reduce the moisture content below this value. Many other drying agents can reduce the partial pressure of water below this value. (See Problem 16.44.) To find out the conditions under which CuS04 . 3H20 would be efflorescent, we would have to know the equilibrium constant for another reaction, which shows the product of dehydration of CuS04 . 3H20. This reaction is CUSO~ 3H2O(s) i C U S O ~H20(s) + 2H20(g) = * * Supplementary Problems Use may be made of data in Tables 7-1 and 16-1 for the solution of the following problems. These data are based on experimental measurements and are subject to reevaluation. Thus, tables in different books may not always agree. The data posted in this book are part of an internally consistent set, and the answers are based on them. THERMODYNAMICS 16.17. Calculate So at 25C for PC15 (g). Ans. 364 J/K - mol 16.18. Calculate AH; for C120 (g) at 25C. Ans. 80.2 kJ/mol 16.19. Predict the phase-transition temperature for the conversion of gray to white tin at one atmosphere, using the tabulated thermodynamic data. Ans. 9 "C (the observed value is 13 "C) 16.20. Consider the production of water gas: C (s, graphite) + H20 ( g ) CO (8) + H2 ( g ) (a) What is AGO for this reaction at 25"C? (b) Estimate the temperature at which AGO for the reaction becomes zero. Ans. (a) 91.44 kJ; (b) 982 K. The extrapolation for this estimate is extended over such a large temperature that an appreciable error might be expected. The actual experiment value, 947K, is not far from the estimate. CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 26 1 G 16.21. A : for the formation of HI (g) from its gaseous elements is -lO.lOkJ/mol at 500K. When the partial pressure of HI is lOatm, and of I2 0.001 atm, what must the partial pressure of hydrogen be at this temperature to reduce the magnitude of AG for the reaction to zero? Ans. 7.8 x 102 atm 16.22. Under what conditions could the decomposition of AgzO (s) into Ag (s) and 0 2 ( g ) proceed spontaneously at 25C Ans. The partial pressure of 0 2 must be kept below 0.090 torr. 16.23. Estimate the minimum temperature at which the decomposition of AgzO(s) (see problem 1622) will proceed spontaneously ( a ) when the oxygen pressure is l.OOatm, and (b) in air, where the oxygen pressure is 0.21 atm. Note: T ( A g ) = 42.72 J/K . mol. Ans. (a) 466K; (b) 425K 16.24. The effect of changing the standard state from 1 atm to 1 bar is to raise the molar entropies, So, all of gaseous substances by 0.109 J/(K mol) at 298.2 K. Convert .4G: of ( a ) CH30H ( I ) , (b) CH30H (g), (c) H20 ( I ) , and ( d ) Sn (s, gray) to the standard state of 1 bar. Assume that within the precision of the tables in this text AH: of all substances is unchanged by this small change in standard state. Ans. (a) -- 166.28 kJ/mol; (b) - 161.95 kJ/mol; (c) -237.14 kJ/mol; ( d ) 0.12 kJ/mol 16.25. N2 + O2 + 2N0. State the effect upon the reaction equilibrium of (a) increased temperature, (b) decreased pressure, (c) higher concentration of 02, ( d ) lower 'concentration of Nz, (e) higher concentra- tion of NO, (f)presence of a catalyst. Ans. Favors (a) forward reaction, (b) neither reaction, (c) forward reaction, ( d ) backward reaction, (e) backward reaction, (f)neither reaction. 16.26. Predict the effect upon the following reaction equilibria of: ( a ) increased temperature, (b) increased pressure 1. CO (g>+ H20 (g>+ CO2 (g>+ H2 (g> 2. 2so2 ( g ) + 0 2 (g>+ 2so3 (d 3. N24 (8) + 2N02 (g> 4. H20 ( g > + H2 (d + 4 0 2 (g> 5. 203 (d + 302 (g> 6. CO (g)+2H2 (g) + CH30H ( g ) 7 . CaC03 (s) + CaO (s)+ CO2 (g) 8. c (4 -tH20 (g>+ H2 (d + CO (d 9. 4HC1 (g) 0 2 (g) + 2H20 (g) + 2C12 (g) 20. C (s, diamond) + C (s, graphite) (This equilibrium can exist only tinder very special conditions.) Densities of diamond and graphite are 3.5 and 2.3 g/cm3, respectively. + = favors forward reaction, B = favors backward reaction. 2. ( a ) B; (b) neither 2. ( a ) B; (b) F 3. ( a ) F; (b) B 4 . ( a ) F; (b) B 5 . ( a ) B; (b) B 6. (a) B; (b) F 7 . (a) IT; (b) B 8. (a) I:; (b) B 9. (a) 13; (b) F 20. (a) B; (b) B 16.27. Assuming that AH" and AS" remain constant, derive from equation (26-22) an expression relating K1 at 2 temperature Tl to K at temperature T2. Ans. ln(K2/K1)= (AH0/R)(T2 T1)/T2T1 - 262 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 16.28. For the neutralization reaction: H'(aq) + OH-(aq) --+ H20 AH" is -55.8 kJ. For the ionization of water, which is the reverse of the above, the equilibrium constant at 25 "C is: K , = [H+][OH-]= 1.0 x lO-I4. This is a very important constant, which appears throughout Chapter 17. Using the result of problem 16.27, calculate K, at (a) 37"C, the physiological temperature, and at (b) 50C. A ~ (a) 2.4 ~ . 10-l~; (b) 5.7 10-l~ EQUILIBRIUM CONSTANTS 16.29. When a-D-glucose is dissolved in water it undergoes a partial conversion to P-D-glucose, a sugar of the same molar mass but of slightly different physical properties. At 25 "C, this conversion, called mutarofation, stops when 63.6 percent of the glucose is in the P-form. Assuming that equilibrium has been attained, calculate K and AGO for the reaction a-D-glucose + P-D-glucose at the experimental temperature. Ans. 1.75, -1.38 kJ 16.30. The equilibrium constant for the reaction H3B03 + glycerin + (H3B03-glycerin)is 0.9. How much glycerin should be added per liter of 0.10 molar H3B03 solution so that 60 percent of the H3B03 is converted to the boric acid-glycerin complex? 1.7mol Ans. 16.31. The equilibrium p-xyloquinone + methylene white + p-xylohydroquinone + methylene blue may be studied conveniently by observing the difference in color between methylene blue and methylene white. One millimole of methylene blue was added to a liter of solution that was 0.24M in p-xylohydroquinone and 0.012 0 M in p-xyloquinone. It was then found that 4.0 percent of the added methylene blue was reduced to methylene white. What is the equilibrium constant for the above reaction? The equation is balanced with one molecule of each of the four substances. Ans. 4.8 x 102 16.32. When SO3 is heated to a high temperature it decomposes 2so3 ( g ) +-+ 2so2 ( g ) + 0 2 (8) A sample of pure SO3 was sealed in a large cylinder fitted with a piston, and heated to a high temperature, T . At equilibrium the ratio of SO2 to SO3 was 0.152 and the total pressure was 2.73 atm. If the piston is pushed in, reducing the volume to half its original value, what will be the new pressure? (Note: because of the degree of the equations involved, a solution is most readily obtained by the method of successive approximations.) Ans. 5.40atm 16.33. Consider the equilibrium described in the previous problem. Instead of pushing in the piston, a quantity of SO3 ( g ) equal to that already present is injected, momentarily doubling the SO3 pressure. When equilibrium is restored (still at temperature T ) , (a) what is the new SO3 pressure, and (b) what is its ratio to that momentarily present at the time of injection? (c) Calculate a similar ratio for the SOZ. Ans. (a) 4.26atm; (b) 0.96; (c) 1.54 CHAP. 161 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM 263 16.34. A saturated solution of iodine in water contains 0.33g I2 per liter. More than this can dissolve in a KI solution because of the following equilibrium: I2 (as) + I - =: 13 A 0.100 M KI solution (0.100 M I-) actually dissolves 12.5g of iodine per liter, most of which is converted to 13. Assuming that the concentration of 12 in all saturated solutions is the same, calculate the equilibrium constant for the above reaction with 1 molar solutions as standard states. What is the effect of adding water to a clear saturated solution of 12 in the KI solution? Ans. 7.1 x 102; backward reaction is favored. 16.35. H2 (g) + I2 ( g ) 2HI (g). When 46g of 12 and l.0Og of H2 are heated to equilibrium at 470 "C, the equilibrium mixture contains 1.9Og 12. (a) How many moles of each gas are present in the equilibrium (b) Compute the equilibrium constant. mixture? Ans. (a) 0.007 5 mol 12, 0.32 mol H2, 0.35 mol HI; (b) K = 50 * 16.36. Exactly 1 mol each of H2 and I2 are heated in a 30-L evacuated chamber to 470 "C. Using the value of K from Problem 16.30, determine (a) how many moles of' 12 remain unreacted when equilibrium is established, (b) the total pressure in the chamber, (c) the partial pressures of I2 and of HI in the equilibrium mixture. ( d ) Now if one additional mole of H2 is introduced into this equilibrium system, how many moles of the original iodine will remain unreacted? Ans. (a) 0.22mol; (b) 4.1 atm; (c) P(H2) =P(Iz)= 0.45 atm, P(HI) = 3.2 atm; ( d ) 0.065mol 16.37. PC15 ( g ) PC13 ( g ) + C12 (g). Calculate the number of moles of C12 produced at equilibrium when 1 mol of PC15 is heated at 250 "C in a vessel having a capacity of 10 1,. At 250 "C, K = 0.041 for this dissociation based on the 1 mol/L standard state. Ans. 0.47 mol * 16.38. Pure PC15 is introduced into an evacuated chamber and comes to equilibrium (see Problem 16.37) at 250 "C at 2.00atm. The equilibrium gas contains 40.7% chlorine by volume. (a) 1. What are the partial pressures of the gaseous compon.ents at equilibrium? 2. From these data calculate K,, at 250 "C, based on the 1 atm standard state, for the reaction as written in Problem 16.37. If the gas mixture is expanded to 0.200 atm at 250 "C, calculate: (b) 1. The percent of PC15 that would be dissociated at equilibrium 2. The percent by volume of C12 in the equilibrium gas mixture 3. The partial pressure of C12 in the equilibrium mixture Ans. (a) 1. P(C1,) =P(PC13) = 0.814atm, P(PC15) = 0.372atm; 2. 1.78 (b) 1. 94.8%; 2. 48.7%; 3. 0.0974 atm 16.39. At 46"C, K,, for the reaction N204 (g) * 2N02 (g) is 0.67, based on the 1 bar standard state. Compute the percent dissociation of N204 at 46C and a total pressure of 0.507 bar. What are the partial pressures of N2O4 and NO2 at equilibrium? Ans. 50%; P(N2O4) = 0.17 bar, P(NO2) = 0.34 bar 16.40. 2NOBr (g) 2 N 0 (g) + Br2 (8). If nitrosyl bromide (NOBr) is 34 percent dissociated at 25 "C and a total pressure of 0.25 bar, calculate K,, for the dissociation at this temperature, based on the 1 bar standard state. Ans. * 1.0 x 10-2 264 THERMODYNAMICS AND CHEMICAL EQUILIBRIUM [CHAP. 16 16.41. The equilibrium constant for the reaction CO (g>+ H20 (8) + CO2 (g>+ H2 (g> at 986C is 0.63. A mixture of exactly 1 mol of water vapor and 3mol of CO is allowed to come to equilibrium at a total pressure of 2atm. ( a ) How many moles of H2 are present at equilibrium? (b) What are the partial pressures of the gases in the equilibrium mixture? Ans. ( a ) 0.68 mol; (b) P(C0) = 1.16atm, P(H20) = 0.16 atm, P(C02) = P(H2) = 0.34 atm 16.42. For the reaction Sn02 (solid) + 2H2 (gas) + 2H20 (steam) + Sn (molten) calculate Kp (a) at 900 K, where the equilibrium steam-hydrogen mixture was 45% H2 by volume; (b) at llOOK, where the equilibrium steam-hydrogen mixture was 24% H2 by volume. (c) Would you recommend higher or lower temperatures for more efficient reduction of tin? Ans. (a) 1.5; (b) 10; (c) higher 16.43. In the preparation of quicklime from limestone, the reaction is CaC03 (s) + CaO (s) + CO2 (g) Experiments carried out between 850C and 950C led to a set of Kp-values fitting an empirical equation 8 500 log Kp = 7.282 - T where T is the absolute temperature. If the reaction is carried out in quiet air, what temperature would be predicted from this equation for the complete decomposition of the limestone? In quiet air assume it is necessary to build up the CO2 pressure to 1 atm to ensure continual removal of the product. Ans. 894C 16.44. The moisture content of a gas is often expressed in terms of the dew point. The dew point is the temperature to which the gas must be cooled before the gas becomes saturated with water vapor. At this temperature, water or ice (depending on the temperature) will be deposited on a solid surface. The efficiency of CaC12 as a drying agent was measured by a dew point experiment. Air at 0 C was allowed to pass slowly over large trays containing CaC12. The air was then passed through a glass vessel through which a copper rod was sealed. The rod was cooled by immersing the emergent part of it in a dry ice bath. The temperature of the rod inside the glass vessel was measured by a thermocouple. As the rod was cooled slowly the temperature at which the first crystals of frost were deposited was observed to be -43 "C. The vapor pressure of ice at this temperature is 0.07torr. Assuming that the CaC12 owes its desiccating properties to the formation of CaC12 e2H20, calculate Kp at 0 "C for the reaction CaC12 . 2 H 2 0 (s) + CaC12 (s) + 2 H 2 0 ( g ) Ans. 8 x 10-9 16.45. At high temperatures the following equilibria prevail in a mixture of carbon, oxygen and their compounds: If it were possible to measure K1 and K2 independently, how could K3 and Ans. K3 = K 2 / K l ; K4 = K 1 / K 3= K:/K2 K4 be calculated? Chapter 17 Acids and Bases The general principles of chemical equilibrium apply equally to reactions of neutral molecules and to reactions of ions. Because of the special interest which chemists have in ionic equilibria in aqueous solutions and because of some common methods of treating some of these problems, this chapter and the following are devoted to the specific applications of chemical equilibrium to ionic reactions. As in Chapter 16, concentrations will always be expressed in mol/L, and [XI will represent the numerical value of the concentration of X. ACIDS AND BASES Arrhenius concept According to the classical definition as formulated by Arrhenius, an acid is a substance which can yield H+ in solution. HC104 and HN03, which are completely ionized in water into H+ and ClO, and into H+ and NO, respectively, are called strong acids. HC2H302, acetic acid, and HN02, nitrous acid, are only partly ionized into HS and C2H3Oi and into HS and NOT, and these substances are called weak acids. The dissociation of a weak acid is reversible in aqueous solutions and may be described by an equilibrium constant, usually designated as K,. Thus ( I 7-1) Similarly, a base is a substance which can yield OH-. NaOH, a strong base, is completely ionized in water into Na' and OH-; even relatively insoluble hydroxides, such as Ca(OH)2, give solutions within their limited solubility range which are completely ionized. A solution of ammonia, NH3, in water also produces hydroxide ions, and so at one time was thought to consist of NH40H in solution. Since the OH- concentration is only a few percent of the ammonia concentration, ammonia is considered a weak base. Bronsted-Lowry concept The Arrhenius concept of acids and bases has been modified and generalized by the Bronsted-Lowry picture, which allows for the specific role of the solvent. The proton is the essential ingredient in the definition of both acids and bases in the Bronsted-Lowry scheme, according to which an acid is a substance which can transfer a proton to another substance, which could be the solvent itself, and a base is a substance, which might be the solvent, which accepts a proton from an acid. The species remaining after an acid has transferred a proton is also a base, since in principle it could regain the proton to form the original acid. Such a grouping of an acid and its deprotonated form is said to constitute a conjugate acid-base pair. The generalized proton-transfer reaction is then represented as HA + B + A- + BH+ ( I 7-2) In this equation, HA-A- and B-BH+ constitute conjugate acid-base pairs. Neither HA nor B is necessarily a neutral species, but in any event the charge on the base conjugate to HA is algebraically one unit less than the charge on HA, and the charge on the acid conjugate to B is one positive unit greater than B. In this notation, (17-1) would be written as follows: (17-3) 265 266 ACIDS AND BASES [CHAP. 17 In a similar fashion the ionization of the weak base, ammonia, is written: (17-4) H 2 0 serves as a base in (17-3) and as an acid in (17-4). Note that the bare H+ of (17-1) becomes the hydrated proton or hydronium ion, H 3 0 + , of (17-3). In the formulation of equilibrium constants, [H'] and [H30'] are always equivalent to each other; the two forms are used interchangeably in most contexts and will be so used in all ionic equilibrium problems in this book. Although the proton is indeed hydrated in aqueous solutions, the notation H+ is often used instead of H 3 0 + because the reader understands the fact of hydration, because he need not worry about specifying the exact extent of hydration (which exceeds one H20 per proton), and because the specific use of the hydrated formula for the proton might obscure the important fact that all ions in water are extensively hydrated. Note also that the denominator in the K, expression in (17-3) is identical with that in (17-1); since applications of these equilibria are intended for dilute solutions, H 2 0 is always taken to be in its standard state and therefore need not be represented by a term in the K, expression. Equation (17-4) avoids describing aqueous ammonia as NH40H, a species that probably does not exist at ambient temperatures. The strengths of acids can be compared in terms of their K,-values; the stronger the acid, the larger its K,. The strong acids, which are completely ionized in water, cannot be described by the K, formalism applicable to dilute aqueous solutions, but other solvents not as basic as water (i.e., not as effective in removing protons) might allow only partial and reversible ionization and thus permit distinctions among the strong acids to be made. For example, H N 0 3 becomes a weak acid with a measurable K, in ethanol solvent, while HC104 and HC1 remain strong. In acetone solvent, HC1 becomes measurably weak and HC104 remains strong. Lewis concept The Lewis categorization of acids and bases uses an even more general picture. In this view an acid is a structure which has an affinity for electron pairs contributed by bases, defined as substances with unshared pairs of electrons. Examples of acids include not only the proton, which can react with the unshared pairs of such bases as H20, OH-, and C2H3O2, but also transition-metal ions, which can react with ligands (bases) to form complexes, and electron-deficient substances such as BF3, which can react with a base like NH3 to form the compound I I H-N-B-F I I H H F F IONIZATION OF WATER Since water is amphiprotic, that is, it can act both as an acid and as a base, every aqueous solution is characterized by the auto-ionization process, in which one H20 molecule transfers a proton to another H 2 0 molecule. The water auto-ionization equilibrium must always be satisfied, whether other acids or bases are present in the solution or not. 2 H 2 0 + H30f + OH- K, = [Hf][OH-] (17-5) The unhydrated form of the proton is written in the K , expression, but the H30+ notation is of course acceptable. Although the number of H20 molecules in the balanced chemical equation depends on whether the proton or hydronium ion is written, [H20] does not appear in the K , expression since applications are usually limited to dilute solutions where H 2 0 remains in its standard state. In the problems of this book, total concentrations of nonelectrolytes do not exceed 1 M, and concentrations CHAP. 171 ACIDS A N D BASES 267 of electrolytes normally do not exceed 0.1 M. For solutions containing larger ionic concentrations, the same laws of equilibrium apply if proper correction is made for the electrical interactions between ions. Such corrections will not be made in this book. Instead, examples will be chosen for which the numerical solutions should be correct to within 10 percent even without corrections for these electrostatic interactions. In general, 10 percent is about the limit of accuracy expected for a calculated concentration in this and in the following chapter. At 25 "C, K, = [H'][OH-] = 1.OO x 10-14; this value should be memorized. In pure water containing no other acid or base, the concentrations of Hf and OH- must equal each other. Therefore, at 25"C, [H'] = [OH-] =Jl.OO x 10-14 = 1.00 x 10-7 (The Hence a neutral solution may be defined as one in which [H'] = OH-] =K. value of Kw depends on temperature. Thus at 0 "C in pure water, [H'] == [OH-] = 0.34 x 10-7.) At 25 "C, an acid solution is one in which the [H'] is greater than 10-7, or one in which the [OH-] is less than 10-7; a basic solution is one in which the [H'] is less than 10-7, or one in which the [OH-] is greater than 10-7. The acidity or alkalinity of a solution is often expressed by its p H . By definition, 1 pH=log -= -log [H'] [H+l or [H'] = 10-pH Thus, if the [H'] of a solution is expressed as a simple power of 10, the pH of the solution is equal to the negative of the exponent. The smaller the pH, the greater the acidity. Similarly, we define pOH= - log [OH-] The smaller the pOH, the greater the alkalinity. The two measures are related by pH + pOH = -log K , = 14.00 Table 17-1 summarizes the pH and pOH scales at 25 "C. Table 17-1 [OH-I PH (at 25 "C) (17-6) 1 0.1 0.001 0.000 01 0.000000 1 0.000000 001 0.000 000 000 01 0.000 ooo ooo ooo 1 = loo = 10--1 = 10-3 = 10--5 = 10-7 = 10-9 = 10-I' = 10--13 10-l~ 10-l~ 10-" 1o - ~ 10-~ 10-~ 10-~ ~ . ~ o o ~ ~ ~ o o=o10--14o o i ~o 10-' 1 0 1 3 5 7 9 11 13 14 14 13 11 9 7 5 Strongly acidic Weakly acidic Neutral Weakly basic I 3 1 0 Strongly basic An uncertainty of 10 percent in [H+],the upper limit for problems in this book, corresponds to an uncertainty of 0.04 pH unit. For convenience, pK,, defined as -log K,, is often used to express the strength of an acid. Thus an acid whose ionization constant is 10e4 has a pK, of 4. Similarly, pKb = -log Kb; and for any equilibrium constant, pK = -log K. 268 ACIDS A N D BASES [CHAP. 17 HYDROLYSIS A salt containing at least one ion which is conjugate to a weak acid or base undergoes a reaction with water of an acid-base nature. For example, sodium acetate, NaC2H302, has an ion which is the conjugate base to acetic acid, a weak acid. The acetate ion is a base and can accept protons from acids, or from the solvent, water: ( 17-7) The above reaction is called hydrolysis. As a result of this process, a solution of sodium acetate in water is basic because an excess of OH- is produced. Note that the reaction (17-5) is the sum of the reactions (17-3) and (27-7). The equilibrium constants of these three reactions must then be related as follows (see Problem 16.14): Kw=K,Kb or Kh=- K W K U ( 17-8) An equation of the type (17-8) applies to the hydrolysis of any species which is the conjugate base to an acid having an ionization constant K,. Some examples are CN-, HS-, SCN-, NOT. Since hydrolysis involves a reverse of acid dissociation, the tendency toward hydrolysis runs counter to the tendency of the conjugate acid toward ionization. The weaker the acid, the greater the difficulty of removing a proton from the acid, and the easier for its anion, or conjugate base, to attach a proton from water (that is, hydrolyze). This relationship appears mathematically as the inverse proportionality between K, of the acid and Kb of the conjugate base. Acetic acid is a moderately weak acid, and the acetate ion hydrolyzes to a slight extent. HCN is a very weak acid and the cyanide ion, CN-, hydrolyzes to a great extent. Chloride ion, on the other hand, does not undergo the hydrolysis reaction at all, because its conjugate acid, HC1, is a strong acid and cannot form to any appreciable extent in dilute aqueous solutions. An analogous reaction will be shown by a positive ion which can act as an acid in water solution to form its weak conjugate base. Thus, a solution of ammonium chloride would be acidic because of the reaction NH,+ + H 2 0 + NH3 + H 3 0 + K, for NH: can be obtained from Kb for NH3, its conjugate base, by means of Equation (27-8) Many heavy-metal cations hydrolyze to some extent in aqueous solution in a reaction characterized by an equilibrium constant, K,. For example, Fe3+ + H 2 0 + Fe(OH)2+ + H+ K, = [Fe(OH)2+] [H+] [ Fe3+] ( 17-9) The above reaction is sometimes written with the hydrated forms of ions to show that ferric ion, like the neutral acids, demonstrates its acidity by loss of a proton. Fe(H20)i' + Fe(H20),(OH)2++ Hf The two formulations of the equation are equivalent. BUFFER SOLUTIONS AND INDICATORS If the [H'] (or pH) of a solution is not appreciably affected by the addition of small amounts of acids and bases, the solution is said to be buffered. A solution will have this property if it contains relatively large amounts of both a weak acid and a weak base. If a small amount of a strong acid is added to this CHAP. 171 ACIDS AND BASES 269 solution, most of the added H+ will combine with an equivalent amount of the weak base of the buffer to form the conjugate acid of that weak base; thus the [H'] of the solution remains almost constant. If a small amount of a strong base is added to the buffer solution, most of the OH- will combine with an equivalent amount of the weak acid of the buffer to form the conjugate base of that weak acid. In this way, the [H+] of the buffer solution is not appreciably affected by the addition of small amounts of acid or base. Any pair of weak acid and base can be used to form a buffer solution, so long as each can form its conjugate base or acid in water solution. EXAMPLE1 A particularly simple and common case of the buffer solution is one in which the weak acid and weak base are conjugates of each other. Thus acetic acid may be chosen as the weak acid and acetate ion as the weak base. Since relatively large amounts of each are needed, it would not be satisfactory to use just a solution of acetic acid in water, in which the concentration of the acetate ion is relatively small. An acetic acid-acetate buffer can be made in any of the following three ways: 1. Dissolve relatively large amounts of acetic acid and an acetate salt (sodium or potassium acetate, for example) in water. 2. Dissolve a relatively large amount of acetic acid in water. Partially neutralize the acid by adding some strong base, like sodium hydroxide. The amount of acetate formed will be equivalent to the amount of strong base added. The amount of acetic acid left in solution will be the starting amount minus the amount converted to acetate. 3. Dissolve a relatively large amount of an acetate salt in water. Partially neutralize the acetate by adding some strong acid, like HCl. The amount of acetic acid formed will be equivalent to the amount of strong acid added. The amount of acetate ion left in solution will be the starting amount minus the amount converted to acetic acid. The ratio of acetic acid to acetate ion in solution can be chosen so a s to give a desired [H+] or pH for the buffer solution. Usually the ratio is kept within the limits 10 and 0.1. According to a rearranged form of the ionization equilibrium equation for acetic acid: More generally, [H'] = K, x [acid] [conjugate base] ( I 7-10) Taking logarithms of (17-10) and reversing the signs yields the convenient form: [base] pH = pK, +log- [acid] An indicator is used to give a visual indication of the pH of a solution. Like a buffer it is a conjugate acid-base pair, but contrary to a buffer it is present in such a small concentration that it does not affect the overall pH. Rather the pH of the solution determines the ratio of acid form to conjugate base form of the indicator placed in that solution. Since at least one form, and usually both, are intensely colored (but different in color), the pH determines the color of the solution. If the acid dissociation of the indicator is written, HIn then [HIri] -[In- I If the ratio exceeds about 10 the color will appear to be purely that of HIn, the acid form, and if it is less than about 0.1 it will be purely that of the base form In-. Hence there will be a gradual change in color over about a 100-fold range in [H'] or about a range of 2 pH units, with pK, near the center of the range. [H+]= &(indicator) + H+ + In- (17-11) 270 ACIDS AND BASES [CHAP. 17 (The greater the difference in color intensity between the acid and base forms, the further off-center will be pK,.) A variety of indicators is known, each with its own pK, and range of color change. WEAK POLYPROTIC (POLYBASIC) ACIDS If multiple ionizations are possible, as in H2S and H2C03 each stage of ionization has its own equilibrium constant. Subscripts are usually used to distinguish the various constants. Primary ionization : Secondary ionization: H2S + H+ + HSHS- K1 = [H+I[HS-I P2SI K 2 = (17-12) [H+lP2-1 (17-13) [HS-I The secondary ionization constant of polyprotic acids is always smaller than the primary (K2 < K1);the tertiary, if there is one, is smaller than the secondary; and so on. It should be clear that [H+] means the actual concentration of hydrogen ions in solution. In an aqueous mixture containing several acids, the different acids all contribute to the hydrogen ion concentration in solution, but there is only one value of [H'] in any given solution and this value must simultaneously satisfy the equilibrium conditions for all the different acids. Although it might seem very complex to solve a problem in which several equilibria are involved, simplifications can be made when all sources but one make only a negligible contribution (less than 10 percent, for the purposes of this book) to the total concentration of a particular ion. In the case of polyprotic acids, K 1 is often so much greater than K2 that only the K1 equilibrium need be considered to compute [Ht] in a solution of the acid. Examples where this assumption may and may not be made will be given in specific problems. Another problem of interest is the calculation of the concentration of the divalent ion in a solution of a weak polyprotic acid when the total [H+] is essentially that due to a stronger acid present in the solution or to a buffer. In such a case the concentration of the divalent ion can best be calculated by multiplying the expressions for K 1 and K 2 . Again illustrating with H2S, we find (17-14) + H+ + S2- TITRATION When a base is added in small increments to a solution of acid, the pH of the solution rises on each addition of base. When the pH is plotted against the amount of base added, the steepest rise occurs at the equivalence point, when the acid is exactly neutralized. This region of steepest rise is called the endpoint, and the whole process of base addition and determination of the end point is called titration. The graph showing the change of pH during the titration is called a titration curve. EXAMPLE 2 Figure 17-la shows the titration at 25C of 50.0cm3 of an acid, either strong or weak, with strong base, and Fig. 17-lb the titration of 50.0cm3 of a strong base and also a weak base with strong acid. All reagents are 0.100 M so that the end point always occurs on the addition of exactly 50.0 cm3 of titrant. The strong acid is HC1, the weak acid HC4H703 (P-hydroxybutyric acid), the strong base NaOH, and the weak base NH3. All curves display a very sharp rise or fall in pH at the end point. When HC1 is titrated (Fig 17-la) the pH rises slowly until very near the end point. The rise at the end point is much greater than for the HC4H703which starts at a much higher pH, has an initial rapid rise, reverses its curvature, and eventually undergoes the very sharp rise typical of the end point. Beginning slightly beyond the end point the two curves are identical. The titration curves for NaOH and NH3 (Fig 17-lb) are almost mirror images of those for HCl and HC4H703 with decreasing pH. There is no practical reason for using a weak reagent as a titrant. If, for instance, NH3 were used in place CHAP. 171 ACIDS AND BASES 27 1 12.0 11.0 10.0 12.0 i . c - 9.0 4.0 8.0 7.0 PH 7.0 6.0 6.0 5.0 4.0 2.0 3.0 1.0 10 L l ' l 1 1 1 30 40 50 60 7 0 80 Volume of 0.100 M NaOH added, in an3 20 I 1 1 1 1 1 1 1 1 1 I 90 10 20 30 40 50 60 70 80 Volume of 0.100 M HCl added, in cm3 90 Fig. 17-la Fig. 17-lb of NaOH in Fig. 17-la the curve would level off beyond the end point at least 3 p H units lower, thus making it more difficult to discern the end point. The points along a titration curve can be calculated by methods previously discussed in this chapter. Four parts of the curve can be distinguished. Consider the titration of acid with strong base. 1. Starting point: 0% neutralization. In the case of the strong acid, [H'] in the initial solution is simply the molar concentration of the acid. In the case of the weak acid, [H'] is calculated by the method used to compute the extent of ionization of any weak acid in terms of its ionization constant and molar concentration. 2. Approach to the end point: 5% to 95% neutralization. For the case of the strong acid, the neutralization reaction, H+ + OH- + H20 may be assumed to go to completion to the extent of the amount of base added. The amount of unreacted H + is then the difference between the initial amount of H + and the amount neutralized. To determine [H+] allowance is made for the dilution effect of increasing the total volume of the solution on addition of base. For the case of the weak acid, the neutralization reaction may be written HC4H703 + OH-+ C4H7C): + H20 The amount of the hydroxybutyrate ion, C4H703, is equal to the amount of base added. The amount of un-ionized acid, HC4H703,is the difference between the initial amount and the amount neutralized. Then, [HC41370 31 [H+I = & ( H c ~ H , o ~ ) [C4H7031 The above is Equation (27-20); all solutions in this region are buffers. 272 ACIDS AND BASES [CHAP. 17 3. The end point: 100% neutralization. The pH at the end point is the same as for a solution of the salt containing the ions remaining at neutralization, NaCl or NaC4H703.NaCl solutions are neutral and have a pH of 7.00. The other salt hydrolyzes, and the pH can be evaluated by solving the hydrolysis equilibria. The HC4H703-NaOH titration has a pH greater than 7 at the end point because of the hydrolysis of the hydroxybutyrate ion. 4. Extension beyond the end point: Over 105% neutralization. With NaOH titrations, the excess OH- beyond that needed for neutralization accumulates in the solution. The [OH-] is computed in terms of this excess and the total volume of solution [H+] can then be computed from the K , relationship. It makes no difference whether the acid titrated was weak or strong. Points within 5% of the starting point or of the end point can be computed by the same equilibria, but some of the simplifying assumptions made above are no longer valid. Polyprotic acids like H3P04may have two or more distinct end points, corresponding to neutralization of the first, second, and subsequent hydrogens. In such a case each end point would occur at a different pH. The calculations for the titration curves of bases with strong acid (Fig. 17-lb) are done by similar methods. The end point of a titration, the region of steepest rise in the titration curve, can be determined experimentally if an instrument is available for measuring the pH after each addition of base. A simpler means is to introduce into the soluiton a small amount of an indicator chosen so that its range lies within the steep vertical portion of the curve, ensuring a sharp color change at the end point. Solved Problems ACIDS AND BASES 17.1. Write the formulas for the conjugate bases of the following acids: (a) HCN, (b) HCOT, (c) N2@, ( d ) C2H5OH, (e) HN03. In each case the conjugate base is derived from the acid by the loss of a proton. In (d)the proton is lost from the highly electronegative oxygen rather than from carbon. (a) CN-, ( d ) C2H50-, (e) NOT. (b) CO:-, ( c ) N2H4, 1 . . Write the formulas for the conjugate acids of the following bases: 72 (4CSHSN, (4 N2HL (4 OH-. (a) HC2H302, (b) HCOT, In each case the conjugate acid is formed from the base by the addition of a proton. The proton is added to oxygen in (a) and (b)or to nitrogen in ( c ) and (d), all of which have unshared pairs available. In (a) it is added to the carbonyl (-C=O) oxygen. (a)H2C2H30t. This is a species which might form in liquid acetic acid upon addition of a strong acid. (b)H2CO3. Note that HC03, can serve as both an acid [Problem 17.l(b)] and a base. (c) C ~ H ~ N H + . (d)N2H;+. Note that bases, like acids, can be polyfunctional. The second proton is accepted by N2H4, however, only with great difficulty. (e) H20. CHAP. 171 ACIDS AND BASES 273 17.3. Liquid NH3, like water, is an amphiprotic solvent. Write the equation for the auto-ionization of NH3. 2NH3 + NH: + NHY 17.4. Aniline, C6H5NH2, is a weak organic base in aqueous solutions. Suggest a solvent in which aniline would become a strong base. Obviously, a solvent is needed which has appreciably stronger acid properties than water. One such solvent, in which aniline is a strong base, is liquid acetic acid; many other acidic solvents would have the same effect. 17.5. NH4C104 and HC104.H20both crystallize in the same orthorhombic structure, having unit-cell volumes of 0.395 and 0.370nm3, respectively. How do you account for the similarity in crystal structure and crystal dimension? Both substances are ionic substances with lattice sites occupied by cations and anions. In "perchloric acid monohydrate," the cation is H30+ and there are no H20 molecules of hydration. The cations in the two crystals, H30+and NH;, should occupy nearly equal amounts of space because they are isoelectronic (have the same number of electrons). The data in this problem have been cited as one of the proofs for the existence of the hydronium ion, H~o+. 17.6. (a) Give an explanation for the decreasing acid strength in the series HC104, HC103, HC102. (b) What would be the relative basic strengths of ClO,, C103, ClO,? (c) Keeping in mind the discussion of (a), how do you account for the fact that there is very little difference in acid strength in the series H3P04, H3P03, H3PO2? (a) The Lewis structures are :ij: :O-L!I-O-H :o: : 0- &I -0- H :O-CI-OH I Since oxygen is more electronegative than chlorine, each terminal oxygen tends to withdraw electrons from the chlorine and in turn from the O - H bond, thus increasing the tendency for the proton to dissociate. In general, the greater the number of terminal oxygens in an oxy-acid, the stronger the acid. ( b ) Members of a conjugate acid-base pair stand in complementary relationship to each other. The stronger an acid, the weaker its conjugate base. Thus the order of decreasing basic strength is Cloy, ClO,, ClO,, the reverse of the order of decreasing strength of the conjugate acids. The hydrogens in these acids are not all bonded to oxygens. The Lewis structures are (c) :0: .. :i i: H-ij-L-ij-H H :o: H-P-0-H H H-O-Y-O-H I :0: I I I H The number of terminal oxygen atoms, one, is exactly the same in all three acids, so that, according to (a), no great differences in acidity are expected. Since the electronegativities of P and H are almost the same, there is no tendency for the phosphorus-bound H to ionize, nor to influence the ionization of the oxygenbound H's. I 274 ACIDS AND BASES [CHAP. 17 17.7. How do you account for the fomation of S20:- from SO:- and S in terms of Lewis acid theory? A sulfur atom is electron-deficient and can thus be regarded as an acid. SO:-, for which an octet structure can be written with an unshared electron pair, is the base. [ :p: l 2 .. :s-s-0: .. :: :0: 17.8. In combustion analysis of organic compounds, solid NaOH is used to absorb the CO2 from the combustion gases. Account for this reaction in terms of Lewis acid theory. The oxygen in the OH- ion has three unshared pairs and is thus a Lewis base. To understand how CO2 functions as an acid, note that the carbon in CO2 is sp hybridized, but by reverting to sp2 hybridization as in HCOT, an orbital is made available to accept the base pair. IONIZATION OF ACIDS AND BASES 17.9. At 25 "C a 0.010 0 M ammonia solution is 4.1 ?hionized. Calculate ( a ) the concentration of the OH- and NH; ions, (b) the concentration of molecular ammonia, ( c ) the ionization constant of aqueous ammonia. ( d ) [OH-] after 0.009Omol of NH4C1 is added to 1 L of the above solution, (e) [OH-] of a solution prepared by dissolving 0.010 mol of NH3 and 0.005 0 mol of HC1 per liter. NH3 + H 2 0 + NH: + OH- The label on a solution refers to the stoichiometric or weight composition and does not indicate the concentration of any particular component of an ionic equilibrium. Thus the designation 0.010 0 M NH3 means that the solution might have been made by dissolving 0.010 0 mol ammonia in enough water to make a liter of solution. It does not mean that the concentration of un-ionized ammonia in solution, [NH3], is 0.0100. Rather, the sum of the ionized and the un-ionized ammonia is equal to 0.0100mol per liter. (4 (b) [NH,f] = [OH-] = (0.041)(0.010 0) = 0.000 41 [NH3] = 0.010 0 - 0.000 41 = 0.009 6 ( d ) Since the base is so slightly ionized, we may assume that (1) the "Hi] is completely derived from the NH4C1 and that (2) the [NH3]at equilibrium is the same as the stoichiometric molar concentration of the base. Then gives CHAP. 171 ACIDS AND BASES 275 The addition of NH4C1 represses the ionization of NH3, thus reducing greatly the [OH-] of the solution; this phenomenon is called the common-ion efect. Assumptions made to simplify the solution of a problem should always be verified after the problem has been solved. In this case we assumed that practically all the NH: came from the added NH4Cl. In addition to this, there is a small amount of NH: resulting from the dissociation of the NH3. From the chemical equation given in the statement of the problem, the amount of NH; coming from the NH3 must equal the amount of OH-, which we now know is 1.94 x 10d5M. The correct total [NH:] is then the sum of the contributions of NH4C1 and NH3, 0.009 0 + (1.94 x 10-5). This sum is indeed equal to 0.0090, well within our 10 percent allowance. We have thus justified our assumption. The reader may wonder why we must make assumptions at all. This problem could have been solved by a more complex analysis as follows, without the assumption: NH3 + H20 + NH: + OH0 0 0.0100 A at start from NH3: 4 M at start from NH:: 0 O.QO90 0 Change by reaction: -X +X +X 0.0100-~ 0.0090+~ x M at equilibrium: Kb[NH3] [OH-] = x = - (1.75 x 10-5)(0.010 0 - x) -"Ha 0.009 0 x + This is a quadratic equation in x and can be solved by the usual methods. Clearing fractions, and ignoring for the moment the rules for significant figures: 0 = x2 + 0.009 017 5x + 1.'75 x 10-7 Using the quadratic formula: + (8.131 5 x 10-'5+ 7.00 x 10-7)*/2} x = (f)( 0.009 017 5 + 0.009 056 2) = 1.94 x 10-' x = (f){-0.009 017 5 Clearly, making simplifying assumptions greatly reduced the labor and the chance of numerical mistakes. Notice how the answer depends on a very small difference between two similar terms. Because of the desirability of applying chemical intuition to problem analysis, we will make simplifying approximation in this book in advance, wherever they seem to make chemical sense. After soIving the problem, we will always check the validity of our assumptions. (e) Since HC1 is a strong acid, the 0.005 0 mol of HCI will react completely with 0.005 0 mol of NH3 to form 0.005 0 mol NH:. Of the original 0.010 0 mol of NH3, only half will remain as un-ionized ammonia. [OH-] = Kb[NH3] - (1.75 x 10-')(0.005 0) = 1.75 10-5 ": HI 0.005 0 ~ Check of assumption: The amount of NH: contributed by dissociation of NH3 must be equal to the amount of OH-, or 1.75 x lOP5mol/L. This is indeed small compared with the 0.005 0 mol/L formed by neutralization of NH3 with HCl. 17.10. Calculate the molar concentration at which an acetic acid solution is 2.0 percent ionized. Ka for HC2H302is 1.75 x lO-' at 25 "C. ' HC2H302 + H + C2H302 Let x = molar concentration of acetic acid solution. Then [H'] = [C2H30J = 0 . 0 2 0 ~ the approximation being well within 10 percent. and [HC2H302] - 0 . 0 2 0 ~ x =x M Solving, (0.020)2x = 1.75 x lO-' or x = 0.044. 276 ACIDS AND BASES [CHAP. 17 17.11. Calculate the percent ionization of a 1.OO M solution of hydrocyanic acid, HCN. K , of HCN is 4.93 x 10-l0. HCN + H+ + CH- Since H + and CN- are present in the solution only as a result of the ionization, their concentrations must be equal. Assume that [H+] from the ionization of water is negligible. Let x = [H'] = [CN-1. Then [HCN] = 1.OO - x. Let us assume that x will be very small compared with 1.OO, so that [HCN] = 1.OO, within the allowed 10 percent error. Then whence x = 2.22 x 10-5. Percent ionization = ionized HCN 2.22 x 1 0 - ~mol/L x 100% = x 100% = 0.002 22% total HCN 1.00 mol/L Check of assumptions: (1) x( = 2.22 x 10-5) is indeed very small compared with 1.00; (2) the [H+] in neutral water is only 1 x 10-7, which is < 1% of x. Furthermore, the ionization of water is suppressed even more by the presence of the acid. 17.12. The [H+]in a 0.020 M solution of benzoic acid is 1.1 x lOP3.Compute K , for the acid from this information. Since the hydrogen ion and benzoate ion come only from the ionization of the acid, their concentrations must be equal. (We have neglected the contribution of the ionization of water to EH+].) [H'] = [C7H50J = 1.1 x lOV3 [HC7H502] = 0.020 - (1.1 x 10-3) = 0.019 17.13. The ionization constant of formic acid, HCOZH, is 1.77 x 10-4. What is the percent ionization of a 0.001 00 M solution of this acid? Let x = [H'] = [HCO?]; then HC02HI = 0.001 00 - x. Assume, as in Problem 17.11, that the percent ionization is less than 10 percent and that the formic acid concentration, 0.001 00-x, may be approximated as 0.001 00. Then K, = [H+l[HCOTI - x2 = 1.77 x 10-4 [HCOZH] 0.001 00 or x = 4.2 x 10-4 In checking the assumption, we see that x is not negligible compared with 0.00100. Therefore the assumption and the solution based on it must be rejected and the full quadratic form of the equation must be solved. = 1.77 x 1 0 - ~ 0.001 00 - x Solving, x = 3.4 x 10-4. (We reject the negative root, -5.2 x 10-4.) X2 Percent ionization = ionized HC02H 3.4 1 0 - ~ x 100% = x 100% = 34% total HC02H 0.001 00 This exact solution (34 percent ionization) shows that the solution based on the original assumption was almost 25% too high. CHAP. 171 ACIDS AND BASES 277 17.14. What concentration of acetic acid is needed to give a [H+] of 3.5 x 10-4? K, is 1.75 x 10-5. Let x = number of moles of acetic acid per liter. [H'] = [C2H302] = 3.5 x 10-4 [HC2H302] =x - 3.5 x 10-4 or x = 7.3 x 10-3 [H'][C2H302] - (3.5 x 10-4)2 = 1.75 x 10-5 x - 3.5 x 10-4 [HC2H302] 17.15. A 0.100 M solution of an acid (density = 1.010 g/cm3) is 4.5 percent ionized. Compute the freezing point of the solution. The molar mass of the acid is 300. We must first determine the molality of the solution, i.e., the number of moles of acid dissolved in 1 kg of water. Mass of 1 L of solution = (1,000 cm3)(1. O g/cm3) = ml O = 1 010 g Mass of solute in 1 L of solution (0.100 mo1)(300 g/mol) = 30 g = Mass of water in 1 L of solution Molality of solution = 1 010 g - 30 g = 980 g 0.100 mol acid - = 0.102 mol/kg 0.980 kg water If the acid were not ionized at all, the freezing-point lowering would be 1.86 x 0.102 = 0.190"C (see Chapter 14). Because of ionization, the total number of dissolved particles is greater than 0.102 mol per kg of solvent. The freezing-point depression is determined by the total number of dissolved particles, regardless of whether they are charged or uncharged. Let a =fraction ionized. For every mole of acid added to the solution, there will be (1 - a ) moles of unionized acid at equilibrium, a moles of H + , and a moles of anion base conjugate to the acid, or a total of (1 + a ) moles of dissolved particles. Hence the molality with respect to all dissolved particles is (1 + a ) times the molality computed without regard to ionization. Freezing-point depression = (1 + a)(1.86)(0.102) = (1.045)( 1.86)(0.102) = 0.198 "C The freezing point of the solution is -0.198 "C. 17.16. A solution was made up to be 0.010 0 M in chloroacetic acid, HC2H202C1, and also 0.002 0 M in sodium chloroacetate, NaC2H2O2C1. for chloroacetic acid is 1.40 x 10-3. What is [H+] in the K, solution? We could proceed as in part ( d )of Problem 17.9 and assume that the concentration of chloroacetate ion can be approximated by the molar concentration of the sodium salt, 0.0020, and that the extent of dissociation of the acid is small. Letting [H'] = x, [C2H202Cl-] = 0 . 0 0 2 0 + ~ 0.0020 M Then Check: We see that the assumption led to a self-inconsistency. The value of x obtained, 7.0 x 10-3, cannot be neglected in comparison with either 0.0020 or 0.0100. We must start again, without making the simplifying assumptions. x = [H'] = (1.40 x 10-3) [HC;!H202Cl]= 0.010 0 - x M 0.010 0 (::E:) :T The solution of the resulting quadratic equation gives the proper value for [H'], 2.4 x lOP3. 278 ACIDS AND BASES [CHAP. 17 Chloroacetic acid is apparently strong enough that the common-ion effect does not repress its ionization to a residual value small enough to be neglected. A hint of this result is the relatively large value of K,. 17.17. Calculate [H+] and [C2H302]in a solution that is 0.100 M in HC2H302 and 0.050 M in HCl. K , for HC2H302is 1.75 x 10-5. The HCl contributes so much more H + than the HC2H302 that we can take [H+]as equal to the molar concentration of the HCI, 0.050; this is another example of the common-ion effect. Then, if [C2H302] x, we have [HC2H302] = 0.100 - x M 0.100 and = Check of assumptions: (1) Contribution of acetic acid to [H +I, x, is indeed small compared with 0.050; (2) x is indeed small compared with 0.100. 17.18. Calculate [H'], [C2H30J, and [CN-] in a solution that is 0.100M in HC2H302 ( K , = 1.75 x lO-') and 0.200M in HCN (K, = 4.93 x 10-l'). This problem is similar to the preceding one in that one of the acids, acetic, completely dominates the other in terms of contribution to the total [H+] of the solution. We base this assumption on the fact that K, for HC2H302is much greater than for HCN; we will check the assumption after solving the problem. We will proceed by treating the acetic acid as if the HCN were not present. Let [H+] = [C2H302]=x; then [HC*H302] = 0.100 - x M 0.100 and [H+I[C2H3021- x2 = 1.75 x IOp5 [HC2H302] 0.100 or x = 1.32 x 10-3 Check of assumption: x is indeed small compared with 0.100. Now we treat the HCN equilibrium established at a value of [H+] determined by the acetic acid, 1.32 x 10-3. Let [CN-] = y ; then [HCN] = 0.200 - y and M 0.200 Check ofassumption: ( I ) y is indeed small compared with 0.200; (2) the amount of H + contributed by HCN ionization, equal to the amount of CN- formed (7.5 x 10-* mol/L), is indeed small compared with the amount of H + contributed by HC2H302 (1.32 x lOP3 mol/L). 17.19. Calculate [H+] in a solution that is 0.100M HCOOH (K, = 1.77 x lOP4) and 0.100M HOCN (K, = 3.3 x 10-~). This is a case in which two weak acids both contribute to [H+], neither contributing such a preponderant amount that the other's share can be neglected. A (initial): 4 Change by reaction : A (equilibrium): 4 Approx. M (equilibrium) : HCOOH 0.100 -X + H+ + HCOT 0 +x x+y x+y 0 +x O.lO0-x 0.100 x x HOCN F= H+ + OCN0.100 0 0 -Y +Y +v 0.100-y x + y y 0.100 x+y y The final line in the above tabulation is based on the assumption that x and y are both small compared with 0.100. Dividing the HOCN equation by the HCOOH equation, CHAP. 171 ACIDS AND BASES 279 3.3 v-- 1.77 = 1.86 x or y = 1.86~ Subtracting the HCOOH equation from the HOCN equation, Y ( x + Y ) - ? & ~ ~ = 1.5 0.100 10-4 or y 2 - x * = 1.5 x 10-5 Substitute y = 1 . 8 6 ~ the last equation and solve to obtain x = 2.5 x 10-3. Then into y = 1 . 8 6 = 4.6 x 10-3 ~ and [H'] = x + y = 7.1 x lOP3 Check qf assumptions: The values of x and y are less than 10 percent of 0.100. (They are not much less than 10 percent. In this case we are just about at the limit of error we are allowing in this chapter.) IONIZATION OF WATER 17.20. Calculate the [H+] and the [OH-] in 0.100M HC2H302which is 1.31 percent ionized. [H'] = (0.013 1)(0.100) = 1.31 x 10-3 [OH-] = LOO 1 0 - l ~__ 1.00 x 1 0 - l ~ - = 7.6 x 10-I2 [H+l 1.31 x 10-' Note that [H+] is computed as if the HC2H302 were the only contributor, whereas [OH-] is based on the ionization of water. Obviously, if water ionizes to supply OH-, it must supply an equal amount of H + at the same time. Implied in this solution is the assumption that water's contribution to [H+], 7.6 x 10-'* mol/L, is negligible compared with that of the HC2H302. This assumption will be valid in all but the most dilute, or weakest, acid solutions. In computing [OH-], however, water is the only source and it therefore cannot be overlooked. 17.21. Determine the [OH-] and the [H '1 in 0.010 0 M ammonia solution which is 4.1 percent ionized [OH-] = (0.041)(0.010 0) = 4.1 x I V 4 [H'] = 1.00 1 0 - l ~ - 1.00 x 10-~- = 2.4 14 4.1 x 10[OH-I lo-ll Here we have made the assumption that the contribution of water to [OH-] (equal to [H+], or 2.4 x 10-l') is negligible compared with that of NH3. K , is used to compute [H+], since water is the only supplier of H . In general, [H '1 for acidic solutions can usually be computed without regard to the water equilibrium; then K,,, is used to compute [OH-]. Conversely, [OH-] for basic solutions can usually be computed without regard to the water equilibrium; then K , is used to compute [Hi]. + 17.22. Express the following H + concentrations in terms of pH: (a) 1 x 10-3 mol/L, mol/L. (a) (b) 5.4 x 10-9 pH = -log [H+]= -log lOP3= 3 pH = (b) -log [H'] = -log (5.4 x IOp9) = -log 5.4 + 9 = -0.73 + 9 = 8.27 Part (b) can be done on an electronic calculator by merely entering 5.4 x 10-9, pressing LOG, and changing the sign of the result. The presentation here (and in subsequent problems) would be necessary if a table of logarithms had to be used. Students (almost all of whom have calculators) should study these examples anyway to gain a better understanding of how logarithms work. 280 ACIDS AND BASES [CHAP. 17 17.23. Calculate the pH-values, assuming complete ionization, of (a) 4.9 x 10-4 N acid, (b) 0.001 6 N base (a) Here [H+] = 4.9 x lOP4 (see Chapter 12 for the definition of normality). pH = -log [H'] = -log (4.9 x 10-4) = -log 4.9 + 4 = -0.69 + 4 = 3.31 pH = -log [H'] = -log 10-l~ 1.6 x 10-3 =-(-14-log 1.6+3)=14+0.20-3~11.20 17.24. Change the following pH-values to [H+]-values: (a) 4, (b) 3.6. (b) [H+]= 10-pH = 10-3.6 = 100.4-4 From a table of logarithms = antilog 0.4 = 2.5 10-~ and so [H+]=2.5 x 10-4. Note that a negative number like -3.6 cannot be found in tables of logarithms, nor can -0.6. Only positive mantissas appear in the printed logarithm tables. The positive mantissa was achieved by adding and subtracting the next higher integer to the negative number: -3.6 = 4.0 - 3.6 - 4.0 = 0.4 - 4.0. Then 0.4 can be found to be the logarithm of 2.5. Using a calculator, simply enter 3.6, press + I-" to change to -3.6, then INV-LOG to give the answer. (Warning: not all calculators are alike. If the instructions given in this text do not work, consult your calculator instruction manual.) " 17.25. What is the pH of (a) 5.0 x 10-* M HC1, (b) 5.0 x lO-'OM HCl? (a) If we were to consider only the contribution of the HC1 to the acidity of the solution, [H+] would be 5.0 x 10T8and the pH would be greater than 7. This obviously cannot be true, because a solution of a pure acid, no matter how dilute, cannot be less acid than pure water alone. It is necessary in this problem to take into account something that we have omitted in all previous problems dealing with acid solutions, the contribution of water to the total acidity. A complete analysis of the water equilibrium is required. H20 M from HCl : Change by ionization of H 2 0 : M at equilibrium: * H+ + 5.0 x 108 X OHX 5.0 x 10-8 + x x K , = [H+][OH-]= (5.0 x 10-8 + x ) x = 1.00 x lO-I4 Solution of the quadratic equation gives the result: x = 0.78 x 10-7. Then [H+]= 5.0 x lops + x = 1.28 x lO-' and pH = -log (1.28 x 10-7) = 6.89 (b) Although the method of (a)could be used here, the problem can be simplified by noting that the HC1 is so dilute as to make only a negligible contribution to [ H f ] as compared with the ionization of water. We may therefore write directly: [H'] = 1.OO x lOP7 and pH = 7.00. CHAP. 171 ACIDS AND BASES 28 1 HYDROLYSIS 17.26. Calculate the extent of hydrolysis in a 0.0100 M solution of NH4Cl. Kb for NH3 is 1.75 x lO-'. NH,f T + NH3 + Ht K, = "H31H'I [NH:] [NH:] - K, = 1.00 Kh 10-. = 5.7 1.75 x 10-5 0.010 0 10-10 By the reaction equation, equal amounts of NH3 and H + are formed. Let x = [NH3]= [H']. Then = 0.010 0 - x % and Solving, x =: 2.4 x 10-6. Check of the approximation: x is very small compared with 0.0100. Fraction hydrolyzed = amount hydrolyzed - 2.4 x mol/L -total amount 0.010 0 mol/L = 2.4 x 10-4 = 0.024% 17.27. Calculate [OH-] in a 1.OO M solution of NaOCN. Ku for HOCN is 3.5 x 10-4. OCN- + H20 F+ HOCN + OH- Kb = [OH-][HOCN]- K , - 1.00 x I O - l 4 [OCN-] K, 3.5 x 10-4 x = [OH-] = [HOCN]. Then - 2.9 x 10-l' Since the source of OH- and HOCN is the hydrolysis reaction, they must exist in equal concentrations; let and whence x =: [OH-] = 5.4 x 10-6 Check of approximation: x is very small compared with 1.00. 17.28. The acid ionization (hydrolysis) constant of Zn2+ is 3.3 x 10-l'. (a) Calculate the pH of a 0.001 OM solution of ZnC12. (b) What is the basic dissociation constant of Zn(OH)+? Let x=[Zn(OH)+] = [H']. Then [Zn2'] ___ = 0.001 0 - x NN 0.001 0 and XL 0.001 0 = = 3.3 x 10-I' or x = [H'] = 5.7 x 10-' pH -log (5.7 x 10-7) = -log 10-6.24= +6.24 Check qfupproximation: x is very small compared with 0.001 0. Zn2+, as an acid, is conjugate to the base Zn(OH)+ . For the basic dissociation, (b) 17.29. Calculate the extent of hydrolysis and the pH of 0.0100M NH4C2H302. KO for HC2H302is 1.75 x 10--5and Kb for NH3 is 1.75 x 282 ACIDS AND BASES [CHAP. 17 This is a case where both cation and anion hydrolyze. For NH,f &=-- K, - K~(NH~) 1.00 x 1 0 - l ~ = 5.7 x 10-I' 1.75 x 10-5 By coincidence, the hydrolysis constants for these two ions are identical. The production of H + by NH; hydrolysis must therefore exactly equal the production of OH- by C2H302 hydrolysis; and the H + and OH- formed by hydrolysis neutralize each other to maintain the water equilibrium. The solution is thus neutral, [H'] = [OH-] = 1.00 x 10-7, and the pH is 7.00. For -NH; hydrolysis, Let x = [NH3].Then 0.0100-x = [NH:] and x(i.oo x 1 0 - ~ ) = 5.7 x 0.0100 - x 10-l' or x = 5.7 x 1 0 - ~ . 5.7 x 1 0 - ~ Percent NH: hydrolyzed = 0.010 0 x 1OOoh = 0.57% The percent hydrolysis of acetate ion must also be 0.57%, because the equilibrium constant for the hydrolysis is the same. In comparing this result with Problem 17.26, note that the percent hydrolysis of NH: is greater in the presence of a hydrolyzing anion. The reason is that the mutual removal of some of the products of the two hydrolyses, H + and OH-, by the water equilibrium reaction allows both hydrolyses to proceed to increasing extent. 17.30. Calculate the pH in a 0.lOOM solution of NH40CN. Kb for NH3 is 1.75 x 10-5 and Ka for HOCN is 3.5 x 10-4. As in Problem 17.29, both cation and anion hydrolyze. Since NH3 is a weaker base than HOCN is an acid, however, NH: hydrolyzes more than OCN-, and the pH of the solution is less than 7. In order to preserve electrical neutrality, there cannot be an appreciable,difference between [NH,f] and [OCN-] . (The balancing of a slight difference could be accounted for by [H+]or [OH-].) Thus [NH3] must be practically equal to [HOCN], and we will indeed assume that they are equal. Let x = [NH3] = [HOCN]; then 0.100 - x = [NH,f] = [OCN-1. ForNH;: and ForOCN-: and Dividing ( I ) by (2), [H+] - 5.7 x 10-I' - 19.7 [OH-] - 2.9 x 10-" Also, [H+] and [OH-] must satisfy the K , relationship: [H+][OH-]= K , = 1.00 x 10-14 Ka= 1.00 x lO-I4 [NH3][H+] = 5.7 x 10-'' [NH,f] Kb 1.75 x lOV5 5 [H'] = (5.7 x lO-'')( 0.100 - x X K K b = [HOCN][OH-] - -,-- 1.00 x 10-14 = 2.9 x 10-I' [OCN-] Ku 3.5 x 10-4 [OH-] = (2.9 x 10-" )(o.lo: - "> CHAP. 171 ACIDS AND BASES 283 Multiplying (3) by (4), we obtain [H']' = 19.7 x lO-I4 [H'] = 4.4 x 10-7 pH = -log [H'] = 6.36 Check qf assumption: Our assumption that [NH3]= [HOCN] is valid only if [H'] and [OH-] are much smaller than [NH3] and [HOCN]. We shall therefore solve for x,the [NH3] or [HOCN]. From ( I ) , X = (5.7 x lo-*o)(o.loo- x) - (5.7 x lo-lo)(o.loo - x) x 1.3 x 1 0 - ~ [H'I 4.4 x 10-7 (5) Both [H+] and [OH-] are small compared with x. It appears from the solution to this problem that the pH is independent of the concentration of NH40CN. Indeed this is true at sufficiently high concentrations. However x decreases with the initial concentration as shown in ( 5 ) so that at much lower concentrations the simplifying assumptions are no longer valid. (The problem can still be solved by explicitly invoking conservation of electrical charge but the solution becomes very complicated.) POLYPROTIC ACIDS 17.31. Calculate the H + concentration of 0.10M H2S solution. K 1 and K2 for H2S are respectively 1.0 x 10-7 and 1.2 x lO-I3 Most of the H + results from the primary ionization: H2S + H' Let x = [H'] = [HS-1. Then [H2S]= 0.10 - x x 0.10. + HSx = 1.0 x 10-4 K1 = [H+l[HS-l F2SI or 1.0 x 10-7 = 0.10 X2 or Check of assumptions: (1) x is indeed small compared with 0.10. (2) For the above value of [H+] and [HS-1, the extent of the second dissociation is given by p - ] = --K2[HS-] - (1.2 x [H+l lO-I3)(l.O x 10-4) -- - 1.2 (1.0 x 10-4) 10-l~ The extent of the second dissociation is so small that it does n,ot appreciably lower [HS-] or raise [H+] as calculated from the first dissociation only. This result, that in a solution of a polyprotic acid the concentration of the conjugate base resulting from the second dissocia.tion is numerically equal to K2, is general whenever the extent of the second dissociation is less than 5 percent. 17.32. Calculate the concentration of C8H40i- (a) in a 0.0lOM solution of H2C8H404, (b) in a solution which is 0.010 M with respect to H&H4o4 i2nd 0.020M with respect to HCl. The ionization constants for H2CSH404, phthalic acid, are H2C8H4O4 Hf + + HC8H404 HC8H40i+ H+ + C8H40i- K1 = 1.3 x 10-3 K2 = 3.9 x 1OW6 (a ) If there were no second dissociation, the [H+]could be computed on the basis of the K , equation. x = [H'] == [HC8H40i] 3.0 x 10-3 = = 1.3 x 10-3 or 0.010 - x Note that it was necessary to solve the quadratic to obtain x. If we assume that the second dissociation does not appreciably affect [H+] or [HC8H40,], then X2 [CgH4O;-] = K2[HCSH40;]- (3.9 x 10-6)(3.0 x lOV3) = 3.9 x 10-6 [H'I 3.0 x 10-3 Check ofassumption: The extent of the second dissociation relative to the first, 3.9 x 10-6 = 1.3 x lOP3=: 0.13% 3.0 x 10-3 is sufficiently small to validate the assumption. 284 ACIDS AND BASES [CHAP. 17 The [H+] in solution may be assumed to be essentially that contributed by the HC1. Also, this large common-ion concentration represses the ionization of the phthalic acid, so that we assume that [H2C8H4O4]= 0.010. The most convenient equation to use is the K1K equation, since all the con2 centrations for this equation are known but one. `H'12[c8H40'-1 [H2C8H404] = K1K2= (1.3 x 10-3)(3.9 x = 5.1 x lOP9 Check o assumptions: Solving for the first dissociation, f The amount of H t contributed by this dissociation, 6.5 x top4mol/L, is indeed less than 10 percent of the amount contributed by HCI (0.020 mol/L). The amount of H contributed by the second dissociation is still less. + 17.33. Calculate the extent of hydrolysis of 0.005M K2Cr04. The ionization constants of H2Cr04are K1 = 0.18, K2 = 3.2 x 10-7. Just as in the ionization of polyprotic acids, so in the hydrolysis of their salts, the reaction proceeds in successive stages. The extent of the second state is generally very small compared with the first. This is particularly true in this case, where H2Cr04 is quite a strong acid with respect to its first ionization. The equation of interest is CrOi- + H 2 0 + HCr04 + OH- which indicates that the conjugate acid of the hydrolyzing Cr0;- is HCr04. As the ionization constant for HCr04 is K2, the basic hydrolysis constant for the reaction is K,/K2. [OH-] [HCrO,] = K - K w lo-l4 = 3.1 -% - 3.2 x 10-7 [CrOj-] Let x = [OH-] = [HCrOT].Then [CrOi-] = 0.005 - x M 0.005 and 10-8 I 2 10-~ Fraction hydrolysis = A = 2.4 x 10-3 = 0.24% 0.005 Check o assumption: x is indeed small compared with 0.005. f 17.34. What is the pH of a 0.005OM solution of Na2S? The ionization constants for H2S are K1 = 1.0 x 10-7 and K2 = 1.2 x As in Problem 17.33, the first stage of hydrolysis, leading to HS-, is predominant. S2Kb = + H 2 0 + HS- + OH- [HS-][OH-] K, - __ [S2- 1 K 2 1.00 x 1.2 x = 8.3 x 10-2 Because of the large value for Kb, it canot be assumed that the equilibrium concentration of S2- is approximately 0.005 0 mol/L, the stoichiometric concentration. In fact, the hydrolysis is so extensive that most of the S2- is converted to HS-. CHAP. 171 ACIDS AND BASES 285 Let x = [S2-], then [HS-] = [OH-] = 0.005 0 - x (0.005 0 - x)* X = 8.3 x 10-2 whence x = 2.7 x 10-4 [OH-] = 0.0050 - 2.7 x 10-4 = 0.004 7 pOH = -log (4.7 x 10-3) = 2.33 pH == 14.00 - 2.33 = 11.67 Check qf assumption: Consider the second stage of hydrolysis. HS- + H20 * H2S + OH- LOO 1 0 - l ~ K = -= b K, = 1.0 x 1 0 - ~ K, 1.0 x 10-7 Solve for [H2S]by assuming the values of [OH-] and [HS-] already obtained. w3) Kb[HS-] - (1.0 X 10-7)(4.7 X -= 1.0 [H2S] = [OH-1 4.7 x 10-3 10-~ The extent of the second hydrolysis compared with the first, (1.6 x 10-7)/(4.7 x lOP3),is indeed small. 17.35. Calculate [H+], [H2POT], [HPO:-], and [PO:-] in 0.0100M H3P04. K1, K2, and K3 are 7.52 x 10-3, 6.23 x 10-*, and 4.5 x lO-I3, respectively. We begin by assuming that H + comes principally from the first stage of dissociation, and that the concentration of any anion formed by one stage of an ionization is not appreciably lowered by the succeeding stage of ionization. H3P04+ H+ + H2POQ KI == 7.52 x lOP3 Let [H'] = [H2POQ] x. Then [H3P04] 0.0100 - x, and = = XL 0.0100 - x = 7.52 x lOP3 or x = 0.005 7 We next take the above value for [ H + ] and [H2POQ]to solve for [HPOi-1. HzPOQ + H+ + HPOi[HPOZ-] = K 2 := 6.23 x 10-' 10-8 K2[H2P0;] - (6.23 x 10-')(0.0057) - 6.23 ---[H+l 0.005 7 6.23 x 10-8 5.7 x 10-3 Check of assumption: The extent of the second dissociation compared with the first, is indeed small. Next we take the above values for [H+] and [HPOZ-] to solve for [PO:-]. H P O ~ -+ H+ +PO:[PO:-] = K3[HPOi-] [H+l K~ == 4.5 x 10-l~ (4.5 x 10-13)(6.23 x 10-8) - 4.9 x = 5.7 10-3 mol/L, is an insignificant Check of assumption: The depletion of HP04 by the third stage, 4.9 x fraction of the amount present as a result of the second step, 16.23 x 10-8mol/L. 17.36. What is the pH of 0.0100M NaHC03? K 1 and K2 for H2C03 are 4.3 x 10-7 and 5.61 x lO-", respectively. (H2C03in aqueous solution is in equilibrium with dissolved C02, the majority species. The value of K 1 given here is based on the total concentration of both neutral species. Since there is no effect on the 286 ACIDS AND BASES [CHAP. 17 stoichiometry or charge balance the problem can be worked as if all the neutral species were in the form H2C03.) This has similarities to Problem 17.30 in that there is one reaction tending to make the solution acidic (the K2 acid dissociation of HCO,) and another reaction tending to make the solution basic (the hydrolysis of HCOT). * H+ + CO:HCOT + H2O * OH- + H2C03 HCOT K = 5.61 x 10-I' 2 Kb = - = (1) = 2.3 K, K~ 1.00 x 1O-I4 4.3 x 10-7 10-8 [Note that the hydrolysis constant for reaction (2) is related to K1, because both hydrolysis and the K 1 equilibrium involve H2C03and HCO, .] We see that the equilibrium constant for (2) is greater than that for ( I ) ; thus the pH is certain to exceed 7. We assume that after self-neutralization both [H+] and [OH-] were so small as to have no appreciable effect on the ionic charge balance. Therefore electrical neutrality can be preserved only by maintaining a fixed total anionic charge among the various carbonate species, since the cationic charge remains at 0.010 0 M, the concentration of Na', regardless of the acid and base equilibria. In other words, for every negative charge removed by converting HCOT to H2C03, another negative charge must be created by converting HCOT to CO:-. This leads to the following conditions: [H*C03]= [CO:-] =x [HCOT] = 0.0100 - 2x M 0.0100 Multiplying ( 3 ) and (4) and recalling that [H'] [OH-] = 1.OO x (l'oo 10-'4)?c2= (5.61 x 10-")(2.3 x IO-') (0.010 o)2 IO-l4, we obtain x = 1.14 x lOP4 or Provisional check: 2x is indeed small compared with 0.0100. We return now to (3). [H'] = (5.61 x lO-")[HCO;] - (5.61 x 10-l')(O.OIOOO) = 4.9 [CO:1.14 10-4 pH = -log [H'] = -log 4.9 + 9 = -0.69 + 9 = 8.31 I 10-9 Final check: Both [H'] and [OH-] are small compared with x, and will have no appreciable effect on the electrical charge balance. Alternatively, [H+] could have been computed from K l equilibrium; the same result would have been obtained. BUFFER SOLUTIONS, INDICATORS, AND TITRATION 17.37. A buffer solution of pH 8.50 is desired. (a) Starting with 0.0100mol of KCN and the usual inorganic reagents of the laboratory, how would you prepare 1 L of the buffer solution? Ka for HCN is 4.93 x 10-". (b) By how much would the pH change after the addition of 5 x lO-' mol HC104 to l00mL of the buffer? (c) By how much would the pH change after the addition of 5 x lOP5 mol NaOH to 100mL of the buffer? ( d ) By how much would the pH change after the addition of 5 x 10-5 mol NaOH to 100mL of pure water? (a) To find the desired [H+]: log [H'] = -pH -8.50 = 0.50 - 9.00 [H+] = antilog 0.50 x antilog( - 9.00) = 3.2 x lOP9 CHAP. 171 ACIDS AND BASES 287 The buffer solution could be prepared by mixing CN- (weak base) with HCN (weak acid) in the proper proportions so as to satisfy the ionization constant equilibrium for HCN. Hence 4.93 x 10-l' - 0.154 [CN-] - K, _----[HCN] [H+]- 3.2 x 10-9 This ratio of CN- to HCN can be attained if some of the CN- is neutralized with a strong acid, like HC1, to form an equivalent amount of HCN. The total cya.nide available for both forms is 0,010 0 mol. Let x --- [HCN]; then [CN-] = 0.0100 - x. Substituting in ( I ) , 0.0100 - x X = 0.154 from which x = 0.008 7 and 0.0100-x = 0.001 3. The buffer solution can be prepared by dissolving 0.0100mol KCN and 0.008 7 mol HCl in enough water to make up 1 L of solution. (b) l00mL of the buffer contains (0.008 7 mol/L)(O.100 L) = 8.7 x lOP4 mol HCN and (0.001 3 mol/L)(O.100 L) = 1.3 x 10-4 mol CNThe addition of 5 x 10-5 rnol of strong acid will convert more CN- to HCN. The resulting amount of HCN will be (8.7 x 10-4) + (0.5 x 10-4) = 9.2 x 10-4 mol (1.3 x 10-4) - (0.5 x 10-4) = 0.8 x 10-4 rnol Only the ratio of the two concentrations is needed. [H+]= K,- [HCNI = (4.93 x 10-lo) [CN-I pH = -log [H'] = 9 - 0.76 = 8.24 (c) and the resulting amount of CN- will be The drop in pH caused by the addition of the acid is 8.50-8.24, or 0.26pH unit. The addition of 5 x 10-5mol of strong base will convert an equivalent amount of HCN to CNHCN : CN- : + Resulting amount = (1.3 x lOw4) (0.5 x 10-4) = 1.8 x 10-4 mol pH = -log [H+]= 9 - 0.34 = 8.66 Resulting amount = (8.7 x 10-4) - (0.5 x 10-4) = 8.2 x 10-4 mol The rise in pH caused by the addition of base is 8.66-8.50, or 0.16pH unit. [OH-] = pOH 5 x 10-~ mol 0.100 L = sx 1 0 - ~M -log [OH-] = 4 - 0.70 = 3.30; pH 14.00 - pOH= 10.70 Pure water has a pH of 7.00. The rise caused by the addition of base is 10.70 - 7.00 = 3.70pH units, in sharp contrast to the small rise in the buffer solution. 17.38. If 0.000 10mol H3P04 is added to a solution well buffered at pH 7.00, what are the relative proportions of the four forms H3P04, H2P0,, HPOi-, PO:-? K 1 , K 2 , and K3 for phosphoric acid are respectively 7.52 x 10-3, 6.23 x 10-*, and 4.5 >: 10-13. 288 ACIDS AND BASES [CHAP. 17 In contrast to the previous problem, in which strong acid or base was added, we can assume that the pH is not changed by addition of the phosphoric acid, especially since this solution was previously well buffered with respect to the very small amount of H3P04added. Then, if [H+]is fixed, the ratio of two of the desired concentrations can be calculated from each of the ionization constant equations. - LOO 1 0 - ~ 7.52 x 10-3 - LOO - 10-~ 6.23 x 10-* - = 1.33 x 1 0 - ~ = 1.61 I LOO 1 0 - ~ 4.5 x 10-13 105 = 2.2 Since the ratio [H3P04]/[H2P04] very small and the ratio [HPOi-]/[POi-] is very large, practically all is of the material will exist as H2POi and HPOi-. The sum of the amounts of these two ions will be practically equal to 0.000 10 mol; and if the total volume of solution is 1 L, the sum of these two ion concentrations will be 0.000 10 mol/L. Let x = [HPOi-1; then [H2P04]= 0.000 10 - x. Then -`H2PoQ1- 1.61 [HPOi-] gives 0.000 10 - x X = 1.61 Solving x = [HPOi-] = 3.8 x 10-5 [H2P0i]= 0.000 10 - x = 6.2 x 10-5 [H2P04] (1.33 x 1O-')[H2POi] = (1.33 x 10-5)(6.2 x 10-5) = 8.2 x 10-'' = [PO:-] = ~ [HPOi-] - 3.8 x lOP5 = 1.7 x 10-'' 2.2 105 2.2 x 105 A total volume of 1 L was selected for convenience in illustrating the relative concentrations. However, the same ratios of concentrations would exist for any reasonable choice of total volume. 17.39. K, for HC2H302 is 1.75 x lO-'. A 40.0mL sample of 0.0100M HC2H302 is titrated with 0.020OM NaOH. Calculate the pH after the addition of (a) 3.0mL, (b) 10.0mL, (c) 20.0mL, ( d ) 30.0mL of the NaOH solution. We can keep track of the changing amounts of the various species and the increasing volume by a scheme such as Table 17-2. (4 Amount of base added/L Total volume/L n(HC2H302), before neutralization n(0H-) added (Row 1x0.0200 M) n(C2H3O:) formed n(HC2H302) remaining n(0H-) excess 0 ib) (4 0.020 0 0.060 0 4.00 x 1 0 - ~ (4 0.030 0 0.070 0 4.00 x 1 0 - ~ 6.00 x 1 0 - ~ 0.003 0 0.043 0 4.00 0.60 0.60 10-~ 10-~ 0.0100 0.050 0 4.00 2.00 2.00 10-~ 10-~ 10-~ 0.040 0 4.00 x 1 0 - ~ 0.0 10-~ 4.00x 1 0 - ~ 0.0 x 1 0 - ~ 10-~ 4.00x 1 0 - ~ X 4.00x 1 0 - ~ Y 4.00x 1 0 - ~ 3.40 x 1 0 - ~ 2.00 x 1 0 - ~ 2.00 10-~ CHAP. 171 ACIDS AND BASES 289 Note that the amount of acetic acid neutralized (amount of C2H3O:) follows the amount of OHadded, up to complete neutralization. Addition OH-, having no more acid to neutralize, accumulates in the solution. Up to the end point, the amount of HC2H302 remaining is obtained by simply subtracting the amount of C2H3OT from the initial amount of HC2H302. At the end point and beyond, however, [HC2H302] cannot be set equal to zero but must be deduced from the hydrolysis equilibrium. (a) and (b). Here absolute concentrations of conjugate acid and base are not needed; only their ratio is required. = 9.9 x 1 0 - ~ p H = -log [H+] M(C2H3O;) = 4.00 1 + = 1.75 1 0 - ~ 4.76 4.00 x 10-4 mol = 6.7 x 10-3 mol/L 0.0600 L The solution at the end point is the same as 6.7 x 10-3 M NaC2H302. Consider the hydrolysis of NaC2H302: C2H3O: H20 + HC2H302 OH- + Let [HC2H302] [OH-] = x, [C2H30;] = 6.7 x 10-3 - x M 6.7 x 10-3; then = Check of assumption: x is indeed small compared with 6.7 x 10-3. pOH = -log [OH-] = -log (1.9 x 10-6')= 6 - 0.28 = 5.72 pH = 14.00 - 5.72 = 8.28 ( ) From the excess OH- beyond that needed to neutralize all the acetic acid, we know that d M(0H-) = Hence pOH = -log [OH-] = -log (2.9 x 10-3) = 2.54 pH = 14.00 - 2.54 = 11.46 2.0 x 1 0 - ~ mol = 2.9 x 10-3 mol/L 0.070 L 17.40. Calculate a point on the titration curve for the addition of 2.0 mL of 0.010 0 M NaOH to 50.0 mL of 0.0100M chloroacetic acid, HC2H202Cl.I, = 1.40 >c: 10-3. ( The simplifying assumption made in Problem 17.39 breaks down here. If the amount of chloroacetate ion formed were equivalent to the amount of NaOH added, we would have the following: Amount OH- added = (0.0020 L)(O.OlO mol/L) = 2.0 x 10-5 mol Total volume = 0.0520 L M(C2H202Cl-) = 2.0 x 1 0 - ~ mol = 3.8 x 10-4 mol/L 0.0520 L [H"] = Ka[HC2H202C1] (1.40 x 10-3)(9.2 x 10-3) = 3.4 [C2H202C1- 1 3.8 x 10-4 10-2 (3) 290 ACIDS AND BASES [CHAP. 17 This answer is obviously ridiculous. [H+] cannot possibly exceed the initial molar concentration of the acid. Apparently the amount of chloroacetate ion is greater than the equivalent amount of base added. This fact is related to the relatively strong acidity of the acid and to the appreciable ionization of the acid even before the titration begins. Mathematically, this is taken into account by an equation of electroneutrality, according to which there must be equal numbers of cationic and anionic charges in the solution. (4) It is safe to drop the [OH-] term in (4) in this case, because it is so much smaller than any of the others. "a+] is obtained from the amount of NaOH added and the total volume of solution. [H'] + "a+] = [C2H202C1-] [OH-] + [C2H202Cl-] = 3.8 x 10-4 + [H'] (6) The undissociated acid concentration is then the total molar concentration of acid (including its ion) minus [C2H202Cl-]. 2.0 1 0 - ~ = 3.8 1 0 - ~ 0.052 0 Chloroacetate ion can be computed from ( 4 ) and (5) with the neglect of [OH-]. "a+] = Note that (6) and ( 7 ) differ from ( I ) and (2) only in the inclusion of the [H+] terms. Now we may return to the ionization equilibrium for the acid. K,[HC2H202C1]- (1.40 x 10-3)(9.2 x 10-3 - [H']) 3.8 x 10-4 + [H+] P 2 H202Cl- 1 The solution of the quadratic equation is [H+]= 2.8 x 10P3,and pH= -log [H'] = 2.55. The complication treated in this problem occurs whenever [H '1 or [OH-] during partial neutralization cannot be neglected in comparison with the concentrations of other ions in solution. This is likely to be the case near the beginning of the titration of an acid which is only moderately weak. [H'] = 17.41. An acid-base indicator has a K, of 3.0 x 10-5. The acid form of the indicator is red and the basic form is blue. (a) By how much must the pH change in order to change the indicator from 75% red to 75% blue? (b) For which of the titrations shown in Fig. 17-la and 17-lb would this indicator be a suitable choice? K, [acid] [H'] = [base] The change in pH is 5.00 - 4.05 = 0.95. (b) The indicator changes its color in the pH range 4 to 5. In both titrations using HCl as titrant in Fig. 17-lb, the pH is falling rapidly in this range; thus the indicator would be suitable for both of them. In the titration of HCl with NaOH in Fig. 17-la the pH is rising rapidly in this range and it could also be done with this indicator. This indicator, however, would not be suitable for the HC4H703titration which would change from red to blue long before the end point is reached. Supplementary Problems Notes: 1. The numerical values tabulated for equilibrium constants differ in various books. The values selected for this book are internally consistent, and the computed answers are based on values CHAP. 171 ACIDS A N D BASES 29 1 given here. The temperature is assumed to be 25 "C unless there is a specific statement to the contrary. 2. Some of the problems at the end of the sections on Acids and Bases, Hydrolysis, Polyprotic Acids, and Buffers involve multiple equilibria. They may be omitted in a simplest treatment of ionic equilibrium. ACIDS AND BASES 17.42. A certain reaction is catalyzed by acids, and the catalytic activity for 0.1 M solutions of the acids in water was found to decrease in the order HC1, HCOOH, HC2H302. The same reaction takes place in anhydrous ammonia, but the three acids all have the same catalytic effect in 0.1 M solutions. Explain. Ans. The order of catalytic activity in water is the same as the order of acidity. In anhydrous ammonia, which is a much stronger base than water, i.e., a more effective proton acceptor, all three acids are strong. 17.43. The amino acid glycine exists predominantly in the form +NE13CH2C00-. Write formulas for conjugate base and (b) the conjugate acid of glycine. (a) the Ans. (U) NH,CH,COO-; (b) 'NH3CH2COOH 17.44. In the reaction of BeF2 with 2F- to form BeFi-, which reactant is the Lewis acid and which is the Lewis base? Ans. BeF2 is the acid and F- is the base. 17.45. The conjugate acid of nitric acid loses a water molecule in certain solvents. Explain how the resulting entity can still act as an acid in the Lewis sense. Ans. H N 0 3 + H+ t H 2 0 + NO; (see structure below) :O =N =O : N can go from sp to sp2 hybridization providing an empty orbital for a base pair. (Note similar behavior of CO2 in Problem 17.8.) 17.46. Is acetic acid a weaker or stronger acid in the following solvents compared to water? (a) hydrazine, N2H4; (6) sulfur dioxide, SO,; (c) methyl alcohol, CH30H; ( d ) liquid hydrogen cyanide, HCN; (e) liquid sulfuric acid, H2SO4. Ans. (a) stronger; (b) weaker; (c) weaker; ( d ) weaker; (e) weaker. 17.47. The self-ionization constant for pure formic acid, K = [HCOOH,f][HCOO-1, has been estimated as 10-6 at room temperature. What percentage of formic acid molecules in pure formic acid, HCOOH, are converted to formate ion'? The density of formic acid is 1.22g/cm3 Ans. 0.004% 17.48. Calculate the ionization constant of formic acid, HCO*H, which ionizes 4.2 percent in 0.10 M solution. A ~ 1.8 . 1 0 - ~ ~ 17.49. A solution of acetic acid is 1.O percent ionized. What must be the molar concentration of acetic acid and the [H+] of the solution? K, of HC2H302 is 1.75 x lO-'. Ans. 0.17M. 1.7 x IOV3 (a) the degree of ionization and 17.50. The ionization constant of ammonia in water is 1.75 x lOP5. Determine (b) the [OH-] of a 0.08M solution of NH3. Ans. 1.5 percent, 1.2 x lOU3 292 ACIDS AND BASES [CHAP. 17 17.51. Chloroacetic acid, a monoprotic acid, has a K, of 1.40 x 10-3. Compute the freezing point of a 0.lOM solution of this acid. Assume that the stoichiometric molar concentration and molality are the same in this case. Ans. -0.21 "C 17.52. Determine [OH-] of a 0.050 M solution of ammonia to which has been added sufficient NH4Cl to make the total [NH:] equal to 0.100. Kb of ammonia is 1.75 x lO-'. Ans. 8.8 x 10-6 17.53. Find the value of [H+] in a liter of solution in which are dissolved 0.080mol HCzH302 and 0.100mol NaC2H302. KO for HC2H302 is 1.75 x 10-5. A ~ 1.4 x 1 0 - ~ ~ . 17.54. A 0.025OM solution of monobasic acid had a freezing point of -0.060"C. What are K, and pK, for the acid? Assume that at this low concentration molality is equal to the molar concentration. Ans. 3.0 x 10-3.2.52 17.55. What is [NH,f] in a solution that is 0.020 0 M NH3 and 0.010 0 M KOH? Kb for NH3 is 1.75 x 10-5. 3.5 x 1 0 - ~ 17.56 What molar concentration of NH3 provides a [OH-] of 1.5 x 10-3? Kb for NH3 is 1.75 x 10-5. Ans. 0.13M 17.57. What is [HCOO-] in a solution that is both 0.015 M HCOOH and 0.020 M HCl? K, for HCOOH is 1.8 1 0 - ~ . \ 1.4 x 1 0 - ~ 17.58. What are [H'], [C3H50T], and [OC6HF]in a solution that is 0.030 M HC3H503 and 0.100 M HOCbHS? K,values for HC3H503and HOC6H5 are 3.1 x 10-5 and 1.05 x 10-lo, respectively. Ans. [H'] = 9.6 x 10-4, [C3H503] = 9.6 x 10-4, [OC&y] = 1.1 x 10-8 17.59. Find the value of [OH-] in a solution made by dissolving 0.005 0 mol each of ammonia and pyridine in enough water to make 200mL of solution. Kb for ammonia and pyridine are 1.75 x 10-5 and 1.78 x lOw9, respectively. What are the concentrations of ammonium and pyridinium ions? Ans. [OH-] = [NH,f] = 6.6 x 10-4, [pyridinium ion] = 6.7 x 10-8 17.60. Consider a solution of a monoprotic weak acid of acidity constant K,. Calculate the minimum concentration, C , for which the percent ionization is less than 10 percent. = 9 0 K, 17.61. What is the percent ionization of 0.006 5 M chloroacetic acid? K, for this acid is 1.40 x 10-3. Ans. 37% 17.62. What concentration of dichloroacetic acid gives a [H+] of 8.5 x lOP3?K, for the acid is 3.32 x 1OV2. Ans. 1.07 x 10-2 M CHAP. 171 ACIDS AND BASES 293 17.63. Calculate [H'] in a 0.200M dichloroacetic acid solution that is also 0.100M in sodium dichloroacetate. K, for dichloroacetic acid is 3.32 x 10-2. Ans. 0.039 17.64. How much solid sodium dichloroacetate should be added to a liter of 0.100 M dichloroacetic acid to reduce [H+] to 0.030? K, for dichloroacetic acid is 3.32 x 10-2. Neglect the increase in volume of the solution on addition of the salt. Ans. 0.047 mol 17.65. Calculate [H+]and [C2H02Clz]in solution that is 0.0100M in HCl and 0.0100M in HC2H02C12. K, for HC2H02C12 (dichloroacetic acid) is 3.32 x 1OY2. Ans. 0.0167, 0.0067 17.66. Calculate [H+], [C2H302], and [C7H50J in a solution that is 0.020OM in HC2H302, and 0.0100M in HC7H502.K,-values for HC2H302and HC7H502are 1.75 x 10-5 and 6 . 4 6 10-5, respectively. ~ Ans. 1.00 x 10-3,3.5 x 10-4,6.5 x 10-4 17.67. Liquid ammonia ionizes to a slight extent. At -50 "C, its ion product is K N = "Hi] [NHJ = 10-30. How ~ ~ many amide ions, NH;, are present per mm3 of pure liquid ammonia? Ans. 6.0 x 102 17.68. Calculate the molar concentration of ammonium ions in a liter of a liquid ammonia at -50 "C (see previous problem for data), in which 10.0 grams each of ammonium chloride and sodium amide (NaNH2) were dissolved. ~ n 1.4 x . 1 0 - ~ ~ ~ M IONIZATION OF WATER 17.69. Calculate the pH and pOH of the following solutions, assuming complete ionization: (a) 0.003 45 N acid, (b) 0.000 775 N acid, (c) 0.008 86 N base. Ans. (a) pH = 2.46, pOH = 11.54; (b) 3.1 1, 10.89; (c) 11.9S, 2.05 17.70. Convert the following pH-values to [H+]-values: (a) 4, (b) 7, (c) 2.50, ( d ) 8.26. A ~ (a) 1 0 - ~ ; (6) 1 0 - ~ ; (c) 3.2 ~ . 10-~; ( d ) 5.5 10-~ 17.71. The [H+] of an H N 0 3 solution is 1 x 10-3, and the [H+] of an NaOH solution is 1 x 10-l2. What are the molar concentration and pH of each solution? Ans. HN03: 0.001 M, pH=3; NaOH: 0.01 M, p H = 12 17.72. Compute [H+] an [OH-] in a 0.001 Omolar solution of a monobasic acid which is 4.2 percent ionized. What is the pH of the solution? What are K, and pK, for the acid? Ans. [] H ' = 4.2 x 10-5, [OH-] = 2.4 x 10-lo, pH = 4.38, K, = 1.8 x 10-6, pK, = 5.74 17.73. Compute [OH-] and [H+] in a 0.10 N solution of a weak base which is 1.3 percent ionized. What is the pH of the solution? Ans. [OH-.] = 1.3 x 10-3, [H'] = 7.7 x 10-'*, pH = 11.11 294 ACIDS AND BASES [CHAP. 17 17.74. What is the pH of a solution containing 0.010mol HCl per liter. Calculate the change in pH if 0.020mol NaC2H302is added to a liter of this solution. K, of HC2H302 is 1.75 x 10-5. Ans. Initial pH=2.0, final pH=4.76 17.75. The value of K,,, at the physiological temperature 37 "C is 2.4 x 10-14. What is the pH at the neutral point of water at this temperature, where there are equal numbers of H + and OH-? Ans. 6.81 17.76. Calculate the pH of 1.0 x 10-7 M NaOH. What percentage of the added base was neutralized by the H + present in the water and what percentage remained to make the solution basic? Ans. 7.21; 38% was neutralized, 62% remained. 17.77. What is the pH of 7.0 x 10-* M acetic acid? What is the concentration of un-ionized acetic acid? K, is 1.75 x 10-5 (Hint: Assume essentially complete ionization of the acetic acid in solving for [H+].) Ans. 6.85,5.6 x 10-l' M 17.78. Calculate OH- in a 0.010 0 M solution of aniline, C6HSNH2. Kb for the basic dissociation of aniline is 4.3 x 10- 16 . What is the [OH-] in a 0.0100M solution of aniline hydrochloride, which contains the ion C~HSNH~? Ans. 2.1 x 10-6,2.1 x 10-" 17.79. Calculate the percent hydrolysis in a 0.0100M solution of KCN. K, for HCN is 4.93 x 10-''. Ans. 4.5% 17.80. The basic ionization constant for hydrazine, N2H4, is 9.6 x 10-7. What would be the percent hydrolysis of 0.100 M N2HSC1, a salt containing the acid ion conjugate to hydrazine base? Ans. 0.032% 17.81. Calculate the pH of a moderately concentrated solution of pyridinium acetate, (C5H5NH)(C2H302).K, for ~ acetic acid is 1.75 x 10-5 and for the pyridinium ion is 5 . 6 10-6. Ans. 5.00 17.82. A 0.25 M solution of pyridinium chloride, C S H ~ N + C ~ - , found to have a pH of 2.93. What is Kb for the was basic dissociation of pyridine, CSHSN? A ~ 1.8 x. 1 0 - ~ ~ 17.83. K, for the acid ionization of Fe3+ to Fe(OH)2+ and H + is 6.5 x IOp3. What is the maximum pH-value which could be used so that at least 95 percent of the total + 111 iron in a dilute solution is in the Fe3+ form? Ans. 0.91 17.84. A 0.010M solution of Pu02(N03)2 was found to have a pH of 3.80. What is the hydrolysis constant, K,, for PuO;+, and what is Kb for PuOZOH+? Ans. K, = 2.5 x 10-6, Kb = 4.0 x 10m9 17.85. Calculate the pH of 1.00 x 10-3 M sodium phenolate, NaOC6HS. K, for HOC6HS is 1.28 x IO-" Ans. 10.39 CHAP. 171 ACIDS AND BASES 295 17.86. Calculate [H'] and [CN-] in 0.010 0 M NH4CN. K , for HCN is 4.93 x 10-l' and & for NH3 is 1.75 x lO? Ans. 5.3 x 10-",4.8 x 10-3 POLYPROTIC ACIDS 17.87. Calculate the [ H f ] of a 0.050M H2S solution. K 1 of H2S is 1.0 x 10-7 A ~ 7.1 . 1 0 - ~ ~ 17.88. What is IS2-] in a 0.050M H2S solution? K2 of HZS is 1.2 x lO-.l3 Ans. 1.2 x IO-l3 17.89. What is [S2-] in a solution that is 0.050 M H2S and 0.010 0 M HCI? Use data from Problems 17.87 and 17.88. Ans. 6.0 x 10-l8 17.90. (a) Calculate the [HS-] in the solution in the preceding problem. ( b ) If sufficient ammonia were added to buffer the solution at pH =4.37, what would be the concentrations of S2- and HS-? Ans. (a) 5.0 x 10-7; (6) [S2-] = 3.3 x lO-I3, [HS-] = 1.2 x 10p4 17.91. K , and K2 for oxalic acid, H2C204, are 5.9 x 10-2 and 6.4 x lOP5 respectively. What is [OH-] in a 0.0050 molar solution of Na2C204? Ans. 8.8 x 10-7 17.92. Malonic acid is a dibasic acid having K 1 = 1.49 x 10-3 and K2 = 2.03 x 10-6. Compute the concentration of the divalent malonate ion in (a) 0.001 0 M malonic acid, (b) a solution that is 0.000 10 M in malonic acid and 0.00040 M in HC1. Ans. (a) 2.0 x 10-6; (b) 3.2 x 10-7 17.93. Compute the pH of a 0.010M solution of H3P04. K , and K2 for H3P04 are respectively 7.52 x 10-3 and 6.23 x 10-8. Ans. 2.24 17.94. What is [H'] in a 0.006 0 M H2S04 solution? The first ionization of H2S04 is complete and the second What is [SO:-] in the same solution? ionization has a K2 of 1.20 x 10--2. Ans. [H'] := 9.4 x lOP3, [SO:-] = 3.4 x 10p3 17.95. Ethylenediamine, NH2C2H4NH2, is a base that can add one or two protons. The successive p&-values for the reaction of the neutral base and of the monocation with water are 3.288 and 6.436, respectively. In a 0.0100M solution of ethylenediamine, what are the concentrations of the singly charged cation and of the doubly charged cation? Ans. 2.03 x 10-3mol/L, 3.66 x 10-7mol/L 17.96. Suppose 0.0100mole of NaOH were added to a liter of the solution in the preceding problem, what would the concentrations of singly charged and doubly charged cations become? Ans. 5.1 x 10-4M; 1.88 x IOP8M 296 ACIDS AND BASES [CHAP. 17 17.97. Suppose that instead of 0.0100 mole of NaOH, 0.0100 mole of HC1 had been added to the solution in the preceding problem. Calculate the molar concentrations of the singly charged cation, doubly charged cation, and the neutral ethylenediamine. A ~ 0.010; 2.7 ~ . 10-~; x 1 0 - ~ 2.7 17.98. pK1 and pK2 for pyrophosphoric acid dre 0.85 and 1.49, respectively. Neglecting the third and fourth dissociations of this tetraprotic acid, what would be the concentration of the doubly charged anion in a 0.050M solution of the acid? Ans. 1.5 x 10-2 M 17.99. What is [CO:-] in a 0.001 00 M Na2C03 solution after the hydrolysis reactions have come to equilibrium? K1 and K2 for H2CO3 are 4.30 x lO-' and 5.61 x 10-", respectively. Ans. 6.6 x 10-4 17.100. Calculate the pH of 0.050M NaH2P04 and of 0.00200M Na3P04. K , , K2, and K3 of H3P04 are 7.52 x lOP3,6.23 x 1OP8, and 4.5 x 10-13, respectively. Ans. 4.7, 11.27 17.101. Citric acid is a polyprotic acid with pK1, pK2, and pK3 equal to 3.15, 4.77, and 6.39, respectively. Calculate the concentrations of H , the singly charged anion, the doubly charged anion, and the triply charged anion in 0.0100M citric acid. A ~ 2.3 x ~ o - ~ M . 3 ~ 1 0 - i~. ~ ,~ 1 0 - 2 .~9, ~ 1 0 - ~ ~ ~ . 2 , 7 ~ 17.102. The amino acid glycine, NH2CH2COOH, is basic because of its NH2- group and acidic because of its -COOH group. By a process equivalent to base dissociation, glycine can acquire an additional proton to form +NH3CH2COOH.The resulting cation may be considered to be a diprotic acid, since one proton from the -COOH group and one proton from the +NH3- group may be lost. The pK, values for these processes are 2.35 and 9.78, respectively. In a 0.010 0 M solution of neutral glycine, what is the pH and what percent of the glycine is in the cationic form at equilibrium? Ans. 6.14, 0.016% BUFFER SOLUTIONS, INDICATORS, AND TITRATIONS 17.103. A buffer solution was prepared by dissolving 0.020 0 mol propionic acid and 0.015 0 mol sodium propionate in enough water to make 1 L of solution. (a) What is the pH of the buffer? (b) What would be the pH change if 1.0 x 10-5 mol HCl were added to lOmL of the buffer. (c) What would be the pH change if 1.0 x 10-5 mol NaOH were added to lOmL of the buffer? K, for propionic acid is 1.34 x 10-5. Ans. (U) 4.75; (b) -0.05; 17.104. The base imidazole had a Kb of 1.11 x 10-7. (a) In what amounts should 0.020 0 M HC1 and 0.020 0 M imidazole be mixed to make 100mL of a buffer at pH 7.00? (b) If the resulting buffer is diluted to 1 L, what is the pH of the dilute buffer? Ans. (a) 34mL acid, 66mL base, (b) 7.00 17.105. In the titration of HC1 with NaOH represented in Fig. 17-la, calculate the pH after the addition of a total of 20.0, 30.0, and 60.0mL of NaOH. Ans. 1.37, 1.60, 11.96 + (c) +0.05 CHAP. 171 ACIDS AND BASES 297 17.106. In the titration of P-hydroxybutyric acid, HC4H703, with NaOH represented in Fig. 17-la, calculate the pH after the addition of a total of 20.0, 30.0, and 70.0mL.of Na0E-E. pK, for HC4H703 is 4.70. Ans. 4.52, 4.88, 12.22 17.107. In the titration of NH3 with HCl represented in Fig. 17-lb, calculate the total volume of the HC1 solution needed to bring the pH to 10.00 and to 8.00. Kh for NH3 is 1.75 x lO-'. Ans. 7.4mId,47.3mL 17.108. The dye bromcresol green has a pK,-value of 4.95. For which of the four titrations shown in Fig. 17-1 would bromcresol green be a suitable end-point indicator? Ans. HCl vs. NaOH, NaOH vs. HCl, NH3 vs. HC1 17.109. Bromphenol blue is an indicator with a &-value of 5.84 x lO-'. What percentage of this indicator is in its basic form at a pH of 4.84? Ans. 80% 17.110. Calculate the pH and [NH3] at the end point in the titration of NH3 with HC1, at the concentrations indicated in Fig. 17-16. Kb for NH3 is 1.75 x lops. Ans. 5.28, 5.3 x 10-6 17.111. If 0.001 00 mol of citric acid is dissolved in 1 L of a solution buffered at pH 5.00 (without changing the volume), what will be the equilibrium concentrations of citric: acid, its singly charged anion, the doubly charged anion, and the triply charged anion? Use pK-values from Problem 17.101. A ~ S . 5.0 x I O - ~ M3.6 x 1 0 - ~M, 6.1 x ~ o - ~ M , x 1 0 - ~M , 2.5 17.112. If 0.000 500 rnol NaHC03 is added to a large volume of a solution buffered at pH 8.00, how much material will exist in each of the three forms H2CO3, HCO,, and CO:-? K , and K2 of H2C03 are, respectively, 4.30 x 10p7 and 5.61 x lop". Ans. 1.14 x lO-' mol, 4.86 x lOP4mol, 2.73 x 10-6 mol 17.113. A buffer solution of pH 6.71 can be prepared by employing solutions of NaH2P04 and Na2HP04. If 0.005Omol NaH2P04 is weighed out, how much Na2MP04 must be used to make 1 L of the solution? Take K-values from Problem 17.100. Ans. 0.001 6 rnol 17.114. How much NaOH must be added to 1 L of 0.010 M H3B03to make a buffer solution of pH 10.10? H3B03 is a monoprotic acid with K , = 5.8 x IO-". Ans. 0.008 8 mol 17.115. A buffer solution was prepared by dissolving 0.050 mol formic acid and 0.060 rnol sodium formate in enough water to make 1 L of solution. K, for formic acid is 1.77 x WO-4. (a) Calculate the pH of the solution. (b) If this solution were diluted to ten times its volume, what would be the pH? (c) If the solution in (6) were diluted to ten times its volume, what would be the pH? Ans. (a) 3.83; (b) 3.85; (c) 4.00 17.116. Consider the titration curve of a 0.100M solution of a weak acid, HA, for which K, = 2.00 x 10-4, using 0.100 M strong base. Calculate the pH at (a) the beginning (before any base), (b) the half-way point, and ( c ) the end point of the titration. Ans. (a) 2.35; ( b ) 3.70; (c) 8.20 298 ACIDS AND BASES [CHAP. 17 17.117. The three points determined in the preceding problem were easy to calculate and enable one to make a good sketch of the titration curve. Other intermediate values, e.g., 10%, 75%, etc., are also easy to calculate, but it may be more challenging to calculate points very near the beginning and end. Determine the pH in this titration (a) after 1% of the requisite base is added, and (b) after 99% of the base is added. Ans. (a) 2.41; (b) 5.7 Chapter 18 Complex Ions; Precipitates COORDINATION COMPLEXES This section deals with the equilibria involving metal ions, their complexing ligands, and the resulting coordination complexes (see Chapter 9). Some of these cornplexes are so stable that salts can be prepared from them that show no appreciable amounts of the separate constituents. An example is the ferricyanide ion, Fe(CN)i-, solutions of which are quite different in properties from those of Fe3+ or of CN-. More commonly, the complex is not so stable and in solution is partly dissociated into its components. In such a case there is an equilibrium constant which regulates the allowable simultaneous values of the concentrations of the various species. An example is FeBr2+, which may be formed or dissociated easily, depending on slight modifications of experimental conditions. Fe3+ + Br- + FeBr2+ l,FeBr2+] K, = [Fe3+]Br-] [ The constant K, is called a stability constant; the larger its value, the stabler the complex. In some cases several ligands may be complexed, one at a time. A separate equilibrium equation may be written for each successive ligand addition; e.g., Cd2++ CN- + CdCNS K1 = : [CdCN'] [Cd2+] [CN-] Analogous equations can be written for the addition of a third and fourth cyanide, with constants K3 and K4. In addition to the stepwise formation equilibria, we can write a single overall equation for the formation of a complex containing several ligands from the fre:e cation and ligands. It can easily be shown that in this case Ks = K1K2K3K4. Equilibria between a complex ion and its components are sometimes written in reverse. The dissociation constant, Kd, is the reciprocal of the overall stability constant, K,. Other symbols commonly used in texts and tables of data are Synonymous with K,: Synonymous with Kd: Kj- or Kfom (formation constant) (instability constant) or Kinst SOLUBILITY PRODUCT Consider the equilibrium between solid AgCl and its dissolved ions in a saturated solution: AgCl (solid) + Ag+ -+ Cl- (in solution) The expression for the equilibrium constant is Ksp = [Ag+I[Cl-l 299 300 COMPLEX IONS; PRECIPITATES [CHAP. 18 where the convention of excluding terms for solids has been honored. This particular case of the equilibrium constant for the solution of ionic substances is called the solubility product (Ksp).Solubility products for some other saturated solutions of a slightly soluble salt are Ksp of BaC03 = [Ba2'][CO:-] Ksp of CaF2 = [Ca2'][F-l2 Ksp of Bi2S3= [Bi3']2[S2-]3 In each of these expressions, the exponents are derived from the balanced equation in the usual manner. As in previous chapters, the temperature will be taken as 25C unless a statement is made to the contrary. APPLICATIONS OF SOLUBILITY PRODUCT TO PRECIPITATION Precipitation The solubility product enables us to explain and predict the degree of completeness of precipitation reactions. Whenever the product of the appropriate powers of the concentrations of any two ions in a solution exceeds the value of the corresponding solubility product, the cation-anion combinations will be precipitated until the product of the concentrations of these two ions remaining in solution (raised to their respective powers) again attains the value of the solubility product. EXAMPLE 1 When some NaF is added to a saturated solution of CaF2, the [F-] is greatly increased and the product of the ion concentrations, [Ca2+][F-I2,may temporarily exceed the value of the solubility product. To restore equilibrium, some Ca2+ unites with an equivalent quantity of F- to form solid CaF2, until [Ca2+][F-I2 for the ions remaining in solution again attains the value of the solubility product. Note that in this case the final value of [F-] is much greater than twice that of [Ca2+],since NaF yields a large contribution to the total [F-1. The ion concentrations that appear in the expression for the solubility product refer only to the simple ions in solution, and do not include the material in the precipitate. Additional equilibria may exist between the simple ions and complexes in solution, in case soluble complexes form; these equilibria would be governed by their own stability constants. Solution of precipitates The product of the concentrations of any two ions (raised to the appropriate powers) in a solution is called the ion product, Q, analogous to Q, the reaction quotient of Chapter 16. Whenever Q is less than the value of the corresponding solubility product, the solution is not saturated. If some of the corresponding solid salt is in contact with the solution. more of the salt will dissolve. Suppose HCl (which supplies H + ) is added to a saturated solution of Mg(OH)2 in equilibrium with some undissolved solute. The H+ removes nearly all the OH- in solution to form H2O. This greatly decreases the [OH-], and more Mg(OH)2 dissolves so that the ion concentration product can again attain the value of Ksp for Mg(OH)2. If all of the Mg(OH)2 dissolves, there is no longer an equilibrium between the ionic solid and the solution, and Q will be less than Ksp. EXAMPLE 2 Prevention of precipitation To prevent the precipitation of a slightly soluble salt, some substance must be added which will keep the concentration of one of the ions so low that the solubility product of the salt is not attained. CHAP. 181 COMPLEX IONS; PRECIPITATES 30 1 EXAMPLE 3 H2S will not precipitate FeS from a strongly acid (HCl) solution of Fe2+. The large [H+] furnished by the hydrochloric acid represses the ionization of H2S (common-ion effect) and thus reduces the [S2-] to so low a value that the solubility product of FeS is not reached. Solved Problems COORDINATION COMPLEXES 18.1. One liter of' solution was prepared containing 0.001 00 mcd of silver in the +1 oxidation state and 1.OO mol o f NH3. What is the concentration of free Ag' in the solution at equilibrium? Kd for Ag(NH3)i is 6.0 x lO-'. Most of the silver, approximately 0.001 00 mol, will be in the form of the complex ion Ag(NH3);. The concentration of free NH3 at equilibrium is practically unchanged from 1.OO mol/L, since only 0.002 00 mol of NH3 would be used up to form 0.001 00 mol of complex. Ag(NH3); F + Ag+ + 2NH3 Solving [Ag'] = 6.0 x 10-l'; i.e., an equilibrium concentration of 6.0 x 10-" mol/L. 18.2. K 1 for the complexation of one NH3 with Ag' is 2.0 x 103. (a) With reference to Problem 18.1, what is the concentration of Ag(NH3)+? (b) What is K2 for this system? (a) K1 refers to the following: Then, from Problem 18.1, [Ag(NH3)+]= K1 ([Ag'] [NH3]) = (2.0 x 1 03)(6.0 x 1O-")( 1.OO) = 1.2 x IOv7 This problem is actually a check on an assumption made in Problem 18.1; namely, that practically all the dissolved silver was in the complex Ag(NH3)2f.If [Ag(NH3)+]had turned out to be greater than about 1 x 10-4, the assumption would have been shown 'to be incorrect. (b) Kl, K 2 , and Kd are interrelated. -- 1 KlKd - 1 = 8.3 x 103 (2.0 x 103)(6.0x 10-8) 18.3. How much NH3 should be added to a solution of 0.001 00 M Cu(NO& in order to reduce [Cu2+] to 10-13? Kd for Cu(NH3)i' is 4.35 x 10-13. Neglect the amount of copper in complexes containing fewer than 4 ammonias per copper. 302 COMPLEX IONS; PRECIPITATES [CHAP. 18 Since the sum of the concentrations of copper in the complex and in the free ionic state must equal 0.001 00 mol/L, and since the amount of the free ion is very small, the concentration of the complex is taken to be 0.001 00 mol/L. Let x = [NH3].Then (10-13)(X4) 4.35 x lO-I3 = 0.001 00 or x4 = 4.35 x lOP3 or x = 0.26 The concentration of NH3 at equilibrium is 0.26 mol/L. The amount of NH3 used up in forming 0.001 00 mol/L of complex is 0.0040 mol/L, an amount negligible compared with the amount remaining at equilibrium. Hence the amount of NH3 to be added in 0.26 mol/L. 18.4. A solution was prepared in which, prior to complexation, Cd(I1) was 0.00025 M and I- was 0.0100 M. K 1 and K2 for complexation of Cd2+ with I- are 190 and 44, respectively. What percentages of Cd(I1) at equilibrium are Cd", CdI', and Cd12? Let [Cd"] = x, [CdI'] = y , and [Cd12]= z. Assume that [I-] remains 0.0100 within the precision of this calculation. (At most only 0.000 5 could be complexed.) Cd2+ + Ix K1 = ~ + CdI+ y CdI' J.' 7 + I- * Cd12 0.0100 0.0100 z z =0.44~ Y 0.010 ox 190 Y = 1.90~ K2 = - 44 = 0.01ooy z = (0.44)(1.90~) 0 . 8 4 ~ = = 6.7 x 1 0 - ~M y = 12.7 x 1 0 - ~ M z = 5.6 x 10-~ M [CdI2] = 5.6 x 1 0 - ~ M 2.5 x 10-4 M 0 looo/ - 22% Note the contrast between this problem and Problem 18.2. In the latter case, the concentration of the complexing agent was so high, and the value of K2 sufficiently greater than K , , that the complexation was driven almost completely to the fully complexed form. SOLUBILITY PRODUCT AND PRECIPITATION 18.5. The solubility of PbS04 in water is 0.038 g/L. Calculate the solubility product of PbS04. PbS04 (solid) + Pb2++ SO:- (in solution) The concentrations of the ions must be expressed in moles per liter. To convert 0.038 g/L to moles of ions per liter, divide by the molar mass of PbS04 (303). 0.038 g/L = 0.038 g/L = 1.23 x 10-4 = mol/L 303 g/mol mol of dissolved PbS04 yields 1.25 x 10h4 mol each of Pb2+ and SO:-, Since 1.25 x lOV4 Ksp = [Pb2'][SO$-] = (1.25 x 10-4)(1.25 x 10-4) = 1.6 x 1Ov8 CHAP. 181 COMPLEX IONS; PRECIPITATES 303 This method may be applied to any fairly insoluble salt whose ions do not hydrolyze appreciably or form soluble complexes. Sulfides, carbonates, and phosphates, and salts of many of the transition metals like iron, must be treated by taking into account hydrolysis and, in some cases, complexation. Some examples will be given in the problems that follow. 18.6. The solubility of Ag2Cr04 in water is 0.022 g/L. Determine the solubility product. Ag2Cr04(s) 2Ag+ + Cr0;- To convert 0.022 g/L to mol/L of the ions, divide by the molar mass of Ag2Cr04 (332). Since 1 mol of dissolved Ag2Cr04 yields 2 mol Ag' and 1 mol CrOi-, [Ag+] = 2(6.6 x lO-') and = 1.3 x 10-4 [CrOi-] = 6.6 x lOV5 Ksp = [Ag'I2CrOi-] = (1.3 x 10-4)2(6.6x 10-5) = 1.1 x 18.7. The solubility product of Pb(IO3)2 is 2.5 x 10-13. Wha-t is the solubility of Pb(103)2 (a) in mol/L and (b) in g/L? (a) Let the solubility of Pb(I03)2 be x mol/L. Then [Pb2'] = A: and [I031 = 2x. [Pb2'][IOJ2 = Ksp gives ~ ( 2 3 : =~2.5 ) x lO-I3 Then 4x3 = 2.5 x 10-13, x3 = 62 x 10-15, and x = 4.0 x (b) w5. Solubility = (4.0 x 10-5 mol/L)(557 g/mol) = 0.022 g/L 18.8. The [Ag'] of a solution is 4 x 10-3. Calculate the [Cl-] that must be exceeded before AgCl can precipitate. The solubility product of AgCl at 25 "C is 1.8 x 10-". [Ag+l[Cl-l = K p s (4 x lo-3)[c1-] = 1.8 x l.O-IO [cl-] 5 x 10-8 = Hence a [Cl-] of 5 x 10-8 must be exceeded before AgCl precipitates. This problem differs from the previous ones in that the two ions forming the precipitate are furnished to the solution independently. This represents a typical analytical situation, in which some soluble chloride is added to precipitate silver ion present in a solution. 18.9. Calculate the solubility of CaF2 in 0.015 M NaF solution. Ksp of CaF2 is 3.9 x 10-l'. Because of the large concentration of the common ion, F - , the solubility will be quite low, and the additional F- from the dissolved CaF2 will be assumed negligible compared to that from the NaF. Hence, [Ca2+][F-l2 Ksp= 3.9 x 10-" = [ca2+]= 3.9 x 10-" = 1.7 (0.015)2 =: [Ca2+](0.015)2 10-~ The solubility will be 1.7 x 10-7 mol/L. The [F-] from the CaF2 will be 2 x 1.7 x 10-7 = 3.4 x 1OP7 M, which can be ignored, as assumed, compared to 0.015 M. 304 COMPLEX IONS; PRECIPITATES [CHAP. 18 18.10. Calculate the concentration of Ag+, CrOi-, NOT, and K+ after 30 mL of 0.010 M AgN03 is mixed with 20 mL of 0.010 M K2Cr04 and equilibrium is attained. Ksp for Ag2Cr04 is 1.1 x 10-l2. If precipitation did not occur the following concentrations would be found, allowing for the dilution upon mixing: [CrOi-] = (g)(O.OlO) = 0.0040 M; [K'] = 2[CrOi-] = 0.0080 M [Ag'] = [NOT] = ($)(0.010) = 0.0060 M Since K' and NO, do not react, the above values of [K'] and [NOT] will prevail whether or not precipitation occurs. To determine whether a precipitate will form calculate the ion product, Q, and compare its value to Ksp. Q = [Ag']2[CrOi-] = (0.0060)2(0.004)= 1.4 x lOP7 Ksp = 1.1 x 10-l2 Since Q >> Ksp, a precipitate will form. Only 0.003 0 mol/L of CrOi- is required to precipitate all the Ag'. The excess CrOi-, 0.001 0 mol/L, ensures that very little Ag' will remain in solution. The [CrOi-] equivalent to the remaining [Ag'] will also be very small compared to the 0.001 0 mol/L excess. Hence, after Ag2Cr04(s)precipitates: Ksp = 1.1 x 10-l2 = [Ag']2[CrOi-] = [Ag']2[0.001 0 1 [Ag'I2 = 1.1 x lOP9; [Ag'] = 3.3 x lOP5 The above value is small enough to justify the assumptions made. The final solution contains 3.3 x lO-'M Ag' and 0.001 0 M CrOi-. 18.11. Calculate the solubility of AgCN in a buffer solution of pH 3.00. Ksp for AgCN is 6.0 x lO-I7 and K , for HCN is 4.93 x 1 0 - l ~ . In this solution the silver that dissolves remains as Ag', but the cyanide that dissolves is converted mostly to HCN on account of the fixed acidity of the buffer. [The complex Ag(CN)F forms appreciably only at higher cyanide ion concentrations.] First calculate the ratio of [HCN] to [CN-] at this pH. The two equilibria can be combined to give an overall K for the dissolution process: AgCN(s) Subtract : * Ag' + CN- Ksp K, K =% !K, ! x = [CN-] ' -I7 HCN*Hf +CNAgCN(s) + H+ $ Ag+ + HCN 4.93 x 10-10 = 1.22 X 1 0 - ~ Let the solubility of AgCN be x mol/L; then x = [Ag') at equilibrium + [HCN] Very little error is made by neglecting [CN-] in comparison to [HCN] (1 part in 2 million) and equating [HCN] to x. 3 K = 1.22 x 1 0 - ~= [Ag'IWNl [H+l 1.00 x 10-3 X- x= 1.1 10-~ 18.12. Calculate the NH: ion concentration (derived from NH4C1) needed to prevent Mg(OH)2 from precipitating in 1 L of solution which contains 0.0100 mol of ammonia and 0.001 00 mol of Mg2+. The ionization constant of ammonia is 1.75 x lOP5.The solubility product of Mg(OH), is 7.1 x 10-l2. CHAP. 181 COMPLEX IONS; PRECIPITATES 305 First, find the maximum [OH-] that can be present in the solution without precipitation of Mg(OH)2. [Mg2+][OH-l2= 7.1 x 10-** [OH-] /- w2+1 $-- d 7.1 x 10-l2 = 7.1 x 10-l2 == 0.001 00 m = 8.4 X Now find the "Hi] (derived from NH4C1) needed to repress the ionization of NH3 so that the [OH-] will not exceed 8.4 x lO-'. From this, "Hi] = 2.1 x 10-3. (Since 0.010 0 M ammonia is only slightly ionized, especially in the presence of excess NH:, the [NH3] may be considered to be 0.0100.) 18.13. Given that 2 x 10-4 mol each of Mn2+ and Cu2+ was contained in 1 L of a 0.003 M HC104 solution, and this solution was saturated with H2S. Determine whether or not each of these ions, Mn2+ and Cu2+,will precipitate as the sulfide. The solubility of H2S, 0.10 mol/L, is assumed to be independent of the presence of other materials in the solution. Ksp of MnS is 3 x 10-14; of CuS, 8 x 10-37. K 1 and K2 for H2S are 1.0 x 10-7 and 1.2 x lO-I3, respectively. [H2S] = 0.10, since the solution is saturated with H2S; [H+] = 0.003, since the H2S contributes negligible H+ compared with HC104. Calculate [S2-] from the combined ionization constants. 0 10 IS2-] = (1.2 x 1 - 0 (1.2 x 10--20)0 2 =m ) - 1.3 x 10-l6 W+J2 (0.003)2 - Now form the ion product, Q, in each case, assuming that no precipitation occurs, and compare it to Ksp . For MnS: For CuS: Q = [Mn2'][S2-] = (2 x 10-4)(1.3 x 10-l6:)= 2.6 x 10-20 < 3 x 10-14 = Ksp Q = [Cu2'][S2-] = (2 x 10-4)(1.3 x 10-l6) = 2.6 x 10-20 > 8 x 10-37 = Ksp Thus MnS remains in solution while CuS precipitates. This example illustrates how two metal ions can be separated by controlling the acidity of the solution while adding a reagent. Actually, as the precipitation of CuS occurs, some additional H+ will be added from the H2S, and the equilibrium concentration of S2- will decrease, but not enough to change the outcome predicted above. This feature will be treated in greater detail in the following problem. 18.14. In Problem 18.13, how much Cu2+ escapes precipitation? Most of the Cu2+ will precipitate, and an increment of 2 :K 2 x 10-4 mol/L of Hf will be added to the solution from the H2S, bringing the total [H'] to 0.0034 M. Making this correction p2-1 = (1.2 x 10-20)- 0.10 = 1.0 x 10-l6 (0.003 4) Thus, 8 x 10-21 mol Cu2+ remains in solution. On a percentage basis the amount of Cu2+ remaining unprecipitated is 306 COMPLEX IONS; PRECIPITATES [CHAP. 18 18.15. If the solution in Problem 18.13 is made neutral by lowering the [H'] precipitate? to lOP7, will MnS [s2-] (1.2 x = 0.10 ~ o - ~ O ) - - = 1.2 (10-~)~ 10-~ Q = [Mn2'][S2-] = (2 x lOP4)(l.2x lOV7)= 2.4 x 10-" Since 2.4 x 10-I' is greater than the Ksp of MnS (3 x 10-14), MnS will precipitate. Lowering the [H'] to 10-7 increases the ionization of H2S to such an extent that sufficient S2- is furnished for the solubility product of MnS to be exceeded. 18.16. How much NH3 must be added to a 0.0040 M Ag' solution to prevent the precipitation of AgCl when [Cl-] reaches 0.001 O? Ksp for AgCl is 1.8 x 10-l' and Kd for Ag(NH& is 6.0 x 10-*. Just as acids may be used to lower the concentration of anions in solution, so complexing agents may be used in some cases to lower the concentration of cations. In this problem, the addition of NH3 converts most of the silver to the complex ion, Ag(NH3);. The upper limit for the uncomplexed [Ag'] without formation of a precipitate can be calculated from the solubility product. [Ag'][Cl-] = 1.8 x 10-" or [Ag+] = 1.8 x 10-Io - 1.8 x 10-" = 1.8 [cl-] 0.001 0 10-7 Enough NH3 must be added to keep the [Agf] below 1.8 x 10-7. The concentration of Ag(NH3)2f at this limit would then be 0.004 0 - (1.8 x lOP7), or practically 0.004 0. and [NH3]= 0.036. The amount of NH3 that must be added is equal to the sum of the amount of free NH3 remaining in the solution and the amount of NH3 used up in forming 0.0040 mol/L of the complex ion, Ag(NH3):. This sum is 0.036 + 2(0.004) = 0.044 mol NH3 to be added per liter. 18.17. What is the solubility of AgSCN in 0.003 0 M NH3? Ksp for AgSCN is 1.1 x 10-l2, and Kd for Ag(NH3)i is 6.0 x 1OP8. We may assume that practically all the dissolved silver will exist as the complex ion, Ag(NH3)z. Then, if the solubility of AgSCN is x mol/L, x = [SCN-] = [Ag(NH3);]. The concentration of uncomplexed Ag' can be estimated from Kd, assuming for the sake of simplification that [NH3]is essentially unchanged. This result validates our first assumption: The ratio of uncomplexed to complexed silver in solution is only 6.7 x 10-3. The two equilibria can be combined to give an overall K for the dissolution process: Subtract: + SCNAg(NH,); + Ag+ + 2NH3 AgSCN(s) + 2NH3 + Ag(NH3),f + SCNAgSCN(s) + Ag' KP S Kd =Ksp= Kd 1.1 x 10-I2 = 1.8 6.0 x 10-* 10-~ The validity of the second assumption is confirmed by the answer. If 1.3 x 10-5 mol of complex is formed per liter, the amount of NH3 used up for complex formation is 2(1.3 x lOP5) = 2.6 x 10-5 mol/L. The concentration of the remaining free NH3 in solution is practically unchanged from its initial value, 0.003 0 mol/L. CHAP. 181 COMPLEX IONS; PRECIPITATES 307 18.18. Calculate the simultaneous solubility of CaF2 and SrF2. Ksp for these two salts are 3.9 x 10-" and 2.9 x 10-9, respectively. The two solubilities are not independent of each other because there is a common ion, F-. We will first assume that most of the F- in the saturated solution is contributed by the SrF2, since its Kspis so much larger than that of CaF2. We can then proceed to solve for the so1ubi:lity of SrF2 as if the CaF2 were not present. If the solubility of SrF2 is x mol/L, x = [Sr2+],2x = [F-I. Then 4x3 = Ksp = 2.9 x lOP9 or x = 9 x 10-4 The CaF2 solubility will have to adapt to the concentration of F- set by the SrF2 solubility. S ! [Ca2+1= - E [F-I2 3.9 x 10-" = 1.2 (2 x 9 x 10-4)2 10-~ i.e., the solubility of CaF2 is 1.2 x 10-5 mol/L. Check cfassumption: The amount of F- contributed by the solubility of CaF2 is twice the concentration of Ca2', or 2.4 x 10-' mol/L. This is indeed small compared with the amount contributed by SrF2, 2 x 9 x lOP4 = 1.8 x 10-3 mol/L. A more general solution requiring no assumptions is as follows: Let x = [Ca2+];x[F-I2 = 3.9 x 10-"; [Sr2'][F-I2 := 2.9 x 10e9. Dividing: [Sr2']H Xpqf - - 2.9 x 10-9 = 74.4 3.9 x 10-11 [sr2+]= 7 4 . 4 ~ [F-] = 2([Ca2+]+ [Sr2+])= 2(x + 7 4 . 4 ~ = 2 x 7 5 . 4 ~ l 5 l x ) = Then substituting in the Ksp for CaF2: ( ~ ) ( 1 5 1 x ) ~ 3.9 x 10-I' = x = 1.2 x 10-' 7 4 . 4 ~ 8.9 x 10-4 = (the solubility of CaF,) (the solubility of SrF,) 18.19. Calculate the simultaneous solubility of AgSCN and AgBr. The solubility products for these two salts are 1.1 x 10-l2 and 5.0 x lO-I3. Since the solubilities are not very different the second approach of Problem 18.18 is required. Let x = [Br-1; then take the ratio of the Ksp's. [Ag'] Then substituting in Ksp for AgBr = [SCN-] + [Br-] = 2 . 2 + x = 3 . 2 ~ ~ [Ag+][Br-]= (3.2x)(x) = 5.0 x lO-I3; 3 . 2 = 5.0 x ~ ~ x = 4.0 x 10-7 (solubility of AgBr) (solubility of AgSCN) 2 . 2 = 8.8 x 10F7 ~ 18.20. Calculate the solubility of MnS in pure water. Ksp is 3 x 10-14. K1 and K2 for H2S are 1.0 x 10e7 and 1.2 x 10-13, respectively. This problem differs from the analogous problems dealing with chromates, oxalates, halides, sulfates, and iodates. The difference lies in the extensive hydrolysis of the sulfide ion. S2- + H20 + HS- + OH- Kb = -. K, 10-l~ = = 0.083 K2 1.2 x lO-I3 308 COMPLEX IONS; PRECIPITATES [CHAP. 18 If x mol/L is the solubility of MnS, we cannot simply equate x to [S2-]. Instead, x = [S2-] + [HS-]+ [H2S].To simplify, we first assume that the first stage of hydrolysis is almost complete and that the second stage proceeds to only a slight extent. In other words, x = [Mn2+] [HS-] = [OH-]. = x2 -[s2-] [HS-] [OH-] - 0.083 = Kb At equilibrium, Check o assumptions: f 1. p2-1 x2 = -= ( l a 4 10-5)2 = 2.4 10-9 0.083 0.083 [S2-] is indeed negligible compared with [HS-1. 2. [H*S]is also small compared with [HS-1. The above approximations would not be valid for sulfides like CuS which are much more insoluble than MnS. First, the water dissociation would begin to play an important role in determining [OH-]. Second, the second stage of hydrolysis, producing [H2S],would not be negligible compared with the first. Even for MnS, an additional complication arises because of the complexation of Mn2+ with OH-. The full treatment of sulfide solubilities is a complicated problem because of the multiple equilibria which must be considered. = 1.7 x 10-7, If hydrolysis had not been considered, the answer would have been simply underestimating the solubility by a factor of 80! (1.4 x 10-5/1.7 x 10-7 E 80.) Thus hydrolysis greatly increases the solubility of sulfides. 18.21. The following solutions were mixed: 500 ml of 0.0100 M AgN03 and 500 mL of a solution that was both 0.010 0 M in NaCl and 0.010 0 M in NaBr. Ksp for AgCl and for AgBr are 1.8 x 10-" and 5.0 x 10-13, respectively. Calculate [Ag'], [Cl-], and [Br-] in the equilibrium solution. If there were no precipitation, the diluting effect of mixing would make [Ag+]= [Cl-] = [Br-] = $(O.OlOO) = 0.005 0 AgBr is the more insoluble salt and would take precedence in the precipitation process. To find whether AgCl also precipitates, we may assume that it does not. In this case, only Ag+ and Br- would be removed by precipitation, and the concentrations of these two ions in solution would remain equal to each other. [Ag+][Br-]= [Ag+I2= Ksp = 5.0 x or [Ag'] = [Br-] = 7.1 x 10-7 We now examine the ion product for AgC1. [Ag+][Cl-]= (7.1 x 10-7)(5.0x lOP3) = 3.6 x 10-9 Since this ion product exceeds Ksp for AgCl, at least some AgCl must also precipitate. In other words, our first assumption was wrong. Since both halides precipitate, both solubility product requirements must be met simultaneously. [Ag+][Cl-] = 1.8 x 10-l' [Ag+][Br-]= 5.0 x 10-13 (1) (2) The third equation needed to define the three unknowns is an equation expressing the balancing of positive and negative charges in solution: "a+] + [Ag'] = [Cl-] + [Br-] + [NOT]. 0.010 0 + [Ag'] = [Cl-] + [Br-] + 0.005 0 CHAP. 181 COMPLEX IONS; PRECIPITATES 309 or [Cl-] + [Br-] - [Ag'] = 0.0050 (3) Dividing ( I ) by (2), [Cl-]/[Br-] = 360, and we see that Br- plays a negligible role in the total anion concentration of the solution. Also, [Ag'] must be negligible in (3) because of the insolubility of the two silver salts. We thus assume in (3) that [Cl-] = 0.0050. From ( I ) , 1.8 x 10-l' - 1.8 x 10-" . = 3.6 x 1OW8 0.005 0 From (2), [Br-] = 5.0 1 0 - l ~- [Ag+] - 5.0 10-l~= 10-5 3.6 x 10-* Check of assumptions: Both [Ag'] and [Br-] are negligible compared with 0.005 0. Note that in general, in the presence of both precipitates the ratio of the anion concentrations must be the same as the corresponding ratio of the Ksp's(as can be confirmed by the above results). Also observe that the addition of a few extra drops of AgN03 (not enough to decrease [Cl-] substantially) will produce more AgCl(s) but will not change the above answers. 18.22. How much Ag' would remain in solution after mixing equal volumes of 0.080 M AgN03 and 0.080 M HOCN? Ksp for AgOCN is 2.3 x 10-7. K , for I-IOCN is 3.5 x lOV4. The overall reaction is Ag' + HOCN * AgOCN(s) + H' (1) Assume this goes nearly to completion, then write the K for the reverse reaction by combining Ksp and K,. AgOCN(s) + Ag' + OCNKP S Subtract: HOCN + H' + OCN+ Agf Ka AgOCN(s) + H" + HOCN K =KP S Ka Because of the dilution upon mixing, the initial [Ag'] was 0.040 M, which would yield 0.040 M H' and leave no appreciable excess HOCN upon completion of reaction ( I ) . Hence if [Ag'] = x, then [HOCN] = x, and [H'] = 0.040 - x. K= x2 - - 10-7 x [Ag'][HOCN] - 2.3 - = 6.6 x 10-4 IH'I 0.040 - x 3.5 x 10-4 Solving the quadratic, x = 4.8 x 10-3 = remaining [Ag+]. Thus although stoichiometrically equivalent amounts of reagents were mixed, about one-eighth (4.8x 10-3/0.040) of the silver failed to precipitate. The H+ generated in the precipitation reaction suppressed the ionization of the HOCN with the result that an insufficient [OCN-] was available. Supplementary Problems COMPLEX IONS 1 . 3 If 0.010 0 mole of pure C U ( N H ~ ) ~ SH 2 0 is dissolved in a liter of pure water, (a) estimate the molar 82. . O~ concentration of Cu2+, neglecting the intermediate stages of dissociation. (b) Estimate the Cu2+ molar concentration if 0.0100 mole of NH3 was also added to the solution. (c) Which is the better estimate, (a) or (b)?Explain. Kd for Cu(NH3);' is 4.35 x 10-13. 310 COMPLEX IONS; PRECIPITATES [CHAP. 18 Ans. (a) 4.43 x 10F4; (b) 4.35 x lOP7; ( c ) Estimate (a) may be high if a significant quantity of NH3 is released in partial dissociation. Estimate (b) should be accurate because the added ammonia far exceeds all the ammonia released by the complex. 18.24. A 0.001 0 mol sample of solid NaCl was added to 1 L of 0.010 M Hg(NO3),. Calculate the [Cl-] equilibrated with the newly formed HgCl'. K1 for HgCl' formation is 5.5 x 106. Neglect the K2 equilibrium. Ans. 2 x 1 0 - ~ 18.25. What is the [Cd2+]in 1 L of solution prepared by dissolving 0.001 0 mol Cd(N03), and 1.5 mol NH3? Kd for the dissociation of Cd(NH3)i+ into Cd2+ and 4NH3 is 3.6 x 10-8. Neglect the amount of cadmium in complexes containing fewer than 4 ammonia groups. Ans. 7 x 1 0 - l ~ 18.26. Silver ion forms Ag(CN)i in the presence of excess CN-. How much KCN should be added to 1 L of a 0.0005 M Ag+ solution in order to reduce [Ag+] to 1.0 x Kd for the complete dissociation of Ag(CN)s is 3.3 x 10-21. Ans. 0.005 mol 18.27. An investigation of the complexation of SCN- with Fe3+ led to values of 130, 16, and 1.0 for K1, K2, and K 3 , respectively. What is the overall formation constant of Fe(SCN)3 from its component ions, and what is the dissociation constant of Fe(SCN)3 into its simplest ions on the basis of these data? Ans. K, = 2.1 x 103, Kd = 4.8 x lOP4 18.28. Sr2+ forms a very unstable complex with NO,. A solution that was nominally 0.001 00 M Sr(C104)2 and 0.050 M K N 0 3 was found to have only 75 percent of its strontium in the uncomplexed Sr2+ form, the , balance being Sr(N03)+.What is K 1 for complexation? Ans. 6.7 18.29. A solution made up to be 0.0100 M C O ( N O ~and 0.0200 M N2H4 at a total ionic strength of 1 was found )~ to have an equilibrium [Co2+]of 6.2 x 10-3. Assuming that the only complex formed was C O ( N ~ H ~ what , )~+ is the apparent K1 for complex formation at this ionic strength? Ans. 38 18.30. Suppose the solution in the preceding problem is diluted by a factor of 2 (while maintaining its ionic strength at l), calculate the molar concentrations of (a) CoN2Hz+, (b) Co2+, and ( c ) N2H4. Ans. (a) 0.001 2; (b) 0.003 8; (c) 0.008 8 18.31. Equal volumes of 0.001 0 M Fe(C104), and 0.10 M KSCN were mixed. Using the data in Problem 18.27, find the equilibrium percentages of the iron existing as Fe3+, FeSCN2+, Fe(SCN),f, and Fe(SCN)3. Ans. 8%, 50%, 40%, 2% 18.32. What is the concentration of free Cd2+ in 0.005 0 M CdC12?K 1 for chloride complexation of Cd2+ is 100; K2 need not be considered. Ans. 2.8 x 10-3 mol/L 18.33. (a) How many moles of NaCl would have to be added to a liter of the solution in problem 18.32 to reduce the free Cd2+ concentration to one-tenth of the total cadmium? (b) At what molar concentration of pure CdC12 in water is the free Cd2+ only one-tenth of the total cadmium? Ans. (a) 0.084 M; (b) 0.082 M CHAP. 181 COMPLEX IONS; PRECIPITATES 31 1 SOLUBILITY PRODUCT AND PRECIPITATION 18.34. Calculate the solubility products of the following compounds. The solubilities are given in moles per liter. (a) BaS04, 1.05 x 10-5 mol/L; (b) TlBr, 1.9 x 10-3 .mol/L; ( c ) Mg(OH)2, 1.21 x 10-4 mol/L; ( d ) Ag2C2041.15 x 10-4 mol/L; ( e ) La(I03)3, 7.8 x lOP4 mol/L. Ans. (a) 1.1 x lO-"; (b) 3.6 x 10-6; (c) 7.1 x ( d ) 6.1 x 10-l2; (e) 1.0 x 10-" 18.35. Calculate the solubility products of the following salts. Solubilities are given in grams per liter. (a) CaC204,0.0055 g/L; (b) BaCr04, 0.0037 g/L; ( c ) CaF:!, 0.017 g/L. Ans. (a) 1.8 x lOP9; (b) 2.1 x 10-''; ( c ) 4.1 x 10-" 18.36. The solubility product of SrF2 at 25 "C is 2.9 x 10-9. (a) Determine the solubility of SrF2 at 25 "C, in mol/L and in mg/mL. (b) What are [Sr2+]and [F-] (in mol/L) in a saturated solution of SrF2? Ans. (a) 9 x lOV4 mol/L, 0.1 1 mg/mL; (b) [Sr2+]= 9 x 10-", [F-] = 1.8 x 10-3 18.37. What [SO:--] must be exceeded to produce an RaS04 precipitate in 500 mL of a solution containing 0.000 10 mol of Ra2+? Ksp of RaS04 is 4 x 10-". A ~ 2~ 1 0 - ~ . 18.38. A solution has an Mg2+ concentration of 0.001 mol/L. Will Mg(OH)2 precipitate if the OH- concentration of the solution is (a) lO-' mol/L, (b) 10-3 mol/L? Ksp of Mg(OH)* is 7.1 x Ans. (a) no; (b) yes 18.39. Radioactive tracers provide a convenient means of measuring the small concentrations encountered in determining Ksp values. Exactly 20.0 mL of 0.0100 M AgN03 solution containing radioactive silver, with an activity of 29610 counts/min per mL, was mixed with 100 mL of 0.0100 M KI03, and the mixture diluted to exactly 400 mL. After equilibrium was reached, a portion of solution was filtered to remove any solids and was found to have an activity of 47.4 counts/min per mL. Calculate KSpof AgI03. Ans. 3.2 x 10-8 18.40. An old procedure for the determination of sulfur in gasoline involves the precipitation of barium sulfate as the final chemical step, and specifies that no more than 1 pg of sulfur be permitted to remain behind in solution. If the precipitation is made from 400 mL what must the concentration of excess barium ion be? Ksp for BaS04 is 1.1 x 10-''. Ans. [Ba2'-] must be at least 1.4 x 10-3 M 18.41. A solution contains 0.0100 mol/L each of Cd2+ and Mg2+. ( U ) To what pH should it be raised to precipitate the maximum amount of one metal (identify which it is) as the hydroxide without precipitating the other? (b) What fraction of the precipitated metal will still remain in solution? ( c ) To avoid accidentally precipitating any of the more soluble metal, a judicious chemist stops the addition of base when the pH is 0.50 units less than that calculated in part (a). Now what fraction of the less soluble metal remains in solution? K , values are 7.1 x 10-I2 for Mg(OH), and 4.5 x 1 0 - l ~for Cd(OH)2. A m . (a) 9.43, to precipitate Cd(OH)2; (b) 6.3 x 10-4; ( c ) 6.2 x 10-3 18.42. When ammonia is used to precipitate a metal hydroxide the solution becomes buffered by the ammonium ion formed, e.g.; Fe2+ + 2NH3 + 2 H 2 0 -+Fe(OI),(s)+ 2NHz To a liter of 0.010 0 M FeS04 0.040 0 moles of concentrated NH3 was added. (Consider any volume change to be negligible.) 312 COMPLEX IONS; PRECIPITATES [CHAP. 18 For Fe(OH)2, Ksp = 2 x 10-15, and for NH3, Kb = 1.75 x lOP5. Calculate (a) the final [Fe2+], and (b) the final pH. Ans. (a) 6 x 10-6; (b) 9.24 18.43. Repeat Problem 18.42 for Mg2+ rather than Fe2+.You will find it much more complicated because Ksp for Mg(OH)2 is 7.1 x 10-l2. (Hint: the higher order algebraic equation can be solved readily by trial and error.) Ans. (a) [Mg2+]= 0.004 1; (b) pH = 9.62 18.44. After seeing the results of Problem 18.43, a chemist decided to precipitate his Mg(OH)2 by simply adding 0.0400 moles per liter of NaOH to his 0.0100 M Mg2+ solution. (Besides, the calculation becomes very easy!). Calculate (a) the final [Mg2+],and (b) the final pH. (c) If the solution also contained 0.020 0 M Ca2+, would Ca(OH)* also precipitate? KSpfor Ca(OH)2 is 6.5 x 1OP6. Ans. (a) 1.8 x 10-'; (b) 12.30; (c) yes 18.45. Returning to Problem 18.43, the magnesium could have been thoroughly precipitated simply by using a sufficiently large amount of ammonia. Recalculate (a) and (b)for the case of 0.400 moles of NH3 rather than 0.0400 moles, and also answer question 18.44 ( c ) for this case. Ans. (a) [Mg2+]= 6.4 x 10d5; (b) pH = 10.52; (c) no, ion product << KSp. 18.46. After solid SrC03 was equilibrated with a pH 8.60 buffer, the solution was found to have [Sr2+]= 1.6 x 10-4. What is the solubility product for SrC03? K2 for carbonic acid is 5.61 x 10-". Ans. 5.6 x 10-l' 18.47. Calculate the solubility at 25 "C of CaC03 in a closed vessel containing a solution of pH 8.60. Ksp for CaC03 is 7.55 x 10F9.K2 for carbonic acid is 5.61 x 10-". ~ n s . 5.9 x 1 0 - ~ mol/L 18.48. How much AgBr could dissolve in 1 L of 0.40 M NH3? Ksp for AgBr is 5.0 x 10-13, and Kd for Ag(NH3): is 6.0 x 10-'. ~ n s . 1.2 x 1 0 - ~ mol 18.49. Solution A was made by mixing equal volumes of 0.001 0 M Cd2' and 0.007 2 M OH-solutions as neutral salt and strong base, respectively. Solution B was made by mixing equal volumes of 0.001 0 M Cd2+ and a standard KI solution. What was the concentration of the standard KI solution if the final [Cd2+]in solutions A and B was the same? KSpfor Cd(OH);! is 4.5 x 10-15, and K, for formation of Cdli- from its simple ions is 4 x 105. Neglect the amount of cadmium in all iodide complexes other than CdIi-. Ans. 2.3 M 18.50. Ag2S04 and SrS04 are both shaken up with pure water. KSp-valuesfor these two salts are 1.5 x 10-5 and 3.2 x 10-7, respectively. Evaluate [As'] and [Sr2+]in the resulting saturated solution. A ~ 3.1 . 10-~,2.1 x 1 0 - ~ ~ 18.51. Calculate [F-] in a solution saturated with respect to both MgF2 and SrF2 (made by dissolving both MgF2 and SrF2 in HzO until an excess of both solids remains). K,-values for the two salts are 6.6 x 10-9 and 2.9 x 10-9, respectively. 10-~ . hs.2.7 CHAP. 181 COMPLEX IONS; PRECIPITATES 313 18.52. Equal volumes of 0.0200 M AgN03 and 0.0200 M HCN were mixed. Calculate [Ag+] at equilibrium. Ksp for AgCN is 6.0 x IO-I7 and K, for HCN is 4.93 x 10-lo. 3.5 10-~ )~ 18.53. Equal volumes of 0.0100 M S T ( N O ~and 0.0100 M NaHS04 were mixed. Calculate [Sr2'] and [H'] at 2 equilibrium. Ksp for SrS04 is 3.2 x 10-7 and K, for HSOY (the same as K for H2SO4) is 1.2 x 10-2. Take into account the amount of H+ needed to balance the charge of the SO:- remaining in the solution. A ~ 6.7 . 1 0 - ~ , 4.8 ~ 10-~ 18.54. Excess solid Ag2C204 is shaken with (a) 0.001 00 M HN03, (b) 0.00030 M H N 0 3 . What is the equilibrium value of [Ag+] in the resulting solution? Ksp for Ag2C20,t is 6 x 10-l2; K2 for H2C204is 6.4 x lops. K l is so large that the concentration of free oxalic acid is of no importance in this problem. 18.55. How much solid Na2S203should be added to 1 L of water so that 0.000 50 mol Cd(OHh could just barely dissolve? Ksp for Cd(OH);? is 4.5 x lO-I5. K , and K2 for S20:- complexation with Cd2+ are 8.3 x 103 and 2.5 x 102, respectively. [As part of the problem, determine whether CdS203 or Cd(S203)i- is the predominant species in solution.] Ans. 0.23 mol 18.56. Precipitation titrations with AgN03 can be carried out using electrometric measurement of the [Ag'] (to be discussed in the next chapter). Consider the feasibility of titrating a solution containing 0.001 0 M NaCl and 0.001 0 M NaI with standard 0.100 M AgN03. Calculate the [Ag+] (a) halfway to the first end point, (b) at the first end point, (c) halfway to the second end point, ( d ) at the second end point, and (e) after the addition of an excess of titrant equivalent to the original NaCI. (Ksp for AgCl is 1.8 x 10-Io, and for AgI is 8.5 x 10-17. For simplicity's sake, ignore volume changes due to addition of ti trant .) Ans. (U) 1.7 x 10-13; (b) 9.2 x 10-9; (c) 3.6 x lO-'; ( d ) 1..3 x lOP5; ( e ) 1.0 x 10-3 In the problems below, use the following physical constants for H2S: Solubility = 0.10 mol/L Kl = 1.0 x 10-7 K2 = 1.2 x 18.57. What is the maximum possible [Ag'] in a saturated H2S solution from which precipitation has not occurred? Solubility product of Ag2S is 6.7 x 10-50. 7.5 10-l~ 18.58. Determine the [S2-] in a saturated H2S solution to which enough HCl has been added to produce an [H+] of 2 10-~. 3 10-1~ 18.59. Will FeS precipitate in a saturated H2S solution if the solution contains 0.01 mol/L of Fe2+ and (a) 0.2 mol/L of H+? (b) 0.001 mol/L of H+? K.7pof FeS is 8 x lO-I9. Ans. (a) no; (b) yes 18.60. Given that 0.001 0 mol each of Cd2+ and Fe2+ is contained in 1 L of 0.020 M HC1 and that this solution is saturated with H2S. Ksp of CdS is 1.4 x 10-29, of FeS, 8 x lO-.l9. ( a ) Determine whether or not each of these ions will precipitate as the sulfide. (b) How much Cd2+ remains in solution at equilibrium? Ans. (a) only CdS precipitates; (b) 4.7 x 10-I2 mol 314 COMPLEX IONS; PRECIPITATES [CHAP. 18 18.61. In an attempted determination of the solubility product of T12S, the solubility of this compound in pure CO2-free water was determined as 3.6 x 10-6 mol/L. What is the computed Ksp? Assume that the dissolved sulfide hydrolyzes practically completely to HS-, and that the further hydrolysis to H2S can be neglected. Ans. 8 x 1OP2' 18.62. Calculate the solubility of FeS in pure water. Ksp = 8 x lO-I9. (Hint: The second stage of hydrolysis, producting HZS, cannot be neglected.) ~ n s . 4 x 1 0 - ~mol/L Chapter 19 EIectrochemistry In this chapter we deal with two aspects of the connection between chemistry and electricity: electrolysis, or the decomposition of matter accompanying the passage of electricity through it, and galvanic cell action, or the role of a chemical reaction as an electric generator. ELECTRICAL UNITS The coulomb (C) is the SI unit of electrical charge. From the point of view of fundamental particles, the elementary unit is the charge of one proton, which is equal in magnitude to the charge of one electron. No chemical particle is known whose charge is not an integral multiple of this elementary charge, the value of which is 1.602 x 10-19C. Electric current is the rate of flow of charge. The SI unit, the ampere (A), is a flow rate of one coulomb per second (1 A = 1 C/s). The electrical potential dzference between two points in a circuit causes the transfer of charge from one point to the other. The volt (V) is the SI unit of electrical potential difference. When a charge of 1 C moves through a potential difference of 1 V it gains 1 J of energy: Energy in joules = (charge in coulombs) x (potential difference in volts) = (current in amperes) x (time in seconds) >: (potential difference in volts) (19-1) The watt (W) is the SI unit of power (electrical and other); one watt is developed when 1 J of work is performed in a time of 1 s. From (Zg-Z), Power in watts = (current in amperes) x (potential difference in volts) (19-2) In electrochemistry we are normally concerned with direct current (DC). However the above discussion of units applies equally well to alternating current (AC), the conventional type of power in the home and laboratory. FARADAY'S LAWS OF ELECTROLYSIS 1. The mass of any substance liberated or deposited at an electrode is proportional to the electrical charge (i.e., the number of coulombs) that has passed through the electrolyte. 2. The masses of different substances liberated or deposited by the same quantity of electricity (i.e., by the same number of coulombs) are proportional to the equivalent masses (Chapter 12) of the various substances. These laws, found empirically by Faraday over half a century prior to the discovery of the electron, can now be shown to be simple consequences of the electrical nature of matter. In any electrolysis, a reduction must occur at the cathode to remove electrons flowing from the external circuit into the electrode and an oxidation must occur at the anode to supply the electrons that leave the electrolytic cell at this electrode. By the principle of continuity of current, electrons must be discharged at the cathode at exactly the same rate at which they are supplied to the anode. By definition of the equivalent mass for oxidation-reduction reactions (that fraction of the molar mass associated with the transfer of one mole of electrons), the number of equivalents of electrode reaction must be proportional to the amount of charge transported into or out of the electrolytic cell and must, indeed, be equal to the number of moles of electrons transported in the circuit. The Faraday constant (9) is equal to the charge of one mole of electrons: 315 316 ELECTROCHEMISTRY [CHAP. 19 9 = (1.602 x C/electron)(6.022~ electrons/mol) = 9 . 6 5 104C/mol 1023 ~ The usual symbol, n(e-), may be used to refer to the number of moles of electronic charge, i.e., the number of equivalents. The equivalent mass needed for electrolytic calculations can be found by writing the balanced halfreactions for the electrode processes. Thus in the electrolytic reduction of Cu2+,the cathode reaction is c u 2 + + 2e+ cu The equivalent mass of copper is the molar mass. If a solution of C u + were electrolyzed, the equivalent mass would be the molar mass of copper, because only one electron is captured per copper atom formed: Cu+ + e- + Cu Special information about the electrode reactions is often needed in order to calculate the equivalent mass for electrolysis, just as in ordinary oxidation-reduction reactions. If a solution containing Fe3+ is electrolyzed at low voltages, the electrode reaction for the iron might be Fe3+ + e-+Fe2+ and the equivalent mass of iron would equal the molar mass. At higher voltages, the reaction might be Fe3+ + 3e+ Fe and the equivalent mass of iron would be one-third of the molar mass. GALVANIC CELLS Many oxidation-reduction reactions may be carried out in such a way as to generate electricity. Such an arrangement for the production of an electric current is called a gulvanic or electrochemical cell. In principle, this may always be done for spontaneous, aqueous, oxidation-reduction reactions for which the following experimental requirements are met. 1. The oxidizing and reducing agents are not in physical contact with each other but are contained in separate compartments, called half-cells. Each half-cell contains a solution and a conductor (electrode), which is usually a metal. 2. The reducing or oxidizing agent in a half-cell may be either the electrode itself, a solid substance deposited on the electrode, a gas which bubbles around the electrode, or a solute in the solution which bathes the electrode. Just as in electrolysis, the electrode at which reduction occurs is termed the cathode and that at which oxidation occurs is termed the anode. 3. The solutions of the two half-cells are connected in some way that allows ions to move between them. Among the possible arrangements to accomplish this are ( a ) careful layering of the less dense solution over the denser one; (b) separation of the two solutions by a porous substance, such as fritted glass, unglazed porcelain, or a fiber permeated with some electrolyte solution; (c) insertion of a connecting electrolyte solution (a salt bridge) between the two solutions. The potential developed across the two electrodes causes an electric current to flow and the half-cell reactions to proceed if the electrodes are connected to each other by an outside conducting circuit. Figure 19-l(a) depicts a simple galvanic cell in which each half-cell consists of a metal in contact with a solution of its ions. The direction of the reaction, direction of current, and the open-circuit voltage have been deduced by methods described in the following sections. There are different structures for halfcells depending on the states of matter of reactants and products. Figure 19-l(b) depicts the hydrogen half-cell, in which the reduced form (H2 gas) is adsorbed onto the surface of platinum, an electrical conductor, which is itself chemically inert. Figure 19-l(c) shows a half-cell in which both the oxidized CHAP. 191 ELECTROCHEMISTRY 317 and reduced forms are in solution and an inert conductor, Pt, carries electrons to or from the external circuit. This chapter deals primarily with reactions in aqueous media, but the principles can be extended to such technical developments as fuel cells and high-temperature batteries with exotic nonaqueous electrolytes. STANDARD HALF-CELL POTENTIALS The reaction occurring in each half-cell may be represented by an ion-electron partial equation of the type described in Chapter 11. The whole-cell operation involves a flow of electrons in the external circuit. The electrons generated in the oxidation half-reaction enter the anode, travel through the external circuit to the cathode, and are consumed at the cathode by the reduction half-reaction. From the electrical principle of equal current at all points in a nonbranching circuit, the number of electrons generated in the oxidation must exactly balance the number of electrons consumed in the reduction. This demands the same rule for combining two half-reactions into a balanced whole equation that was used in Chapter 11. In the half-cell of the reducing agent, the oxidation product accumulates during operation of the cell. The reducing agent together with its oxidation product, known as a couple, are thus found in the same compartment during cell operation. Similarly, the other half-cell contains a couple consisting of the oxidizing agent and its reduction product. An arbitrary couple, consisting of both product and reactant of an oxidation-reduction half-reaction, may sometimes be the reducing part of a galvanic cell and sometimes the oxidizing part, depending on what the other couple is. The (Fe3+/Fe2+)couple, for example, takes an oxidizing role when paired against the strongly reducing (Zn2+/Zn) couple, and a reducing role when paired against the strongly oxidizing (Ce4+/Ce3+) couple. (The oxidized form is listed first.) Each couple has an intrinsic ability to consume electrons. This ability can be assigned a numerical value, called the electrode potential. Electrode potentials may be represented by symbols such as E(Fe3+/Fe2+)or E(Zn2+/Zn). When two couples are combined into a whole cell, the couple with the higher electrode potential provides the oxidizing agent, absorbs electrons from the external circuit at its electrode, and is positive.The other couple provides the reducing agent, delivers electrons from its electrode to the external circuit and is negative. The driving force for the flow of current is the algebraic 318 ELECTROCHEMISTRY [CHAP. 19 Table 19-1. Standard Electrode Potentials at 25C Reaction E"JV 2.87 1.96 1.92 1.763 1.70 1.51 1.358 1.25 1.23 1.229 1.065 1.002 0.915 0.799 1 0.771 0.695 0.535 0.520 0.361 0.34 0.34 0.247 0.18 0.15 0.017 0.000 0 0.00 -0.126 -0.14 - 0.257 -0.336 - 0.403 -0.44 - 0.762 6 -2.713 - 3.040 Pd2+ + 2e- -+ Pd Ag+ + e- -+ Ag Fe3+ + e - -+ Fe2+ 0, + 2H' + 2e- -+ H,O, 12(s)+ 2e- -+ 21Cu+ + e- -+Cu Fe(CN)2- + e - -+ Fe(CN):Co(dip): + e - -+Co(dip): Cu2+ + 2e- - + C u G e 2 + + 2e- - + G e PdIi- + 2e- -+ Pd + 41Sn4+ + 2e- -+ Sn2+ Ag(S203);- + e- Ag + 2S20:2 H + + 2e- -+ H, Ge4+ + 2e- -+ Ge2+ Pb2+ + 2e- --+ Pb Sn2+ + 2e- -+ Sn Ni2+ + 2e- -+Ni Tlf + e - +T1 C d 2 ++ 2e- + C d Fe2+ + 2e- -+ Fe Zn2+ + 2e- - t Z n Na+ + e- -+ Na Li+ + e- -+ Li + + + 2e- 2F+ 2e- 2SO:c o 3 + + e- CO2+ 2H,O H,02 + 2 H + + 2e(in 1 M HClO,) Ce4+ + e- - i c e 3 + MnO, + 8 H + + 5e- M n 2 + + 4 H 2 0 C1, + 2e- 2C1T13++ 2e- T1+ MnO, + 4 H + + 2e- -+ Mn2+ + 2 H 2 0 0, + 4 H + + 4e- 2H,O Br, + 2e- 2BrAuCl, + 3e- -,Au + 4ClF, -+ S,Oi- -+ -+ -+ -+ + -+ + -+ dzflerence between the two electrode potentials and is equal numerically to the voltage output of the cell (measured under the limiting operating condition of zero current). The numerical value of an electrode potential depends on the nature of the particular chemicals, on the temperature, and on the concentrations of the various members of the couple. For purposes of reference, half-cell potentials are tabulated for standard states of all the chemicals, defined as 1 atm + CHAP. 191 ELECTROCHEMISTRY 319 pressure for each gas,* the pure substance for each liquid or solid, and 1 mol/L for every nongaseous solute appearing in the balanced half-cell reaction. Such reference potentials are called standard electrode potentials and are designated by the symbol E". The same symbol is used for the standard potential of the whole cell, the value that could be realized in measurement at the standard states of all reactants and products. A partial listing of standard electrode potentials at 25C is given in Table 19-1. Since only dzferences between two electrode potentials can be measured b:y the cell voltage, there is an arbitrary zero point on this scale, assigned to the potential of the standard hydrogen electrode, (H+/H2). Some values in Table 19-1 are not in complete agreement with those in other tabulations, but the numerical work in this book is all consistent with the values listed. A couple with a large positive E", like (F2/F-), is strongly oxidizing, captures electrons from the electrode, and is experimentally positive with respect to a standard hydrogen electrode. Conversely, a couple with a large negative E", like (Li"/Li), is strongly reducing; that is, it undergoes oxidation and thereby transfers electrons to the electrode. Such a standard half-cell is experimentally negative with respect to a standard hydrogen electrode, since the external circuit receives electrons from the half-cell. COMBINATIONS OF COUPLES There is another way of combining two half-cell reactions, in which the electrons do not cancel. This cannot correspond to a whole cell, where the electrons always cancel. The situation is a hypothetical one often used to calculate an unknown half-cell potential on the basis of two known half-cell potentials. In such a case the electron number cannot be left out of consideration. As discussed in the next section, if n is the number of electrons in the half-reaction, nE" is proportional to the free energy associated with it. As in Chapter 16, when one adds two reactions one may also add their free energies to get the free energy of the overall reaction. The rule for this case is as follows: If two reduction half-reactions are added or subtracted to give a third reduction half-reaction, the two nE" products are added or subtracted correspondingly to give the nE"-value for the resulting half-reaction. EXAMPLE 1 Calculate E" for the (Fe3+/Fe) couple using values from Table 19-1. E" n V - nE" - 0.88 Fe2+ + 2eFe3+ + eSUM: -+ -+ Fe Fe2+ Fe - 0.44 0.77 2 1 3 V Fe3+ + 3e- - 0.77 -0.11 + v Since nE" for the resulting half-reaction is -0.1 1 V, and n is 3 , E" must be -0.1 1/3 = -0.04V This rule makes it possible to reduce the length of compiled tables, since many half-reactions can be computed even if they are not tabulated. FREE ENERGY, NONSTANDARD POTENTIALS, AND THE DIRECTION OF OXIDATION-REDUCTION REACTIONS The discussion following (16-4) indicated that the decrease in free energy of a system can be equated to the maximum amount of work the system can perform at constant temperature and pressure in forms other than expansive or contractive work. At this point, we can make use of that principle by noting from (19-1) that the electrical work performed by a galvanic cell is equal to the voltage times the electrical charge transferred at either electrode. For the passage of n moles of electrons, the charge transferred is n F . Thus, the electrical work in joules equals n 9 E . The maximum amount of work * The convention of a 1 bar standard state is recognized but changes in the values would be exceedingly small. 320 ELECTROCHEMISTRY [CHAP. 19 performed by a galvanic cell is the value of n F E under conditions where the electrode processes are reversible. Reversibility may be approximated for many electrodes if the cell potential is measured at very low current; such measurements indeed are the basis of the values used in the construction of tables like Table 19-1. Since the electrical work is the only form of work other than expansive or contractive work performed in the typical galvanic cell, the free-energy principle may be stated as AG = - n F E (19-3) The dependence of a cell potential on the concentrations of the reactants and products may be derived from the known dependence of AG upon concentration, (16-6). (19-4) where, as in Chapter 16, R = 8.3145 J/K when AG is expressed in joules. When all reactants and products are in their standard states, Q = 1 (see Chapter 16) and E = E"; thus A O = -nFE". SubG stituting back into (19-4), we obtain the Nernst equation: RT E=E"--1nQ (19-5) nF When the constants are combined and the natural logarithm converted to a common logarithm, the expression for the potential (in volts) at 25 "C becomes In (19-6), n has been made dimensignless. For a half-reaction, n is the number of electrons in the halfequation; for a whole-cell reaction, n is the number of electrons in one of the multiplied half-equations before canceling the electrons. The Nernst equation is closely related to the laws of chemical equilibrium. Le Chatelier's principle applies to the potential of a cell in the same sense as it applies to the yield of an equilibrium process. Since Q has product concentrations in the numerator and reactant concentrations in the denominator, an increased concentration of product reduces the potential and an increased concentration of reactant raises the potential. The same type of equation may be used to describe the concentration dependence of the potential of a single half-cell, i.e., of an electrode potential. In this case Q , the reaction quotient, contains terms in the numerator corresponding to the products of the balanced half-cell reaction written as a reduction, terms in the denominator corresponding to the reactants, but no terms for the electron. For example: [Fe2+] E( Fe3+/Fe2+)= E"( Fe3+/Fe2+) 0.059 2 log [Fe3+] 0.059 2 [Mn2+] E( MnO; /Mn2+) = E" (Mn0; /Mn2+)- 7 log [MnO;] [H+I8 Spontaneous oxidation-reduction reactions If the potential of a whole cell is positive, the free-energy change is negative, by (19-3). Then the corresponding oxidation-reduction reaction is spontaneous as written, and a galvanic cell would operate spontaneously, with electrons provided to the external circuit at the half-cell where oxidation takes place. If the potential is negative, the free-energy change is positive, and the corresponding reaction does not take place spontaneously. These statements about the direction of spontaneous reaction are valid whether the reaction is set up as a galvanic cell or as an ordinary process where the reactants and products are all mixed in the same vessel, since the free-energy change for a reaction depends on the concentrations but not on the way in which the reaction is carried out. In particular, if both reduced and oxidized members of the two couples are mixed, all substances being at standard states (unit relative concentrations), any reducing agent can reduce an oxidizing agent occurring higher in the table of CHAP. 191 ELECTROCHEMISTRY 32 1 standard electrode potentials. The same rule of relative position may be applied for general values of the concentrations, with E for each half-cell, as computed from the Nernst equation, replacing E". Usually the qualitative prediction based on E"-values is not changed even for moderate deviations from standard states if the two E"-values are separated by at least several tenths of a volt. It should be noted that predictions under this rule indicate what reactions might occur but say nothing about the rate at which they do occur. Electrode reactions in electrolysis A nonspontaneous oxidation-reduction reaction, for which the calculated cell potential is negative, may be induced by electrolysis, i.e., by using an external electrical potential to force electrons into the couple undergoing reduction and to extract electrons from the couple undergoing oxidation. The minimum external potential required for electrolysis has the magnitude of the computed cell potential of the reaction. (The actual electrolyzing potential exceeds this minimum because of the irreversibility of electrode processes occurring at nonzero rates.) Under nearly reversible conditions, we have the following rule: Of all possible reductions that might occur electrolytically at a cathode, the one for which the corresponding electrode potential is algebraically greatest is the most likely. Conversely, the most probable anodic oxidation is that for which the corresponding electrode (reduction) potential is algebraically the least. In applying the rule, the student should keep in mind that: (a) a solute molecule or ion may undergo oxidation or reduction; (b) the anode electrode may itself undergo oxidation; (c) the solvent may undergo oxidation or reduction. EXAMPLE 2 Let us illustrate possibility (c) for water at 25 "C. Reduction to molecular hydrogen is found in Table 19-1: 2H+ + 2e-+ H2 E" = 0.OOOOV Assume that H2 gas is allowed to accumulate to a partial pressure of 1 atm. In neutral solutions, where [H+]= 10-7 and the H2 pressure retains its unit value, E=E"-- 1 0.0592 P(H2) log - 0.000 0 - 0.029 6 log= 2 P+I2 (10-7)' = 0.000 0 - 0.029 6 x 14 = -0.414 V Water is thus more difficult to reduce, but hydrogen easier to oxidize, in neutral solutions than in acids. For the oxidation of water to molecular oxygen, the most appopriate entry in Table 19-1 is O2 + 4H++ 4e4 2H20 .E"= 1.229 V The standard states to which this E"-value refers are 1 atm for oxygen gas and 1 mol/L for H + . We can calculate E for the above half-cell for neutral solutions, in which [H'] = 10-7, by using the Nernst equation. Assuming that the oxygen remains at its standard state, P ( 0 2 ) = 1 atm, E=E"-1 = 1.229 - 0.0148 log P+ ~ ( 0 , 3 ) = 1.229 - 0.0148 x 28 = 0.815 V 0.059 2 4 1% 1 ~ (10-7)4 It is thus harder for oxygen to be reduced, but easier for water to be oxidized, in neutral than in acid solutions. USE OF E"'-VALUES The above examples show that practical values for electrode potentials may be very different from E'-values for reactions in almost-neutral solutions involving H or OH-. Since most reactions in + 322 ELECTROCHEMISTRY [CHAP. 19 cellular biology occur in solutions near pH 7, it has become customary for biochemists to define E'' as an electrode potential at which all reactants but H + and OH- are at unit concentration and [H+] is 10-7. The values calculated in the previous section, -0.414V and 0.815 V, may be considered the E'' values for the Hf/H2 and the 0 2 / H 2 0couples, respectively. Then, for any reaction carried out in a buffer solution at pH 7, a modified form of the Nernst equation may be given for the electrode potential at 25 "C: At p H 7 : E=Eo'-- 0.059 2 [reduced] n log [oxidized] (19-7) In (19-3, [reduced]/[oxidized] is similar to Q but does not include [H+] or [OH-] terms. Since in practice the concentrations of the principal oxidized and reduced species (other than H + and OH-) are often of the same order of magnitude, E-values do not differ much from E''-values. Another advantage of using an E''-value that incorporates as a standard state the [H+] at a pH of 7 is that we need not worry about the various states of ionization of the oxidized or reduced species. The relative distribution of ionization states is fixed at constant pH, and it has become customary in biochemical applications to define [reduced] or [oxidized] as the relative stoichiometric concentration, i.e., the numerical value of the sum of the molar concentrations of all the ionization states of the reductant or oxidant. Solved Problems ELECTRICAL UNITS 19.1. A lamp draws a current of 2.OA. Find the charge in coulombs used by the lamp in 30s. Charge in coulombs =(current in amperes) x (time in seconds) = 2.0 A) (30 s) = 60 C 19.2. Compute the time required to pass 36 000 C through an electroplating bath using a current of 5 A. Time in seconds = charge in coulombs - 36000C = 7200 s = 2 h current in amperes 5A ~ 19.3. A generator delivers 15A at 120V. (a) Compute the power in kilowatts supplied by the generator. (b) How much electric energy, in kilowatt-hours, is supplied by the generator in 2h? (c) What is the cost of this energy at 6,4 per kilowatt-hour? (4 (b) Power = (15A)(120 V) = = 1800 W = 1.8 kW Energy (1.8 kW)(2 h) = 3.6 kW . h 22# (4 Cost = (3.6 kW h)(6#/kW. h) 19.4. A resistance heater was wound around a 50-g metallic cylinder. A current of 0.65A was passed through the heater for 24s while the measured voltage drop across the heater was 5.4V. The temperature of the cylinder was 22.5 "C before the heating period and 29.8 "C at the end. If heat CHAP. 191 ELECTROCHEMISTRY 323 losses to the environment can be neglected, what is the specific heat of the cylinder metal in J/g K? * BY ( 1 9 - 0 , Energy input = (0.65 A) (5.4 V) (24 s) = 84 J But, also, Energy input = (mass) x (specific heat) >: (temperature rise) Therefore, 84 J = (50 g)(specific heat)[(29.8 - 22.5) K] from which specific heat= 0.23 J/g - K. 19.5. How many electrons per second pass through a cross section of a copper wire carrying 10-l6 A? Since 1 A = 1 C/s, Rate = 10-l6 c / s = 600 electron/s 1.6 x 10-19 C/electron FARADAY'S LAWS OF ELECTROLYSIS 19.6. Exactly 0.2 mol of electrons is passed through three electrolytic cells in series. One contains silver ion, one zinc ion, and one iron(II1) ion. Assume that the only cathode reaction in each cell is the reduction of the ion to the metal. How many grams of each metal will be deposited? One mol of electrons liberates 1 eq of an element. Equivalent masses of Ag', -- Zn2+, and Fe3+ are 107.9 - 107.9 1 -= 65 39 2 32.70 -= 18.62 55.85 3 respectively. Ag deposited = (0.2 mole-)( 107.9g/mol e-) = 21.58 g Zn deposited = (0.2 mol e-)(32.70 g/niol e - ) = 6.54 g Fe deposited = (0.2 mole-)( 18.62g/niol e-) = 3.72 g 19.7. A current of 5.00 A flowing for exactly 30 min deposits 3.048 g of zinc at the cathode. Calculate the equivalent mass of zinc from this information. Number of coulombs used = (5.00 A)[(30x60)s] = 9 . 0 0 103C ~ n ( e - ) used = 9*00 lo3 = 0.093 3 mol e9.65 x 104C/mol e- 3.048 g Equivalent mass =mass deposited by 1 mol e- = - 3 mole- = 32.7g/eq 0.093 324 ELECTROCHEMISTRY [CHAP. 19 19.8. A certain current liberates 0.504 g of hydrogen in 2 h. How many grams of oxygen and of copper (from Cu2+ solution) can be liberated by the same current flowing for the same time? Masses of different substances liberated by the same number of coulombs are proportional to their equivalent masses. Equivalent mass of hydrogen is 1.008; of oxygen, 8.00 (see below); of copper, 31.8. 0.504 g = 0.500 eq Number of eq of hydrogen in 0.504 g = 1.008g/eq Then 0.500eq each of oxygen and copper can be liberated. Mass of oxygen liberated = 0.500eq (8.00 g/eq) = 4.00 g Mass of copper liberated = 0.500eq (31.8 g/eq) = 15.9 g The equivalent mass of any substance in an electrolysis is determined from the balanced electrode reaction. For the liberation of oxygen from water the anode reaction is 2 H 2 0 -, O2 + 4H' + 4e- The molar mass of 0 2 is 32.00. The equivalent mass is the molar mass divided by the number of electrons which must flow to produce one molecule, or 32.00/4 = 8.00. 19.9. The same quantity of electricity that liberated 2.158 g of silver was passed through a solution of a gold salt and 1.314 g of gold was deposited. The equivalent mass of silver is 107.9. Calculate the equivalent mass of gold. What is the oxidation state of gold in this gold salt? Number of eq of Ag in 2.158 g = 2*1 58 107.9g/eq Then 1.314g of Au must represent 0.020 00 eq, and so Equivalent mass of Au = 1.314g 65.70 g/eq 0.020 00 eq = 0.020 00 eq Ag Oxidation State = number of electrons needed to form one gold atom by reduction - molar mass of Au 197.0 - -=3 65.7 equivalent mass of Au 19.10. How long would it take to deposit 100 g A1 from an electrolytic cell containing A1203at a current of 125A? Assume that A1 formation is the only cathode reaction. Equivalent mass of A1 = n(e-) = (i molar mass) = f (27.0) = 9.0 g Al/mol e- l o O g A 1 = 1l.lmol e9.0g Al/mol e- charge - (1 1.1 mol e-)(9.65 x 104C/mol e - ) = 8.6 x 103s = 2.4 h Time = 125A current 19.11 A current of 15.0A is employed to plate nickel in a NiS04 bath. Both Ni and H2 are formed at the cathode. The current efficiency with respect to formation of Ni is 60 percent. (a) How many grams of nickel are plated on the cathode per hour? (6) What is the thickness of the plating if the cathode consists of a sheet of metal 4.0 cm square which is coated on both faces? The density of nickel is 8.9g/cm3. ( c ) What volume of H2 (S.T.P.) is formed per hour? (4 Total number of coulombs used= (15.0 A) (3 600 s) = 5.40 x 104C CHAP. 191 ELECTROCHEMISTRY 325 n(e-) used = 5'40 lo4 = 0.560mol e9.65 x 104C/mole- Number of eq Ni deposited = (0.60)(0.560mol e - ) ( 1 eq Ni/mol e-) = 0.336 eq Ni Equivalent mass of Ni = f(mo1ar mass) = i(58.69) = 29.3 g/eq Mass of Ni deposited = (0.336 eq)(29.3 g/eq) = 9.8 g Area of two faces = 2(4.0 cm)(4.0 cm) = 32 cm2 mass 9.,8g = - 1.10cm3 = Volume of 9.8 g Ni = - densltY 8.9 g/cm3 volume 1.10 _--_ - 0.034cm Thickness of plating = area 32cm2 (4 Number of eq H2 liberated = (0.40)(0.560 F ) ( l eq H 2 / F ) = 0.224 eq H2 Volume of 1 eq (i.e., i mol) H2 = i (22.4 L) = 11.2 L H2 Volume of H2 liberated = (0.224eq)(11.2 L/eq) = 2.51 L H2 Note the use of the symbol 9, Faraday unit, which is synonymous with the unit, mol ethe 19.12. How many coulombs must be supplied to a cell for the electrolytic production of 245 g of NaC104 from NaC103? Because of side reactions, the anode efficiency for the desired reaction is 60 percent. First it is necessary to know the equivalent mass of NaC104 for this reaction. The balanced anode reaction equation is ClO, + H20 --+ ClO, + 2H'+ + 2e- Equivalent mass of NaC104 = molar mass 122.5 -= 61.2 number of electrons transferred - 2 245 g Number of eq NaC104 = - 4.00eq NaClQ 61.2 g/eq n (e- ) required = 4.00 eq = 6..7mole0.60 eq anode product/mol e- Number of coulombs required = (6.7 mol e-)(9.6 x 104C/mole-) = 6.4 x 105C. GALVANIC CELLS AND ELECTRODE PROCESSES 19.13. What is the standard potential of a cell that uses the (Zr?+/Zn) and (Ag+/Ag) couples? Which couple is negative? Write the equation for the cell reaction occurring at unit relative concentrations (the standard states). The standard electrode potentials for (Zn2+/Zn) and (Agf/Ag) are, from Table 19-1, -0.763V and +0.799V, respectively. The standard potential of the cell is the difference between these two numbers, 0.799-(-0.763) = 1.562V. The silver electrode potential is higher, and thus silver ion is the oxidizing agent. The zinc couple provides the reducing agent and is the negative electrode. The equation is Zn + 2Ag+ 3 Zn2+ + 2Ag 326 ELECTROCHEMISTRY [CHAP. 19 19.14. Can Fe3+ oxidize Br- to Br2 at unit relative concentrations? From Table 19-1, the (Fe3+/Fe2') couple has a lower standard electrode potential, 0.771 V, than the (Br2/Br-) couple, 1.065V. Therefore Fe2+ can reduce Br2, but Br- cannot reduce Fe3+; i.e., Fe3+ cannot oxidize Br-. On the other hand, I-, occurring at a much lower standard electrode potential, 0.535 V, is easily oxidized by Fe3+ to 12. 19.15. What is the standard electrode potential for (MnO;/Mn02) in acid solution? ([H+]= l.0OM). The reduction half-reaction for this couple is MnO, + 4H'+3e- --+ M n 0 2 + 2H20 which can be written as the difference of two half-reactions whose electrode potentials are listed in Table 19- 1 . nE"-values may be correspondingly subtracted. n E" nE" MnO, + 8H+ MnOz + 4H+ Difference : MnO, Rearranging, MnO, - + 5e+2e- --+ --+ Mn2+ + 4 H 2 0 Mn2+ + 2 H 2 0 Mn02 + 4H++3e- --+ + 4H' + 3e- 2 H 2 0 3 5.09 V M n 0 2 + 2H20, the desired reaction, in which n = 3. --+ - 5 2 1.51 V 7.55 1.23 2.46 5 09 E" for the desired reaction = -= 1.70 V 3 19.16. Predict the stabilities at 25 "C of aqueous solutions of the uncomplexed intermediate oxibation states of (a) thallium and (b) copper. (a) The question is whether the intermediate state, Tl', spontaneously decomposes into the lower and higher states, T1 and T13+.The supposed disproportionation reaction, 3Tl' --+ 2T1+ T13+ E" (red) for the couple (Table 19-1) could be written in the ion-electron method as 2 x (T1+ + e- --+ T1) TI+ --+ ~ 1 +~2e-+ -0.336 V 1.25 V In (I), the (Tl'/Tl) couple functions as oxidizing agent; in (2), the (T13'/Tl+) couple functions as reducing agent. The reaction would occur at unit concentrations if E" for the reducing couple were less than E" for the oxidizing couple. Since 1.25 V is greater than -0.336 V, the reaction cannot occur as written. We conclude that Tl' does not spontaneously decompose into T1 and T13? On the contrary, the reverse reaction is a spontaneous one: 2T1+ T13+ -+ 3T1+ This means that Tl(II1) salts are unstable in solution in the presence of metallic T1. The supposed disproportionation of Cu+ would take the following form: 2cu+ -+ c u + CU2+ The ion-electron partial equations are E" (red) for the couple (Table 19-1) Cu+ + eCU+ --+ --+ Cu c u 2 + + e- 0.520 V 0.160 V (computed) CHAP. 191 ELECTROCHEMISTRY 327 This process could occur if E" for the supposed reducing couple, (Cu2+/Cut), were less than E" for the oxidizing couple, (Cu+/Cu). Indeed, +0.16V (computed by the method of Problem 19.15) is less than 0.52 V. Therefore Cu+ is unstable to disproportionation in solution. Compounds of Cu(1) can exist only as extremely insoluble substances or as such stable complexes that only a very small concentration of free CU+can exist in solution. 19.17. (a) What is the potential of the cell containing the (Zn2+/Zn)and (Cu2+/Cu)couples if the Zn2+ and Cu2+ concentrations are 0.1 M and 10-9 M, respectively, at 25 "C? (b) What is AG for the reduction of 1 mol of Cu2+ by Zn at the indicated concentrations of the ions, and what is AGO for the reaction, both at 25"C? (a) The cell reaction is Zn + Cu2+ + Zn2+ + Cu, with an n-value of 2. E=&'"- 0.059 2 log (2 n E", the standard cell potential, is equal to the difference between the standard electrode potentials, 0.34 - (-0.76) = 1.10 V Q,the concentration function, does not include terms for the solid metals, because the metals are in their standard states. E = 1.10-0.059 2 [Zn2+] 10-' log-= 1.10.-0.029610g[CU2+] 10-9 = 1.10 - (0.029 6)(8) = 0.86 V (6) From (19-3) AG = - n 9 E = -(2mol e-)(9.65 x 104C/mol e-)(0.86 V) AG" = -nFE" = -(2 x 9.65 x 104 x 1.10)J = -212kJ = -166 x 103 c . v = -166kj 19.18. By how much is the oxidizing power of the (Mn0;/Mn2,+) couple decreased if the H + concentration is decreased from 1 M to lOP4 M at 25 "C? The half-cell reaction for the reduction is MnO, 0.059 2 5 + 8H+ + 5elog -+Mn2' +4H20 with an n-value of 5. Assume that only the H + concentration deviates from 1 mol/L. E - E" - ___ 1 [Mn2+] = -0.011 8 log [Mn0;][H+l8 ( I ) ( 10-4)8 = -(0.011 8)(32) = -0.38 V The couple has moved down the table 0.38 V (to a position of less oxidizing power) from its standard value. 19.19. In the continued electrolysis of each of the following solutions at pH 7.0 and 25"C, predict the main product at each electrode if there are no (irreversible) electrode polarization effects: (a) 1 M NiS04 with palladium electrodes; (b) 1 M NiBr2 with inert electrodes; (c) 1 M Na2S04 with Cu electrodes. (a) Cathode Reaction. The following two possible reduction processes may be considered: ( I ) Ni2+ + 2e(2) 2H++2e- -+ Ni -+ H2 E" -0.25 V 0.00 328 ELECTROCHEMISTRY [CHAP. 19 By the rule that the most probable cathode process is that for which the corresponding electrode potential is algebraically greatest, the hydrogen couple is favored at unit concentrations. Allowing for the effect of the pH 7.0 buffer, however, E for (2) is lowered to -0.41 V, as calculated in Example 2. The reduction of nickel then becomes the favored process. Anode Reaction. Three possible oxidation processes may be considered, the reverses of the following reduction halfreactions: E" ( 3 ) O2 + 4H+ + 4e- 3 2 H 2 0 1.23 V (4) Pd2+ + 2e- 3 Pd 0.915 (5) S20g- + 2e- -+ 2SO;1.96 The standard potentials are reasonable values to take in considering (4) and (5). Although the initial concentration of Pd2+ and S20:- are zero, they would increase during prolonged electrolysis if these species were the principal products. In the case of (3), however, the buffering of the solution prevents the buildup of [H+], and it would be more appropriate to take the E-value calculated for pH 7.0 in Example 2, 0.82V. It is apparent that of the three, ( 3 ) has the smallest E-value and the reverse of (3) would thus occur most readily among the possible anode reactions. In conclusion, the electrode processes to be expected are Anode: Cathode: Overall: 2H20 3 O2 4H++ 4et + Ni2++2e- Ni 2H20 + 2Ni2+ 3 O2 + 4H' + 2Ni (b) Cathode Reaction. Anode Reaction. As in (a), Ni reduction would occur. The expression "inert electrode" is often used to indicate that we may neglect reaction of the electrode itself, either by virtue of the intrinsically high value of its electrode potential or because of polarization effects related to the preparation of the electrode surface. The remaining possible anode reactions are the reverses of the following: E" ( 6 ) Br2 + 2e-+ t 2Br2H20 1.065 V 1.23 (3) 0 2 + 4H+ + 4e- When the E-value for (3) is computed as 0.82V for pH 7.0, as in (a), oxygen evolution takes precedence. [In practice, "overvoltage" or polarization is more difficult to avoid in the case of reactions involving gases (02) as compared with liquids and dissolved solutes, so that electrolysis of NiBr2 at most electrodes would probably lead to Br2 formation.] (c) Cathode Reaction. The new couple to be considered is the sodium couple, the reduction reaction for which is (7) Na++e-+ Na E" -2.71 V This E-value is much lower algebraically than that of (2), the evolution of H2 at pH 7.0, -0.41V. Therefore, hydrogen evolution will occur at the cathode. Anode Reaction. In addition to (3) and (3,the reaction of the Cu anode must be considered, the reverse of which is (8) Cu2++2e--+ Cu E" 0.34 V Process (8) has the lowest E-value, and copper dissolution, the reverse of (8), would thus take precedence over oxygen evolution. CHAP. 191 ELECTROCHEMISTRY 329 In this example the sodium sulfate solute was not at all involved in the reaction, but served as an electrical conductor. Eventually as Cu2+ builds up and copper migrates to the cathode region, reaction (8) will replace reaction (2) as the cathode reaction. 19.20. Knowing that Ksp for AgCl is 1.8 x 10-'*, calculate E' for a silver-silver chloride electrode immersed in 1 M KCl at 25 "C. The electrode process is a special case of the (Ag+/Ag) couple, except that silver in the + I state collects as solid AgCl on the electrode itself. Even solid AgC1, however, has some Ag' in equilibrium with it in solution. This [Ag'] can be computed from the Ksp equation: [Ag+]= lC1-1 Ag+ + eKsP = 1.8 x 10-l' 1 = 1.8 x 10-I' This value for [Ag+] can be inserted into the Nernst equation for the (Ag+/Ag) half-reaction. -+ Ag E" = 0.799 V 0.222 V E =z 1 0.059 2 1 E" - -log -= 0.799 - 0.059 2 log - = 0.799 - 0.577 1 [&+I 1.8 < 10-I' : 19.21. From data in Table 19-1, calculate the overall stability constant, K,, of Ag(S2O3);- at 25 "C. There are two entries in the table for couples connecting the zero and +1 oxidation states of silver. ( I ) Ag' e(2) Ag(S203)i-+eE" 0.799 V + --$ --+ Ag Ag + 2S20:- 0.017 Process ( I ) refers to the couple in which Agf is at unit concentration; the Ag+ concentration to which the E" for (2) refers is that value which satisfies the complex ion equilibrium when the other species are at unit concentration. The formation reaction for this complex is: Ag+ + 2S20:- + Ag(S2O3);- In other words, the standard conditions for couple (2) may lbe thought of as a nonstandard condition, [Ag'] = l/Ky, for couple (I). Then the Nernst equation for ( I ) gives 1 E = E" - 0.0592 log lAg+l or 0.017 = 0.799 - 0.059 2 log K, from which log K, = 0.799 - 0.017 = 13.21 0.059 2 and K, = 1.6 x 1013 19.22. (a) At equimolar concentrations of Fe2+ and Fe3+,what must [Ag+]be so that the voltage of the galvanic cell made from the (Ag+/Ag) and (Fe3'/Fe2+) electrodes equals zero? Fe2+ + Ag' (a) For the reaction as written, + Fe3+-tAg (b) Determine the equilibrium constant at 25 "C for the reaction. E" = E"(Ag+/Ag)-E"(Fe3+/Fe2+) 0.799 - 0.771 = 0.028 V = 330 ELECTROCHEMISTRY [CHAP. 19 From the Nernst equation, E = E" - 0.0592 log [Fe3+] [&+I [Fe2+] 1 0 = 0.028 - 0.059 2 log -= 0.028 + 0.059 2 log [Ag'] [&+I 0.028 log [Ag'] = - - -0.47 0.059 2 [Ag+] = 0.34 To find the equilibrium constant, we must combine the relationship between K and AGO, (26-8), with the relationship between A O and E" obtained from (19-3), AG" = -nFE". G logK=-A O - nFE" - nE" G --2.303RT - 2.303RT 0.0592 Note that the same combination of constants, of value 0.059 2 at 25 "C, occurs here as in the Nernst equation. 0.028 = log K = - 0.47 0.059 2 whence K [Fe3+] [Fe2+] [Ag+l = 3.0 Part (a) above could have been solved alternatively by using the equilibrium constant, noting that [Fe2+]= [Fe3+]and solving for [Ag'] : The two methods must be equivalent because the voltage of a galvanic cell becomes zero when the two couples are at equilibrium with each other. 19.23. An excess of liquid mercury was added to an acidified 1.00 x 10-3 M solution of Fe3+. It was found that only 5.4 percent of the iron remained as Fe3+ at equilibrium at 25C. Calculate E"(Hg;+/Hg), assuming that the only reaction that occurred was 2Hg + 2Fe3+ -+Hgi+ + 2Fe2+ First we calculate the equilibrium constant for the reaction. At equilibrium, = [Fe3+] (0.054)(1.00 x 10-3) = 5.4 x 10a5 [Fe2+]= (1 - 0.054)(1.00 x 10-3) = 9.46 x 10-4 [Hg;+] = [Fe2+] 4.73 x 10-4 = Liquid Hg is in excess and is in its standard state. K = [H$][Fe2+l2 [Fe3+] - 4 (4.73 x 10d4)(9.46x 10-4)2= o.145 (5.4 x 10-512 The standard potential of the cell corresponding to the reaction may be computed from the relation found in Problem 19.22(b). 0.059 2 E" = logK = (0.059 2) (-0.839) = -0.025 2 For the reaction as written, E" = E"(Fe3'/Fe2+) - E"(Hg:+/Hg) v or E"(Hg:+/Hg) = E"(Fe3+/Fe2+)-E" 5 0.771 - (-0.025) = 0.796 V CHAP. 191 ELECTROCHEMISTRY 33 1 19.24. Eo'-values for the (pyruvate/lactate) and (NAD+/NADH) couples at pH 7 and 25 "C are -0.19 V and -0.32V, respectively. In the enzymatic reduction of pyruvate to lactate (2 electrons per molecule) by NADH (2 electrons per molecule), what is E"' for the reaction and what is AGO' at pH 7 per mole of pyruvate reduced? Except for H20 and its ions, the reaction may be written Pyruvate + NADH Then or E'' * lactate + NAD' -0.19 - = E"'(pyruvatej1actate) - E"'(NAD+/NADH:) = (-0.32) = 0.13 V AGO' = -n9Eo' = -(2mol e-)(9.65 x 104C/mol e-)(0.13 V) = -2.5 x 104J = -25 kJ Supplementary Problems ELECTRICAL UNITS 19.25. How many coulombs per hour pass through an electroplating bath which uses a current of 5A? Ans. 1.8 x 104C/h { 19.26. Compute the cost at 5 per kilowatt-hour of operating for 8 hours an electric motor which takes 15 A at 110v. Ans. 66# 1 . 7 A tank containing 0.2 m3 of water was used as a constant-temperature bath. How long would it take to heat 92. the bath from 20 "C to 25 "C with a 250-W immersion heater? Neglect the heat capacity of the tank frame and any heat losses to the air. Ans. 4.6h 19.28. The specific heat of a liquid was measured by placing lOOg of the liquid in a calorimeter. The liquid was heated by an electrical immersion coil. The heat capacity of the calorimeter together with the coil was previously determined to be 31.4 J/K. With the 100-g sample in place in the calorimeter, a current of 0.500 A was passed through the immersion coil for exactly 3 min. The voltage across the terminals of the coil was measured to be 1.5OV. The temperature of the sample rose by 0.800"C. Find the specific heat capacity of the liquid. Ans. 1.37 kJ/kg. K 19.29. The heat of solution of NH4N03 in water was determined by measuring the amount of electrical work needed to compensate for the cooling which would otherwise occur when the salt dissolves. After the NH4N03 was added to the water, electric energy was provided. by passage of a current through a resistance coil until the temperature of the solution reached the value it had prior to the addition of the salt. In a typical experiment, 4.4 g NH4N03 was added to 200 g water. A current of 0.75 A was provided through the heater coil, and the voltage across the terminals was 6.0 V. The current was applied for exactly 5.2 min. Calculate A H for the solution of 1 mol NH4N03in enough water to give the same concentration as was attained in the above experiment. Ans. 25.5kJ 332 ELECTROCHEMISTRY [CHAP. 19 FARADAY'S LAWS In the electrolysis problems, assume that the electrode efficiency is 100 percent for the principal electrode reaction, unless a statement is made in the problem to the contrary. 19.30. What current is required to pass 1 mol of electrons per hour through an electrodeposition cell? How many grams of aluminum and of cadmium will be liberated by 1 mol of electrons? Ans. 26.8 A, 8.99 g Al, 56.2 g Cd 19.31. What mass of aluminum is deposited electrolytically in 30 min by a current of 40 A? Ans. 6.7g 19.32. How many amperes are required to deposit on the cathode 5.00g of gold per hour from a solution containing a salt of gold in the +3 oxidiation state? Ans. 2.04A 19.33. How many hours will it take to produce lOOlb of electrolytic chlorine from NaCl in a cell that carries lOOOA? The anode efficiency for the chlorine reaction is 85 percent. Ans. 40.4h 19.34. A given quantity of electricity passes through two separate electrolytic cells containing solutions of AgN03 and SnC12, respectively. If 2.00 g of silver is deposited in one cell, how many grams of tin are deposited in the other cell? Ans. 1.lOgSn 19.35. An electrolytic cell contains a solution of CuS04 and an anode of impure copper. How many kilograms of copper will be refined (deposited on the cathode) by 150 A maintained for 12.0h? Ans. 2.13 kg Cu 19.36. A fuel cell aboard a spaceship is powered by the overall reaction: 2H2 + 0 2 -+ 2H20 It delivers a steady current of 10.0 amperes and is fed H2 gas from a 3.5 ft3 tank at 2 000 psi and 31 "C. How long will it run? ("psi" = lbf/in2, see Problem 1.9.) Ans. 1.04 x 107seconds, or 121 days 19.37. Hydrogen peroxide can be prepared by the successive reactions 2NH4HS04 -+ H2 (NH4)2S208 + 2H20 -+ + (NH4)2S208 2NH4HS04 + H202 The first reaction is an electrolytic reaction, and the second a steam distillation. What current would have to be used in the first reaction to produce enough intermediate to yield 100 g of pure H202 per hour? Assume 50 percent anode current efficiency. Ans. 315A CHAP. 191 ELECTROCHEMISTRY 333 19.38. An important "figure of merit" for a battery is the energy density, that is, the energy delivered divided by the mass of reactants consumed. Compare the energy density of (z the 2.00 V (approx.) lead storage battery, () for which the discharge reaction is: Pb + PbO2 + 2H2S04 -+ 2PbS04 + 2H20 the (b) the 1.3OV (approx.) nicad battery, for which the discharge reaction is: Cd + 2NiOOH + 2H20 --+ Cd(OH), + 2Ni(OH), Note that these calculations do not take into account such practical factors as the fraction of reactants not utilized, or nonreactive structural materials. A m . (a) 601 kJ/kg; (6) 756kJ/kg 19.39. Engineers interested in developing low-weight batteries have turned to the Group I elements Li and Na, which in addition to their low equivalent mass have very negative reduction potentials. (Because the electrolytes are not aqueous the battery voltages cannot be computed accurately from tabulations such as Table 19-1. The voltages given below are very rough estimates.) Compute the energy densities of (a) the -3.00 volt lithium-titanium(1V) sulfide battery, and (b) the ~ ~ 2 . volt sodium-sulfur battery, for which 60 the discharge reactions can be taken as: (a) Li + TiS2 -+ (Li+)(TiS;) ( b ) 2Na + S2 -+(Naf)2(S:-) Ans. (a) 2.43 x 103kJ/kg; (b) 4.56 x 103kJ/kg 19.40. Neglecting electrode polarization effects, predict the principal product at each electrode in the continued electrolysis at 25 "C of each of the following: (a) 1 M Fe2(S04)3with inert electrodes in 0.1 M H2S04. (b) 1 M LiCl with silver electrodes, (c) 1 M FeS04 with inert electrodes at pH 7.0, (d) molten NaF with inert electrodes. Ans. (a) Fe2+ and 0,; (b) H2 and AgC1; (c) H2 and Fe3+; and ( d ) Na and F2 19.41. A galvanic cell was operated under almost ideally reversible conditions at a current of 10-l6 A. (a) At this current, how long would it take to deliver 1 mol of electrons? (b) How many electrons would be delivered by the cell to a pulsed measuring circuit in 10ms of operation'? Ans. (a) 3 x lOI3 years; (b) 6 19.42. The quantity of antimony in a solution can be determined by converting it to the +3 oxidation state and titrating with standard iodine in bicarbonate solution: H2SbOY + 3HCOY + I2 --+ HSbOz- + 21- + 3co2 + 2H20 In a variation of the method, particularly applicable to very small samples, excess iodide ion is added to the solution and the requisite iodine is generated by electrolysis: 21 + 12 + 2e- Determine the mass of antimony in a solution which required a steady current of 23.2milliamperes for exactly 182 seconds to reach the end point in the above "coulometric" titration. Ans. 2.66mg of Sb GALVANIC CELLS AND OXIDATION-REDUCTION All problems refer to 25 "C 19.43. (a) What is the standard potential of the cell made up of the (Cd2+/Cd) and (Cu2+/Cu) couples? (b) Which couple is positive? 334 ELECTROCHEMISTRY [CHAP. 19 Ans. (a) 0.74V; (b) (Cu2+/Cu) 19.44. What is the standard potential of the cell containing the (Sn2'/Sn) and (Br2/Br-) couples? Ans. 1.20v 19.45. Why are Co3+ salts unstable in water? Ans. Co3+ can oxidize H 2 0 , the principal products being Co2+and 02. 19.46. If H202is mixed with Fe2+,which reaction is more likely, the oxidation of Fe2' to Fe3+, of the reduction of Fe2+ to Fe? Write the reaction for each possibility and compute the standard potential of the equivalent electrochemical cell. Ans. More likely: H 2 0 2+ 2H+ + 2Fe2+ -+ 2 H 2 0 + 2Fe3+; E" = 0.99 V. Less likely: H202 + Fe2+ -+ Fe + O2 + 2H+. The reverse of this reaction occurs with a standard potential of 1.14V. 19.47. What substance can be used to oxidize fluorides to fluorine? Ans. Fluorides may be oxidized electrolytically, but not chemically by any substance listed in Table 19-1. 19.48. Are Fe2+ solutions stable in the air? Why can such solutions be preserved by the presence of iron nails? Ans. They are not stable because 0 2 oxidizes Fe2+ to Fe3+, but Fe reduces Fe3+ to Fe2+. 19.49. What is the standard potential of the (T13+/T1)electrode? Ans. 0.72V 19.50. Which of the following intermediate oxidation states is stable with respect to disproportionation in oxygen- free noncomplexing media: germanium(II), tin(II)? Ans. tin(1I) 19.51. Would H 2 0 2 behave as oxidant or reductant with respect to each of the following couples at standard concentrations: (a) (12/1-), (6) (S,Oi-/SOi-), (c) (Fe3+/Fe2+)? Ans. (a) oxidant, (b) reductant, (c) both; in fact very small amounts of iron salts in either +2 or +3 state catalyze the self-oxidation-reduction of H202. 19.52. What is the potential of a cell containing two hydrogen electrodes, the negative one in contact with 10-8 molar H' and the positive one in contact with 0.025molar H + ? Ans. 0.379V 19.53. The cell in the preceding problem is, in a sense, a pH meter. Suppose one half-cell contained 1.OO M H + , the standard state. Deduce the relation (at 25 "C) between the pH of the other half-cell and the observed voltage (connecting the positive terminal of the meter to the standard half-cell). Ans. V = 0.059 2 pH 19.54. A voltmeter can replace the traditional indicator in titrations by making the titration vessel a half-cell (with an appropriate electrode) and connecting it to a reference half-cell via a salt bridge. To follow the titration in Problem 18.56, a silver/silver chloride electrode is inserted into the halide solution, and the reference half-cell is a silver/silver chloride electrode immersed in l.00M KCI. The reference electrode goes to the positive terminal of the voltmeter. Calculate the voltage reading at each of the five points in the titration specified in Problem 18.56. Temperature is 25 "C. Ans. (a) 0.179V; (6) -0.101 V; ( c ) -0.195V; ( d ) -0.288V; (e) -0.399V CHAP. 191 ELECTROCHEMISTRY 335 19.55. The burning of methanol could be the basis of a convenient fuel cell. First find a good catalytic surface for the anode reaction: CH30H(l) + 60H- -, COZ(g) + 5H20(1) + 6e- The cathode reaction is the same as in the successful hydrogen--oxygen fuel cell: 02(g) + 2H20(1) + 4e- -+ 40H- Calculate the voltage of this fuel cell (for the standard states). Hint: use data from Table 16-1. Ans. E" = 1.213V 19.56. Compute the potential of the (Ag+/Ag) couple with respect to (Cu2+/Cu) if the concentration of Ag+ and Cu2+are 4.2 x 10-6 M and 1.3 x 10-3 M, respectively. What is the value of AG for the reduction of 1 mol of Cu2+ by Ag at the indicated ion concentrations? Ans. 0.23V, -44kJ 19.57. Copper can reduce zinc ions if the resultant copper ions can be kept at a sufficiently low concentration by the formation of an insoluble salt. What is the maximum concentration of Cu2+ in solution if this reaction is to occur, when Zn2+ is lmolar? Ans. 7 x 10-38M 19.58. Evaluate the equilibrium constant for the reaction Fe(CN):Ans. 0.5 + Co(dip)i+ + Fe(CN)i- + Co(dip):+ 19.59. When a rod of metallic lead was added to a 0.010 0 M solution of Co(en):+, it was found that 68 percent of the cobalt complex was reduced to Co(en):+ by the lead. (a) Find the value of K for Pb + 2Co(en):+ (6) What is the value of E'(Co(en):'/Co(en):+)? A ~ s . (a) 0.0154; (b) -0.180V 19.60. A (Tl+/Tl) couple was prepared by saturating 0.1 M KBr with TlBr and allowing the T1+ from the relatively insoluble bromide to equilibrate. This couple was observed to have a potential of -0.443 V with respect to a + Pb2+ + 2Co(en)t+ (Pb2+/Pb) couple in which Pb2+ was 0.1 molar. What is the solubility product of TlBr? Ans. 3.7 x I O - ~ 19.61. K d for complete dissociation of Ag(NH3): into Ag' and NH3 is 6.0 x 10-8. Calculate E" for the following half-reaction by reference to Table 19-1: Ag(NH3)i+eAns. 0.372V 19.62. Calculate K, for formation of PdIf -+ Ag + 2NH3 from Pd2+ and I-. Ans. 1025 19.63. Reference tables give the following entry: HOT + H20 +2e- --f 30H- E" = 0.88 V 336 ELECTROCHEMISTRY [CHAP. 19 Combining this information with relevant entries in Table 19-1, find K 1 for the acid dissociation of H202. Ans. 10-l2 19.64. Calculate AGO' for the oxidation of 1mol of flavin (2 electrons per molecule) by molecular oxygen (1 atm) at pH 7. The Eo'-value for the flavin couple is -0.22V. (Hint: Be sure to use E"' rather than E" for the reduction of 0 2 to H 2 0 . ) An$. -201 kJ Chapter 20 Rates of Reactions The discussion of chemical equilibrium or of the direction of spontaneous processes is often accompanied by the statement that thermodynamics does not explain the rate at which spontaneous processes occur or chemical equilibrium is achieved. The branch of chemistry that treats the rates of reactions is called chemical kinetics. There are two main objectives in this chapter: the systematization of data dealing with the dependence of the rates of reactions on controllable variables, and the inference of the molecular mechanism by which reactions occur from the observed rates. This chapter deals largely with systematics. RATE CONSTANTS AND THE ORDER OF REACTIONS Reactions that occur completely within a single phase of matter, particularly a liquid or gaseous phase, are called homogeneous reactions. Reactions that take place at least in part at the interface between two phases, such as solid and liquid or solid and gas, are called heterogeneous reactions. The discussion and problems in this chapter will concern homogeneous reactions, unless a specific statement to the contrary is made. According to the law o mass action, the rate of a chemical reaction at a given temperature, expressed f as the amount reacting per unit volume per unit time, depends only on the concentrations of the various substances influencing the rate (and not, for example, on the size of the reaction vessel). The substances that influence the rate are usually one or more of the reactants, occasionally one of the products, and sometimes a catalyst that does not appear in the balanced overall chemical equation. The dependence of the rate on the concentrations can be expressed in many cases as a direct proportionality, in which the concentrations may appear to the zero, first, or second power. The power to which the concentration of a substance appears in the rate expression is called the order of the reaction with respect to that substance. Some examples follow. ( I ) Rate =kl [A] (2) Rate = k2[A][B] First-order in A First-order in A ancl first-order in B Second-order in A Second-order in A and first-order in B Zero-order (3) Rate = k3[AI2 ( 4 ) Rate = k,[Al2[B] ( 5 ) Rate = k5 Note: In all expressions involving rates, [XI denotes the concentrations of X (not the relative concentration) and carries the units of concentration (e.g., mol/L). The order cannot be deduced from the chemical equation but must be ascertained by experiment. The overall order of a reaction is the sum of the orders with respect to the various substances. In the above examples, the values of the overall order are 1, 2, 2, 3, and 0, respectively. The proportionality factors, k , called the rate constants or the spec@ rates, are constant at a given temperature but may depend on the temperature. There are dimensions to k , and the units should be expressed when k-values are tabulated. The rate itself is defined as the change of concentration of a reactant or product per unit time. If A is a reactant and C a product, the rate might be expressed as A[A1 or as Rate =- I W Rate = -At At where Am] is the change in the concentration of X and A t is the time interval over which the change is measured. If the rate changes rapidly, A t should be small. The law of mass action applies only at very small values of A t , in which case the derivative notation of calculus, 337 338 RATES OF REACTIONS [CHAP. 20 replaces the ratio expression. The minus sign is used to express the rate in terms of a reactant concentration (which decreases during the reaction) and a plus sign for a product concentration (which increases during the reaction), so that the rate is always a positive quantity. EXAMPLE 1 For some reactions, there is no ambiguity in defining the rate. For example, in CH30H + HCOOH the rate could be expressed as - A[CH,OH] --+ HCOOCH3 + H20 A[HCOOCH,] At At or - A[HCOOH] At or since these three ratios are all equal when concentrations are in molar units and there are not competing or side reactions. In the following reaction, however, N2 + 3H2 --+ 2NH3 the coefficients for the different substances are not the same; the concentration of H2 decreases three times as fast as that of N2 in the same molar units, and NH3 is formed twice as fast as N2 is used up. Any one of the three ratios ____ A "21 At a[H21 At A"H3I At could be used to specify the rate, but the choice should be specified because the numerical value of k (though not the order) depends on the choice. First-order reactions In the particular case of a first-order reaction, where the rate is proportional to the concentration of the reactant [A], as in ( I ) above, the methods of integral calculus give [A] = (20-1) (See Appendix C for the definition of e, the base of natural logarithms.) [A], is the value of [A] at the beginning of the experiment, when t = 0. The logarithmic form can be derived from the above. [AI 2.303 log -= -kt [AI, (20-2) The amount of time, tl12, a first-order reaction to achieve 50 percent completion, is given by (20-2)as for 2.303 log - 0.693 (20-3) k k Note that t l p , called the half-life, is independent of the initial concentration of A; this independence of [A], is characteristic only of first-order reactions. Note also that t l / 2as well as k are independent of the units in which [A] is expressed. tl/2 =- Other rate laws Fractional orders also occur (for example, $- or $-order, in which the rate is proportional to [A]'I2 or [A]'//', respectively). Some reaction rates may not be expressible in the proportional form altogether. An example of a complex rate expression is the following: (6) Rate = kl [AI2 1 + k2PI CHAP. 201 RATES OF REACTIONS 339 The rate of a heterogeneous reaction may be proportional to the interfacial area of contact between the phases, as well as to concentrations of reactants within a particular phase, as in the case of many reactions catalyzed at surfaces. ENERGY OF ACTIVATION The temperature dependence of a reaction rate may be represented by the Arrhenius equation: (20-4) The pre-exponential factor, A , is also called the frequency factor, and E, is called the energy of activation; the units of E, (and of R T ) are J/mol or cal/mol. Both A and E, may be considered to be constant, at least over a narrow temperature range. From (20-4) the rate constants at two different temperatures are related by (20-5) MECHANISM OF REACTIONS It has been said that a chemical equation accurately portrays the nature of the starting and final materials in a chemical reaction, but that the arrow conveniently covers our ignorance of what happens in between. How many successive steps are there in the overall process? What are the spatial and energetic requirements for the interactions in each step? What is the rate of each step? Although rate measurements usually describe only the overall reaction, the measurement of rates under different conditions often contributes to the understanding of the mechanism. Molecularity The molecularity specifies the number of molecules interacting in an individual mechanistic step. A unimolecular reaction is a step in which a single molecule spontaneously undergoes a reaction; a bimolecular reaction requires the coming together of two molecules; a termolecular reaction is one in which three molecules interact in a single step. There are no known single steps of higher molecularity than termolecular . A unimolecular reaction is first-order in the species that undergoes the spontaneous rearrangement or decomposition. This is true because there is an intrinsic probability that a molecule will undergo the reaction in a unit time interval, and the overall rate per unit volume is the product of this probability per unit time and the total number of molecules per unit volume (which in turn is proportional to the concentration). A bimolecular reaction rate is proportional to the frequency of collisions between two molecules of the reacting species. It is known from kinetic theory that the frequency of collisions between two like molecules, A, is proportional to [AI2, and the frequency of collisions between an A and a B molecule is proportional to [A][B]. If the species whose molecules collide are starting materials in limited concentrations, the reaction is second-order, following the rate equation of either type (3) or type (2). A termolecular reaction rate is proportional to the frequency of three-body collisions, which in turn is proportional to [AI3,[AI2[B]or [A][B][C], depending on whether the three colliding molecules represent one, two, or three species. If these species are starting materials, the reaction is third-order. Although the order of a single mechanistic step can be predicted from the molecularity, the molecularity of a step, or steps, cannot be predicted from the order of the overall reaction. There are a number of complications which make it impossible to conclude automatically that a first-order reaction is unimolecular, that a second-order reaction is bimolecular, or that a third-order reaction is termolecular. In many cases, the reaction is a sequence of steps, and the overall rate may be governed by the slowest step. Experimental conditions might interchange the relative speeds of different steps, and the 340 RATES OF REACTIONS [CHAP. 20 order would appear to change. Another complication arises because the slowest, or rate-limiting, step may involve the reaction of an unstable intermediate species, and it is necessary to express the concentration of this species in terms of the reactants before the overall order of the reaction can be deciphered. The solved problems will give examples of some of these complexities. Energetics Most molecular collisions do not result in reaction. Even if the requisite number of molecules came together, only those possessing enough energy, usually far in excess of the average energy, can undergo the violent distortions of bond lengths and angles necessary for the rearrangements that lead to chemical reaction. The energy of activation, E,, is a measure of the excess energy needed for molecules to react, and the exponential term in the Arrhenius equation, (20-4), is of the order of magnitude of the fraction of molecules possessing this amount of excess energy. The activated state has been defined as that distorted combination of reacting molecules having the minimum amount of excess energy so that the combination, or complex, could as easily rearrange to form the products as revert to the reactants. The energy of the activated state may be likened to the potential energy of a mountain pass which a climber must cross in order to go from one side of the mountain to another. For a set of reversible reactions, the same activated state must be crossed for the reactions in the two directions. Thus, for reactions carried out at constant pressure, (1 mol) [&(forward) - E,(reverse)] = A H (20-6) The factor (1 mol) has been inserted in (20-6) so that the enthalpy change of reaction will, as usual, have the units of energy. These relationships are shown graphically in Fig. 20-1 below. When two fast-moving reactant molecules collide, their kinetic energy is converted into the high energy of the complex. When the complex breaks up, the fragments (products or reactants) fly apart, their kinetic energy coming from that stored in the complex. It Energy Reactants Collision (activated complex) Products Progress of reaction - Fig. 20-1 Energy relationships in a reaction (forward reaction is exothermic). CHAP. 201 RATES OF REACTIONS 34 1 Solved Problems RATE CONSTANTS AND THE ORDER OF REACTION 20.1. In a catalytic experiment involving the Haber process, was measured as N 2 + 3H2 + 2NH3, the rate of reaction - 2.0 = at 1 0 - ~moi L-1 - s-' , If there were no side reactions, what was the rate of reaction expressed in terms of (b) H2? (a) (a) NZ, From the coefficients in the balanced equation, An(N2) == - An(NH3).Therefore, 4 (b) Similarly, 20.2. What are the units of the rate constant, k , for (a) a zero-order reaction, (b) a first-order reaction, (c) a second-order reaction, ( d ) a third-order reaction, (e) a half-order reaction, when concentrations are expressed in molar units and time in seconds? In each case, we write out the full equation and find the units of k that will satisfy the equation. [AI - Units of k = units of -- mol/L = mol . L-' . s-l t S Note that the units of A[A], the change in concentration, are the same as the units of [A] itself; similarly for At. 1 mol/L Units of k = - x - =s mol/L s First-order reactions are the only reactions for which k has the same numerical value, regardless of the units used for expressing the concentrations of the reactants or products. A[A1= k[A]* At I - k[A][B] -nt-- 1 mol/L Units of k = - - L . mo1-I . s-' x := (mol/L)* s 342 RATES OF REACTIONS [CHAP. 20 Note that the units of k depend on the total order of the reaction, not on the way the total order is composed of the orders with respect to different reactants. Units of k = 1 (mol/L) ' I 2 X-- m o w - mo11/2 . L4/2 . s-l s 2 . . In a certain polluted atmosphere containing 0 3 at a steady-state concentration of 2.0 x 03 10-* mol/L, the hourly production of O3 by all sources was estimated as 7.2 x 10-13 mol/L. If the only mechanism for destruction of O3 is the second-order reaction 2 0 3 --+ 3 0 2 , calculate the rate constant for the destruction reaction, defined by the rate law for -A[03]/At. At the steady state, the rate of destruction of 0 7.2 x 10-13 mole L-' . h-'. From the second-order rate law, 3 must equal the rate of its generation, 20.4. A viral preparation was inactivated in a chemical bath. The inactivation process was found to be first-order in virus concentration, and at the beginning of the experiment 2.0 percent of the virus was found to be inactivated per minute. Evaluate k for the inactivation process. From the first-order rate law, It is seen that only the fractional change in concentration, -A[A]/[A], is needed; namely, 0.020 when A t = 1 min = 60 s. This form of the equation may be used for the initial rate when the value of [A] is not changing appreciably; that condition is certainly met when only 2 percent is inactivated in the first minute. 0.020 k=-- 3.3 60 s 10-~ s-l 20.5. For the process described in Problem 20.4, how much time would be required for the virus to become (a) 50 percent inactivated, (b) 75 percent inactivated? The method used in Problem 20.4 cannot be used here because [A] changes appreciably over the course of the reaction. Equation (20-Z), (20-2), or (20-3) is appropriate here. (a) The time for 50 percent reaction is the half-life, and (20-3) gives 0.693 0.693 t'/2 = -k 3.3 x 10- s = 2.1 x 103 s = 35 min CHAP. 201 RATES OF REACTIONS 343 (b) Equation (20-2)may be used. If 75 percent of the virus is inactivated, the fraction remaining, [A]/[Al0, An alternate solution is to apply the half-life concept twice. Since it takes 35 min for half of the virus to become inactivated regardless of the initial concentration, the time for the virus to be reduced from 50 percent to 25 percent full strength will be another half-life. The total time for reduction to strength is thus two half-lives, or 70min. Similarly, the total time required to reduce the initial activity to is three half-lives; to four halflives; and so on. This method can be used only for first-order reactions. h, 20.6. In the fermentation of sugar in an enzymatic solution that is initially 0.12 M the concentration of the sugar is reduced to 0.06 M in 10 h and 0.03 M in 20 h. What is the order of the reaction and what is the rate constant? This problem is analogous to Problem 20.5. Since doubling the time reduces the fractional survival of the reactant by half, the reaction must be first-order. Alternatively, the reduction of the sugar concentration from 0.06 M to 0.03 M may be thought of as a new experiment with an initial concentration of 0.06 M. Since the same half-life (10 h) was observed in both experiments, the reaction must be first-order because only in a first-order reaction is the half-life independent of the initial concentration. The rate constant may be evaluated from the half-life via (20-3). 20.7. A reaction between substances A and B is represented stoichiometrically by A + B --+ C . Observations on the rate of this reaction are obtained in three separate experiments as follows: Initial Concentrations "o/M Duration of Experiment Atlh 0.50 0.50 2.00 Final Concentration, [Alf I M 0.097 5 0.090 0 0.045 0 [BIo/M 0.1000 0.1000 0.050 0 1.o 2.0 1.o What is the order with respect to each reactant, and what is the value of the rate constant? First, let us tabulate the initial rate of the reaction for each experiment, noting that A[A] = [AIf - [A], is in each case small enough in magnitude to allow the rate to be expressed in terms of changes over the entire time of the experiment. Initial Rates Initial Concentrations I I I ------I 0.005 0 0.020 0 0.002 5 0.1000 0.050 0 -0.0100 -0.005 0 1 344 RATES O F REACTIONS [CHAP. 20 In comparing experiments ( I ) and (2), we note that [A] is the same in both, but [B] is twice as great in (2). Since the rate in (2) is 4 times that in ( I ) , the reaction must be second-order in B. In comparing experiments ( I ) and (3),we note that [B] is the same in both, but [A] is twice as great in ( I ) . Since the rate in ( I ) is twice that in (3), the reaction must be first-order in A. The rate equation may be written accordingly as --'IA1 At - k[A][BI2 and k may be evaluated from any of the experiments; we take ( I ) as an example, using average values of [A] and [B]. ([B] does not change appreciably during the experiment since it is in such large excess). or 1.4 x 10-5 L2 mol-2 - s-' . The reader should confirm that the same result will be obtained by using data from either of the other two experiments. 20.8. The esterification of acetic anhydride by ethyl alcohol can be represented by the following balanced equation: (CH3CO),O + C2HSOH + CH3COOCZHS + CH3COOH A B When the reaction is carried out in dilute hexane solution, the rate may be represented by k[A][B]. When ethyl alcohol (B) is the solvent, the rate may be represented by k[A].(The values of k are not the same in the two cases.) Explain the difference in the apparent order of the reaction. When a solvent is also a reactant, its concentration is so large compared with the extent of reaction that it does not change. (Compare the convention we had used, in studying the thermodynamics of aqueous solutions, of viewing water as always being in its standard state in all dilute solutions.) Thus, the dependence of the rate on the concentration of ethyl alcohol cannot be determined unless ethyl alcohol becomes a solute in some other solvent, so that its concentration can be varied. 20.9. At elevated temperatures gaseous chloroethane decomposes into ethene and HCl according to the following equation. CH3CHZC1-+ CH2CHZ HCl + At a certain temperature, in a constant-volume experiment, the following data were obtained by observing the ethene photometrically: Time [Ethene]/ (mol/L) 0 1 3 5 10 20 30 50 > 100 0 8.3 x 10-4 2.3 x 10-3 3.6 x 10-3 6.1 x 10-3 9.0 x 10-3 1.05 x 10-* 1.16 x 1 K 2 1.19 x 10-2 Test the data to determine whether or not the reaction is first order, and if it is, determine the rate constant. Since the reaction is essentially complete after 100 hours, the initial concentration of chloroethane, [Ale, was 1.19 x 10-2 M and subsequent concentrations, [A], can be obtained by subtracting the ethene concentrations from this. Then one forms the ratio, [A]/[Al0, and takes the log. These steps are tabulated below. (The 50-hour point, which corresponds to the least accurate value of the ratio, is omitted.) A graph of log([A]/[AIo)vs. t is shown in Fig. 20-2. Since the points fall on a straight line, the reaction is first order. The slope of the graph, calculated on the figure, is -0.031 2 h-', which in accordance with equation (20-2) is equal to -k/2.303, giving k = 7.2 x 10-2h-' or 2.0 x 10-5s-'. Our numerical work can be checked roughly by noting that t1/2= 0.693/k = 9.6h, and in the original table of data, in 10 h the reaction is slightly more than half completed. CHAP. 201 RATES OF REACTIONS 345 -0.2 -0.4 -0.6 -0.8 - 1 .o 10 20 30 40 tlhours Fig. 20-2 First-order test for data of Problem 20-9. Time, t [A] x 102 [AI/ IAI, log( [AI/ [A101 0 1 3 0.96 0.83 0.697 -0.156 0.487 -0.312 0.244 -0.613 1.19 1 .oo 1.11 0.933 0.807 -0.093 0.1 18 -0.929 0 -0.030 20.10. The complexation of Fe2+ with the chelating agent dipyridyl (abbreviated dip) has been studied kinetically in both the forward and reverse directions. For the complexation reaction Fe2+ + 3 dip --+ Fe(dip):+ the rate of formation of the complex at 25C is given by Rate = (1.45 x 1013 L3- m ~ l -s-')[Fe2+][dipl3 ~ and for the reverse of the above reaction, the rate of disappearance of the complex is (1.22 x 10-4 s-')[Fe(dip):+] What is k , the stability constant for the complex? l, Not all reactions can be studied conveniently in both directions. When this is possible, we know that at dynamic equilibrium the rate of formation of the complex must equal the rate of decomposition, since concentrations of the various species stay fixed. 346 RATES OF REAC'TIONS [CHAP. 20 Rate of forward reaction = rate of reverse reaction (1.45 x 1013 L3 . molP3 . s-') [Fe2'] [dip13= (1.22 x 10-4 s-') [Fe(dip):+] Solving K, = [Fe(dip):+] - 1.45 x 1013 = 1.19 [Fe2+][dipI3 1.22 X - 1017 (In the equilibrium constant equation, as opposed to rate equations, [XI is dimensionless, being the concentration relative to the standard state of 1 mol/L.) ENERGY OF ACTIVATION AND MECHANISM OF REACTIONS 20.11. The decomposition of N20 into N2 and 0 in the presence of gaseous argon follows second-order kinetics, with k = (5.0 x 10'' L . mo1-l . ~ - ' ) e - ~ ~ ~ ~ ~ ~ / ~ What is the energy of activation of this reaction? Comparing the equation for k in this case with ( 2 0 4 , we note that the exponent of e is -&/RT. E,RT- 29 000 K T = 241 E, = (29 000 K)R = (29 000 K)(8.314 5 J . K-I . mol-l) kJ . mol-' 20.12. The first-order rate constant for the hydrolysis of CH3C1in H20 has a value of 3.32 x 10-l' s-l at 25 "C and 3.13 x 10-9 s-l at 40 "C. What is the value of the energy of activation? Equation (20-5) may be solved for E,. E, =2 . 3 0 3T2-( q 1) ( l o g 2 ) R 298 x 313 3.13 10-~ = (2.303)(8.314 5 J K-I . mol-I) (313 - 2 9 8 ) (log3.32 x 10-Io = ( 1 19 kJ mol-')(log 9.4) = 116 k J . mol-' 20.13. A second-order reaction whose rate constant at 800 "C was found to be 5.0 x 10-3 L . mol-' - s-l has an activation energy of 45 kJ - rnol-'. What is the value of the rate constant at 875 "C? Equation (20-5) may be solved for k 2 , the rate constant at the higher temperature. - s=1.39 kl (4.5 x 104 J . mo1-')[(875 - 800) K] (2.303)(8.3145 J . K-I . mol-')( 1 073 K ) ( 1 148 K) = 0.143 k2 = (1.39)(5.0 x lOP3 L . mol-' . s-') = 7.0 x lOP3 L . mol-' . s-l 20.14. The tram -+ cis isomerization of 1,2-dichloroethylene proceeds with an energy of activation of 55.3 kcal - mol-'. A H associated with the reaction is 1.0 kcal. What do you predict is the value of E, for the reverse cis -+ tram isomerization? CHAP. 201 RATES OF REACTIONS, 347 From (2049, &(reverse) = &(forward) -= AH 1 mol 55.3 .- 1.0 = 54.3 kcal . mol-I 20.15. A gaseous molecule A can undergo a unimolecular decomposition into C if it acquires a critical amount of energy. Such a suitably energized molecule of .A, designated as A*, can be formed by a collision between two ordinary A molecules. Competing with the unimolecular decomposition of A* into C is the bimolecular deactivation of A* by collision with an ordinary A molecule. ( a ) Write a balanced equation and the rate law for each of the above steps. (b) Making the assumptions that A* disappears by all processes at the same rate at which it is formed, and that [A*] is much less than [A], what would be the rate law for the formation of C, in terms of [A] and constants of individual steps only? (c) What limiting order in A would the formation of C exhibit at low pressures of A, and what limiting order at high pressures? (4 (I) Activation : A+A+A*+A A* + A +A ~ at ~ = kl [AI2 = k2[A*][A] - k3[A*] (2) Deactivation : (3) Reaction : +A At A* -+ C at (b) Note that A* appears in each of the three separate steps. The net change in [A*] can be evaluated by summing over all three steps. From the assumption that the net rate of change of [A*] is zero, the right-hand side can be equated to zero. kl [AI2 k l [A12-k2 - 3 [A*] = ` Or k3 k2[A] + Inserting this value into the rate law for step (3),and recognizing that -A[A*] = A[C] for this step, (c) Thus, the formation of C follows a complex kinetic formulation, not represented by a simple order. At very low pressures (i.e., small [A]), the second term in the denominator on the right side of ( 4 ) becomes negligible in comparison with the first. Low-pressure limit: -= - at The order appears to be second, with k l the second-order rate constant. At very high pressures, on the other hand, the first terim of the denominator becomes negligible in comparison with the second. k3kl[A]2 k3kl [A] High-pressure limit: -= - =At k2[A] k2 Now the order appears to be first, with k3kl/k2 as the apparent rate constant. This problem illustrates the concept of the rate-limiting step. In the familiar example of a bucket brigade, in which a water bucket passes sequentially from one person to his neighbor until it finally reaches the reservoir that is to be filled, the overall rate of transfer of ,water cannot be greater than the rate of the slowest step. Here the activation step (I) becomes the slowest step when [A] is low and the reaction step (3) 348 RATES OF REACTIONS [CHAP. 20 becomes the slowest step when [A] is high. Step (I), depending on the square of [A], is more sensitive to pressure than step (3). 20.16. For the hydrolysis of methyl formate, HCOOCH3, in acid solutions, Rate = k(HCOOCH3] [H'] The balanced equation is HCOOCH3 + H 2 0 -+ HCOOH + CH30H. Why does [H+] appear in the rate law, when it does not appear in the balanced equation? H + is a catalyst for the reaction. It is actually a reactant in an early intermediate stage of the reaction and is then released back into the solution at a later stage. 20.17. The conversion of the D-optical isomer of gaseous I I CH3 I into the L-isomer in the presence of iodine vapor follows the law rate = I c P ( A ) P ( I ~ ) ~ / ~ , A where represents the D-isomer. (Partial presures are a legitimate form for expression of concentrations of gases in rate laws.) Suggest a mechanism that could account for the fractional order. I2 can undergo a slight dissociation into I atoms. This system easily comes to equilibrium. The partial pressure of iodine atoms can be evaluated in terms of this equilibrium as P(1) =K;" x P(12)1/2 If an intermediate stage of the reaction involves the addition of an iodine atom to A, followed by a subsequent loss of the iodine atom that was initially a part of the A molecule, and if the addition of I to A is bimolecular (with a rate constant k2) and rate-determining, then: Observed rate = rate of I addition = k 2 P ( A ) P ( I )= k , P ( A ) x , Ki'2 x P(12)1'2= (k2Ki'2)P(A)P(12)''2 The numerical value of the term in parentheses, (k2KdI2), the numerical value of the apparent rate constant is for the overall $order reaction. The above mechanism is plausible and is consistent with the observations. We cannot be sure from the kinetic data alone, however, that there is not some other mechanism which is also consistent with the observations. Other types of experiment are needed to confirm a mechanism based on rate data. However, a proposed mechanism that yields a rate law other than that observed rate law can surely be rejected. Supplementary Problems RATE CONSTANTS AND THE ORDER OF REACTION 20.18. For the reaction 3Br0- --f Br07 + 2Br- CHAP. 201 RATES OF REACTIONS 349 in alkaline aqueous solution, the value of the second-order (in BrO-) rate constant at 80 "C in the rate law for -A[BrO-]/At was found to be 0.056L .mol-' . s-'. What is the rate constant when the rate law is written for (a) A[Br03]/At, (h) A[Br-]/At? Ans. (a) 0.018 7 L mol-' . s-'; (b) 0.037 L . mol-' . s-' 20.19. The hydrolysis of methyl acetate in alkaline solution, CH3COOCH3 + OH- + CH3COO- + CH30H followed rate = k[CH300CH3][OH-], with k equal to 0.137 L - mol-' . s-' at 25 "C. A reaction mixture was prepared to have initial concentrations of methyl acetate and OH- of 0.050 M each. How long would it take for 5.0 percent of the methyl acetate to be hydrolyzed at 25"C? Ans. 7.7s 20.20. A first-order reaction in aqueous solution was too fast to be detected by a procedure that could have followed a reaction having a half-life of at least 2.0ns. What is the minimum valve of k for this reaction? Ans. 3.5 x 108 s-l 20.21. Gaseous cyclobutene isomerizes to butadiene in a first-order process which has a k-value at 153"C of 3.3 x 10-4 s-I. How many minutes would it take for the isomerization to proceed 40 percent to completion at this temperature? Ans. 26min 20.22. The approach to the following equilibrium was observed kinetically from both directions: PtClz- + H 2 0 + Pt(H20)Cl; + C1L . mol-' s-'[Pt(H20)Cl;][Cl-] At 25"C, it was found that at 0.3 ionic strength: - A? = (3.9 x lO-' s-')[PtCl;-] - (2.1 x What is the value of Ki for the complexation of the fourth C1- by Pt(I1) (the apparent equilibrium constant for the reverse of the reaction as written above) at 0.3 ionic strength? Ans. 54 20.23. The following reaction was studied at 25 "C in benzene solution containing 0.1 M pyridine: CH3OH + ( C ~ H S ) ~ C C ~ ~ O C : ( C ~ H S HCl --+ C H )~ A B c + The following sets of data were obtained in three separate experiments. Initial Concentrations [AIo/M 0.1000 0.100 0 0.200 0 [BIo/M 0.050 0 0.100 0 0.1000 [CIo/M 0.000 0 0.000 0 0.000 0 At/min 2.5 15.0 7.5 Final [C]/M 0.003 3 0.003 9 0.007 7 What rate law is consistent with the above data and what is the best average value for the rate constant, expressed in seconds and molar concentration units? Ans. rate ==k[A2][B], = 4.6 x lOP3L2 . molm2. s-' k 350 RATES OF REACTIONS [CHAP. 20 t/min [A]/(mol/L) 0 20 40 60 80 100 120 140 160 180 200 220 240 0.462 0.449 0.437 0.426 0.416 0.406 0.396 0.387 0.378 0.370 0.362 0.354 0.347 . Ans. Second order ENERGY OF ACTIVATION AND MECHANISM OF REACTIONS 20.25. The rate constant for the first order decomposition of ethylene oxide into CH4 and CO may be described by the equation log k(in s-') = 14.34 (a) What is the energy of activation for this reaction? 1.25 x 104 K T (b) What is the value of k at 670K? Ans. (a) 239 kJ mol-'; (b) 5 x 10-5 s-l 20.26. The first-order gaseous decomposition of N2O4 into NO2 has a k-value of 4.5 x 103 s-' at 1 "C and an energy of activation of 58 kJ . mol-'. At what temperature would k be 1.OO x 104 s-'? Ans. 10C 20.27. Biochemists often define Qlo for a reaction as the ratio of the rate constant at 37 "C to the rate constant at 27C. What must be the energy of activation for a reaction that has a Qlo of 2.5? Ans. 71 kJ - mol-' 20.28. In gaseous reactions important for the understanding of the upper atmosphere, H20 and 0 react bimolecularly to form two OH radicals. A H for this reaction is 72 kJ at 500 K and E, is 77 kJ . mol-'. Estimate E, for the bimolecular recombination of two OH radicals to form H20 and 0. Ans. 5 kJ ' mol-' 20.29. H2 and I2 react bimolecularly in the gas phase to form HI, and HI in turn decomposes bimolecularly into H2 and 12. The energies of activation for these two reactions were observed to be 163 kJ . mol-' and 184 kJ . mol-' , respectively, over the same temperature ranges near 100C. What do you predict from these data for A H of the gaseous reaction H2 + I2 2HI at lOO"C? * Ans. -21 kJ ---j 2 . 0 Predict the form of the rate law for the reaction 2A + B 03. products, if the first step is the reversible dimerization of A, followed by reaction of A2 with B in a bimolecular step. Assume that the equilibrium concentration of A2 is very small compared with [A]. Ans. rate = k[AI2[B] 2 . 1 The following reaction was observed in aqueous solution: 03. 2Cu2++ 6CN-+ 2Cu(CN), + (CN), and the rate was found to be of the form ~ [ C U ~ + ] ~ [ CIf the~ . step is the rapid development of the N - ] first complexation equilibrium to form the relatively unstable (with respect to reversal of the complexation step) Cu(CN), , what rate-limiting step could account for the observed kinetic data? Ans. bimolecular decomposition: 2Cu(CN), -+ 2Cu(CN), + (CN), CHAP. 201 RATES O F REACTIONS 35 1 20.32. The hydrolysis of (i-C3H70),POF was studied at different acidities. The apparent first-order rate constant, k, at a particular temperature was found to depend on pH but not on the nature or concentration of the buffer used to regulate the pH. k was fairly constant from pH 4 to pH 7, but rose from this constant value with decreasing pH below 4 or with increasing pH above 7. What is the nature of the phenomenon responsible for this behavior? Ans. The reaction is catalyzed by both H + and OH- 20.33. It has been found that the rates of reaction of a ketone in mildly basic solution are identical for the following three reactions: (a) reaction with Br2 leading to the substitution of an H on the ketone by a Br, (b) conversion of the D-isomer of the ketone into an equimolar rnixture of the D- and L-isomers, (c) isotopic exchange of a hydrogen atom on the carbon next to the C = O group of the ketone by a deuterium atom in the solvent. The rate of each of these reactions is equal to k[ketone][OH-] and is independent of [Br2]. What can be concluded about the mechanism from these observatioris? Ans. The rate-determining step for all three reactions must be the preliminary reaction of the ketone with OH-, probably leading to the conjugate base of the ketone. The conjugate base subsequently reacts very rapidly with (a) Br2, (b) some acid in the medium, or (c) the deuterated solvent. 20.34. Many free radical reactions (e.g., polymerization) can be initiated by a light-sensitive substance, P, the molecules of which dissociate into two free radicals upon absorption of a photon. Subsequent reactions are either propagation, in which one radical, R-, is produced for each radical consumed, or termination, in which two radicals combine to form nonradical products. The bulk of the chemical change is due to the propagation step. A small steady-state concentration of free radicals prevails throughout. Explain how such a mechanism accounts for the dependence of the rate on the square root of the light intensity, I . Ans. Since propagation does not change the free radical concentration, the rate of initiation must equal the rate of termination. If the rate of initiation is first order in P and proportional to the light intensity, ki[P]I, and the rate of termination is a bimolecular reaction with a rate, kt[R*I2, then at the steady state, P*I = [(ki/kt) PIII If the propagation step is first order in [Re], i.e., proportional to [R-1, then the observed rate is as shown above, proportional to 20.35. The addition of a catalyst to a certain reaction provides an alternative pathway, with an energy of activation which is less than that of the uncatalyzed reaction by 14.7kJ/mol. By what factor is the rate constant increased by the catalyst at a temperature of 420 K? By what factor is the rate constant of the reverse reaction increased? (Assume that the frequency factor is not afrected by the catalyst). Ans. 67; same for the reverse reaction This page intentionally left blank Chapter 21 Nuclear Processes In ordinary chemical reactions, the atoms of the reactant molecules regroup themselves to form the product molecules. In such reactions, the outer electrons of the atoms undergo rearrangements in being transferred wholly or in part from one atom to another. The atomic nuclei, on the other hand, change their positions with respect to each other but are themselves unchanged. There are other reactions in which the nuclei themselves are broken down, and in which the product materials do not contain the same elements as the reactants. The chemistry of such nuclear reactions is called nuclear chemistry. Spontaneous disintegration of individual nuclei is usually accompanied by the emission of highly penetrating radiation and is known as radioactivity. Other nuclear changes result from the interactions of a neutron or high-energy photon with a nucleus, or from the impact of some high-velocity charged particle upon a target nucleus. The results of such nuclear bombardment are highly dependent on the energy. FUNDAMENTAL PARTICLES For the purposes of this chapter, the discussion of the fundamental particles occurring in nuclear reactions will be limited to the nucleons (the proton and neutron), the (negative) electron, and the positive electron (orpositron). In Table 21-1 the masses of these particles are given in atomic mass units (Chapter 2) and their charges are expressed as multiples of the elementary charge e = 1.602 x 10-19 C . Table 21-1 Particle Proton Neutron Symbol Mass/s 1.007276 5 1.008664 9 0.000 548 6 -1 0.000 548 6 BINDING ENERGIES The mass of an atom, in general, is not equal to the sum of the masses of its component protons, neutrons, and electrons. If we could imagine a reaction in which free protons, neutrons, and electrons combine to form an atom, we would find that for all nuclides except 'H the mass of the atom is slightly less than the mass of the component parts and also that a tremendous amount of energy is released when the reaction occurs. The loss in mass is exactly equivalent to the released energy, according to the Einstein equation, E=mc2 Energy = (change in mass) x (velocity of light)* 353 354 NUCLEAR PROCESSES [CHAP. 21 This energy-equivalent of the calculated loss of mass is called the binding energy of the atom or nucleus, When m is expressed in kilograms and c in meters per second, E is in joules. A more convenient unit of energy for nuclear reactions is the MeV (see Chapter 8). The Einstein equation gives Energy = = (change in mass in u)(1.6605 x 10-27 kg/u)(2.998 x 10* m/s)' 1.6022 x lO-I3 J/MeV (931.5 MeV) x (change in mass in U) NUCLEAR EQUATIONS The rules for balancing nuclear equtions are different from the rules for balancing ordinary chemical equations. 1. 2. 3. 4. 5. 6. 7. 8. Each particle is assigned a superscript equal to its mass number or number of nucleons, A , and a subscript equal to its atomic number or nuclear charge, Z . A free proton is the nucleus of the hydrogen atom, and is therefore assigned the notation !H. A free neutron is assigned zero atomic number because it has no charge. The mass number of a neutron is 1. The full notation for a neutron is hn. An electron (e- or p-) is assigned the mass number zero and the atomic number - 1; hence the full notation -ye. A positron (e+ or p') is assigned the mass number zero and the atomic number +I; hence the full notation +ye. An alpha particle (a-particle) is a helium nucleus, and is therefore represented by the full notation ;He (or :a). A gamma ray (y) is a photon (Chapter 8); it has zero mass number and zero charge, and is notated $y. In a balanced equation the sum of the subscripts (atomic numbers), written or implied, must be the same on the two sides of the equation. The sum of the superscripts (mass numbers), written or implied, must also be the same on the two sides of the equation. Thus the equation for the first step in the radioactive decay of 226Rais Many nuclear processes may be indicated by a shorthand notation, in which a light bombarding particle and a light product particle are represented by symbols in parentheses between the symbols for the initial target nucleus and the final product nucleus. The symbols n, p , d , a , p-, p+, y are used to represent neutron, proton, deuteron (YH), alpha, electron, positron, and gamma rays, respectively. The atomic numbers are commonly omitted because the symbol for any element implies its atomic number. Examples of the corresponding long- and shorthand notations for several reactions follow. '!N+ !H + 'AC +;He TiMg :;Fe 14N(p, a ) "C :;A1 :zMn + An + + :H -+ + 2;n + !H '7Al(n, p)27Mg 55Mn(d,2 1 z ) ~ ~ F e Just as an ordinary chemical equation is a shortened version of the complete thermochemical equation which expresses both energy and mass balance, so each nuclear equation has associated with it a term, either written or implied, expressing energy balance. The symbol Q is usually used to designate the net energy released when all reactant and product particles of matter are at zero velocity. Q is the energy equivalent of the mass decrease accompanying the reaction, and is usually expressed in MeV. CHAP. 211 NUCLEAR PROCESSES 355 RADIOCHEMISTRY Special properties of radioactive nuclides make them useful tracers for following complex processes. Radiochemistry is that branch of chemistry which involves the applications of radioactivity to chemical problems, as well as the chemical processing of radioactive substances. A radioactive nuclide is spontaneously converted to another nuclide by one of the following processes, in which there is an overall decrease in mass. 1. Alpha decay: An a-particle is emitted and the daughter nucleus has an atomic number, 2, two units less, and a mass number, A , four units less than the parent's. Thus 2. P- decay: An electron is emitted and the daughter has a Z-value one unit greater than the parent's, with no change in A . Thus 3. pf decay: A positron is emitted and the daughter has a Z-value one unit less than the parent's, with no change in A . Thus 40 Z~SC --+ :Ca + +;P The emitted positron is itself unstable in the neighborhood of electrons and is normally consumed, after being slowed down by collisions, in the following annihilation reaction: 4. K-electron capture: By capturing an orbital electron within its own atom, a nucleus can reduce its 2-value by one unit without a change in A . Thus :Be + -ye(K-capture) -+ :Li A radioactive nucleus decays by a first-order process, so thLat (20-I), (20-2), and (20-3) apply. The stability of a nucleus with respect to spontaneous decay may be indicated by its first-order rate constant, k , or by the half-life, t1/2. Radioactivity is measured by observing the high-energy particles produced directly or indirectly as a result of the disintegration process. A convenient unit of radioactivity is the curie, defined by I Ci = 3.700 0 x 10" disintegration:; per second The activity of a sample, expressed in curies, depends both on the number of atoms of the radioactive nuclide (i.e., the mass of the sample) and on the half-life (or disintegration rate constant). See Problem 21.14. The subunits millicurie (mCi), microcurie (pCi), etc., are also used. Solved Problems 21.1. How many protons, neutrons, and electrons are there in each of the following atoms: (b) 12C, (c) 206Pb? (a) (a) 3He, From the atomic mass table (pages 385-386) or the periodic chart, we see that the atomic number of He is 2; therefore the nucleus must contain 2 protons. Since the mass number of this isotope is 3 , the sum of the numbers of protons and neutrons is 3; therefore there is 1 neutron. The number of electrons in the atom is the same as the atomic number, 2. (b) The atomic number of carbon is 6; hence the nucleus must contain 6 protons. The number of neutrons is 12 -- 6 = 6. The number of electrons is the same as the atomic number, 6. 3 56 NUCLEAR PROCESSES [CHAP. 21 (c) The atomic number of lead is 82; hence there are 82 protons in the nucleus. The number of neutrons is 206 - 82 = 124. There are 82 electrons. 21.2. Complete the following nuclear equations: The sum of the subscripts on the left is 7 + 2 = 9. The subscript of the first product on the right is 8. Hence the second product on the right must have a subscript (nuclear charge) of 1 . The sum of the superscripts on the left is 14 + 4 = 18. The superscript of the first product on the right is 17. Hence the second product on the right must have a superscript (mass number) of 1 . The particle with nuclear charge 1 and mass number 1 is the proton, iH. (b) The nuclear charge of the second product particle (its subscript) is (4 2) - 6 = 0. The mass number of the particle (its superscript) is (9 + 4) - 12 = 1 . Hence the particle must be the neutron, An. The reactants, :Be and iH, have a combined nuclear charge of 5 and mass number of 10. In addition to (c) the a-particle, a product will be formed of charge 5 - 2 = 3, and mass number 10 - 4 = 6. This is !Li, since lithium is the element of atomic number 3. ( d ) The nuclear charge of the second particle is 15 - 14 = + l . The mass number is 30 - 30 = 0. Hence the particle must be the positron, +ye. (e) The nuclear charge of the second particle is 1 - 2 = - 1 . Its mass number is 3 - 3 = 0. Hence the particle must be an electron, -ye. ( f ) The reactants, iiCa and !He, have a combined nuclear charge of 22 and mass number of 47. The ejected product will have a charge 22 - 21 = 1, and mass number 47 - 46 = 1 . This is a proton and should be represented within the parentheses by p . (a) + 21.3. What is the total binding energy of 12C and what is the average binding energy per nucleon? Although "binding energy" is a term referring to the nucleus, it is more convenient to use the mass of the whole atom (nuclide) in calculations, since these are the masses that are given in tables. If M ( X ) is the atomic mass of nuclide X, M(nuc1eus) = M ( X ) - Z M ( e - ) BE = { Z M ( p )+ ( A - Z ) M ( n ) }- M(nuc1eus) (4 (2) The nucleus of X consists of Z protons and A - Z neutrons. Hence its binding energy, BE, is given by Applying (I) both to the nucleus of X and to the proton, which is a iH nucleus, and substituting in (2), BE = { z [ M ( : H ) - ~ ( e - 1 + ( A - z ) M ( ~ )- [ M ( x )- ~ ~ ( e - 1 1 1 ) = {ZM(tH) + ( A - Z ) M ( n ) }- M ( X ) In other words, nuclear masses can be replaced by atomic (nuclidic) masses in calculating the binding energy. Whole atom masses can, in fact, be used for mass-difference calculations in all nuclear reaction types discussed in this chapter, except for ,@ processes where there is a resulting annihilation of two electron masses (one /3+ and one p-). The data needed for 12C can be obtained from Tables 2-1 and 21-1. Mass of 6 `H atoms = 6 x 1.00783 = 6.046 98 Mass of 6 neutrons = 6 x 1.00866 = 6.051 96 Total mass of component particles = 12.09894 Mass of `*C = 12.00000 LOSS in mass on formation of I2c = 0.09894 U Binding energy = (931.5 MeV)(0.098 94) = 92.1 MeV CHAP. 211 NUCLEAR PROCESSES 357 Since there are 12 nucleons (protons and neutrons), the average binding energy per nucleon is (92.1 MeV)/ 12, or 7.68 MeV. 21.4. Evaluate Q for the 7Li(p,n)7Bereaction. The change in mass for the reaction must be computed. Reactants ;Li 7.01600 iH 1.00783 8.023 83 Products An :Be 1.00866 7.01693 8.025 59 Increase of mass = 8.025 59 - 8.023 83 = 0.001 76 U A corresponding net amount of energy must be consumed, equal to (931.5)(0.00176) MeV, or 1.64 MeV; thus Q = -1.64 MeV. This energy is supplied as kinetic energy of the bombarding proton and is part of the acceleration requirement for the proton supplied by the particle accelerator. 21.5. The Q-value for the 3He (rz,p) reaction is 0.76 MeV. What is the nuclidic mass of 3He? The reaction is :He+hn--+iH+:H The mass loss must be 0.76/931.5 = 0.000 82 U. The mass balance can be calculated on the basis of whole atoms. Reactants 3He n x Then (x X Products 'H 1.00783 + 1.00866 1.00866 3H 3.01605 4.023 88 + 1.00866) - 4.023 88 = 0.00082, or x = 3.01604 U. 21.6. Calculate the maximum kinetic energy of the p- emitted in the radioactive decay of 6He. Assume that the p- has its maximum energy when no other emission accompanies the process. The process referred to is !H --+ 2Li + - y ~ In computing the mass change during this process, only the whole atomic masses of 6He and 6Li need be considered, since, if one adds two electrons to each side of the above (nuclear) reaction, one has enough for a whole helium atom on the left and a whole lithium atom on th.e right. Mass of 6He = 6.018 89 Mass of 6Li = 6.015 12 Loss in mass = 0.003 77 U Energy equivalent = (931.5)(0.00377) MeV = 3.51 MeV The maximum kinetic energy of the 0-particle is 3.51 MeV. 21.7. 13N decays by ,8+ emission. The maximum kinetic energy of the p+ is 1.20 MeV. What is the nuclidic mass of I3N? The reaction is 358 NUCLEAR PROCESSES [CHAP. 21 This is the type of process, mentioned in Problem 21.3, in which a simple difference of whole atom masses is not the desired quantity. Mass difference = [M(nucleus) for I3N]- [M(nucleus) for I3C]- M(e) = [M(I3N)- 7M(e)] - [M(I3C)- 6M(e)] - M(e) = M(13N) - M(I3C) - 2M(e) = M(I3N) - 13.00335 - 2(0.000 55) = M(I3N) - 13.00445 This expression must equal the mass equivalent of the maximum kinetic energy of the ,f?+. 1.20 MeV = 0.001 29 U 93 1.5 MeV/u Then 0.001 29 = M(I3N) - 13.00445 or M(I3N)= 13.00574 U 21.8. Consider the two nuclides of mass number 7, 7Li and 7Be. Which of the two is stabler? How does the unstable nuclide decay into the stable one? Table 2-1 shows that 7Be has a larger mass than 7Li. Thus :Be can decay spontaneously into iLi, but not vice versa. There are two types of decay process in which 2 is decreased by one unit without a change in mass number A : p emission and K-capture. These two processes have different mass balance requirements. ' Assume that the process is p+ emission. It was shown in Problem 21.7 (third line of mass difference equation) that a positron emission can occur (i.e., Q is positive and the reaction is spontaneous) only if the nuclidic mass of the parent species exceeds the nuclidic mass of the daughter by at least twice the rest mass of the electron, 2(0.000 55) = 0.001 10 U. In the present case the actual mass difference between parent and daughter nuclides is 7.016 937.01600 = 0.00093 U. We thus see that positron emission in this case is impossible. By elimination, we conclude that 7Be undergoes K-capture. Note that we have predicted only that 7Be should decay by K-capture into 7Li. We have said nothing about the rate of such a process. Measurements show the half-life of the process to be 53 days. 21.9. An isotopic species of lithium hydride, 'Li2H, is a potential nuclear fuel, on the basis of the following reaction: !Li H 2iHe +: --+ Calculate the expected power production, in megawatts, associated with the consumption of 1.00 g of 'Li2H per day. Assume 100 percent efficiency in the process. The change in mass for the reaction is first computed Mass of !Li = 6.015 12 Mass of :H = 2.014 10 Total mass of reactants = 8.02922 Mass of products = 2(4.002 60) = 8.00520 Loss in mass = 0.02402 U Energy per atomic event = (0.02402 u)(931.5 x 106 eV/u)(1.6022 x lO-I9 J/eV) = 3.584 x 10-l2 J Energy per mol LiH = (3.584 x 10-l2 J)(6.02 x 1023mol-') = 2.16 x 10l2J/mol Power production = [(2.16 x 10'2 J/mo1)/(8.03 g/mol)]( 1.00 g/d) (24 h/d)(3.6 x 103 s/h) = 3.11 x 106 W = 3.11 MW CHAP. 211 NUCLEAR PROCESSES 359 21.10. 18F is found to undergo 90 percent radioactive decay in 366 min. What is its computed half-life from this observation? The rate constant for the decay can be found from (20-2). 90 percent decay corresponds to 10 percent, or 0.10, survival. In dealing with radioactive decay the total population of radioactive element is used in place of its concentration. (Remember that in first-order reactions the rate constant, and thus the half-life as well, are independent of the concentration units.) So in place of the concentration ratio [A]/[AIo,put the ratio of the numbers of atoms N / N o , or moles, or masses, of radioactive element. The mass of radioactive element encountered in the laboratory is exceedingly small; a typical sample can be measured only by its activity. Since its activity is proportional to its population the observed ratio of activities A / A o can be used in place of the number ratio N / N o . Then the half-life can be computed from (20-3). 0.693 0.693 t1/2 = -k 6.29 x 10-3 - 110 min 21.11. Estimate the age of an Egyptian mummy from which a piece of the linen wrapping was analyzed and found to have a 14C activity of 8.1 counts (disintegrations) per minute, per gram of carbon. The half-life of 14Cis 5 730 years. It is generally assumed that over at least the past 30000 years the 14C content of atmospheric carbon (as C02) has been roughly constant. Living plants, which obtained their carbon from the air by photosynthesis, thus have had over this period a constant activity, whose value has been found to be 15.3 counts per minute, per gram of carbon. After the flax was harvested and made into linen there was no longer any absorption of I4C from the atmosphere and the activity decayed. Use (20-3) to find k, then (20-2) to find the time, t, for the activity to decay to its present level. - 0.693 - 1.209 10-4 y - ~ k = -0.693 - ---t112 5730 Y -2.303 (-2.303)( -0.276) = 5260 y = 1.209 x 10-4 y-' log(%) = 1.209 x 10-4 y-I t log(^) k -2.303 21.12. A sample of uraninite, a uranium-containing mineral, was found on analysis to contain 0.214 g of lead for every gram of uranium. Assuming that the lead all resulted from the radioactive disintegration of the uranium since the geological formation of the uraninite and that all isotopes of can uranium other than 238U be neglected, estimate the date when the mineral was formed in the earth's crust. The half-life of 238Uis 4.5 x 109 years. The radioactive decay of 23*Uleads, after 14 steps, to the stable lead isotope, 206Pb.The first of these steps, the a-decay of 238Uwith a 4.5 x 109 y half-life, is intrinsically more than 104times as slow (measured in terms of half-life) as any of the subsequent steps. As a result, the time required for the first step accounts for essentially all the time required for the entire 14-step sequence. In a sample containing 1 g U, there is 0*214 Pb = 1.04 x 10-3 mol Pb 206 g/mol and l'ooo 238 g/mol = 4.20 x 10-3 mol U If each atom of lead in the mineral today is the daughter of a uranium atom that existed at the time of the formation of the mineral, then the original number of moles of uranium in the sample would have been (1.04 + 4.20) x 10-3 = 5.2.4 x lOP3 360 NUCLEAR PROCESSES [CHAP. 21 Then the fraction remaining is N - 4.20 x 1 0 - ~ = 0.802 N~ 5.24 x 10-3 Letting t be the elapsed time from the formation of the mineral in the earth's crust to the present time, we have t=- N 2*303log-No k - -2.303 log0.802 - 1.4 x 109 1.54 x 1O-l0 y-I - 21.13. A sample of 14C02 was to be mixed with ordinary CO2 for a biological tracer experiment. In order that 10 cm3 (S.T.P.) of the diluted gas should have 104 disintegrations per minute, how many microcuries of radioactive carbon are needed to prepare 60 L, of the diluted gas? Total activity = 104 dis/min 1ocm3 (60 L)(103cm3/L) 60 s/min To find the mass of I4C needed to provide the 27 pCi, the procedure of Problem 21.14 would be followed. 21.14. The half-life of 40K is 1.25 x 109 years. What mass of this nuclide has an activity of 1 pCi? Let us first calculate the rate constant and express it in s-'. 0.693 k=-t1/2 0.693 = 1.76 (1.25 x 109 y)(365 d/y)(24 h/d)(3.6 x 103 s/h) 1 0 - l ~s-l The disintegration rate is an instantaneous rate measured under conditions of essential constancy of the concentration (i.e., the population) of 40K atoms. The form of the rate equation is that used in Chapter 20, with the numerical value of the rate taken from the definition of the curie. AN Rate = -At = kN = (3.70 x 10" dis . s-l . Ci-')(10-6 Ci - = 3.70 x 104 dis . s-l . pCi-' N = -rate - 3.70 x 104 atoms . s-l . pCi-I = 2.10 x 1 0 ~ 'atoms . pci-' k 1.76 10-17 s-1 and the corresponding mass is (2.10 x 1021atoms. pCi-')(40 g 40K/rnol) = 0.140 g 40K/pCi 6.02 x 1023atoms/mol 21.15. 2 2 7 Ahas a half-life of 21.8 years with respect to radioactive decay. The decay follows two parallel ~ paths, one leading to 227Thand one leading to 223Fr.The percentage yields of these two daughter nuclides are 1.4 percent and 98.6 percent, respectively. What is the rate constant, in y-', for each of the separate paths? The rate constant for the decay of Ac, kAc,can be computed from the half-life. CHAP. 211 NUCLEAR PROCESSES 361 The overall rate constant for a set of parallel first-order reactions must be equal to the sum of the separate rate constants, kAc = kAc(Th) + kAc(Iir) and the fractional yield of either process is equal to the ratio of the rate constant for that process to the overall rate constant. kAc(Th) = (fractional yield Of Th) X kAc = (0.014)(3.18 X y-') = 4.5 X Y-l kAcp+)= (fractional yield of Fr) x kAc = (0.986)(3.18x 1Ov2 y-') = 3.14 x 10-2 y-I Supplementary Problems 21.16. Determine the number of (a) nuclear protons, (b) nuclear neutrons, following atoms: (1) 70Ge, (2) 72Ge, (3) 9Be, (4) 235U. (c) electrons, in each of the Ans. (1): (a) 32; (b) 38; (c) 32 (2): (a) 32; (b) 40; (c) 32 (3): (4 4; (4 5; (c> 4 (4): (a) 92; (b) 1413; (c) 92 21.17. Write the complete nuclear symbols for natural sodium, phosphorus, and iodine. Each has only one stable isotope. Ans. ::Na, ;iP, $71 21.18. By natural radioactivity, 238Uemits an a-particle. The heavy residual nucleus is called UXl. UXI in turn emits a P--particle. The heavy residual nucleus from this radioactive process is called UX2. Determine the atomic numbers and mass numbers of (a) UXl and (b) UX2. Ans. (a) 90, 234; (b) 91, 234 21.19. By radioactivity, ';;Np emits a p--particle. The residual heavy nucleus is also radioactive and gives rise to 235Uby its radioactive process. What small particle is emitted. simultaneously with the formation of 235U? Ans. a-particle 21.20. Complete the following equations. (a) :;Na + :He --t (b) ;@U +:a+ ? -+ :;Mg +? (c) lo6Ag--+ Io6Cd ? ( d ) '!B + $He + -+ I;N +? 21.21. Complete the notations for the following nuclear processes. (a) 24Mg(d,a)? (b) 26Mg(d,p)? (c) 40Ar(a,p)? ( d ) I2C(d,n)? (e) I3'Te(d, 2n)? (g) 59Co(n, a)? (f) 55Mn(n,y)? I3OI; Ans. (a) "2Na; (b) 27Mg; (c) 43K; ( d ) I3N; (e) (1")56Mn; (g) 56Mn 21.22. If a nuclide of an element in Group I (alkali metals) of the periodic table undergoes radioactive decay by emitting positrons, what is the chemical nature of the resulting element? Ans. noble gas (chemically inert, or nearly so) 21.23. An alkaline earth element is radioactive. It and its daughter elements decay by emitting 3 alpha particles in succession. In what group should the resulting element be found? Ans. Group IV (C group) 362 NUCLEAR PROCESSES [CHAP. 21 21.24. A nuclear power reactor gets its energy from neutron-induced fission, usually of 235U. The splitting is not symmetrical and different pairs of products result. Determine the missing fission product for the two examples given below. (a) n + 2 (6) +2 3 5 + 4n ~ 3 5 ~ 5n + 1 3 9 ~ s. . . + + 1 3 5 + ... ~ Ans. (a) 93Rb; (b) 96Y 21.25. In every case in the previous problem notice how much higher the fission product's mass number A is, compared to the stable element's average atomic mass. All will decay by beta (electron) emission to achieve a more normal A to Z ratio. Write the decay reactions for the four fission products above. Ans. 139Cs 0- '39Ba -+ + 93Rb-+ p- + 93Sr 1 3 5 -+ 0- ' 3 5 ~ e ~ + 9 6 t 0-+ 9 6 ~ r ~ 21.26. If an atom of 235U,after absorption of a slow neutron, undergoes fission to form an atom of 13'Xe and an atom of 94Sr, what other particles are produced, and how many? Ans. 3 neutrons 21.27. Which is the more unstable of each of the following pairs, and in each case what type of process could the unstable nucleus undergo: ( a ) 16C, I6N; (b) I8F, "Ne? Ans. (a) 16C, P--decay; (b) "Ne, both P+-decay and K-electron capture are possibilities on the basis of data furnished here. 21.28. One of the stablest nuclei is 55Mn. Its nuclidic mass is 54.938 U. Determine its total binding energy and average binding energy per nucleon. Ans. 483 MeV, 8.78 MeV per nucleon 21.29. How much energy is released during each of the following fusion reactions? (a) i H Ans. (a) 17.4 MeV; (b) 17.6 MeV + :Li -+ 2;He (h) :H + :H -+ !He +An 21.30. If the energy released in reaction ( a ) of the previous problem is divided equally between the two alpha particles, what is their speed? Ans. 2.0 x 107m/s (No correction was made for relativity.) 21.31. 14C is believed to be made in the upper atmosphere by an ( n , p ) process on 14N.What is Q for this reaction? Ans. 0.62 MeV 21.32. In the reaction 32S(n,y)33Swith slow neutrons, the y is produced with an energy of 8.65 MeV. What is the nuclidic mass of 33S? Ans. 32.971 46 U 21.33. If a pf and a p- annihilate each other and their rest masses are converted into two y-rays of equal energy, what is the energy in MeV of each y? Ans. 0.51 MeV CHAP. 211 NUCLEAR PROCESSES 363 21.34. AE for the combustion of a mole of ethylene in oxygen is - 1.4 x 103 kJ. What would be the loss in mass (expressed in U) accompanying the oxidation of one molecule of ethylene? Ans. 1.6 x 10-8 U. (This value is so small compared to the molecular mass that the change in mass, as in all chemical reactions, is ordinarily not taken into account.) 21.35. The sun's energy is believed to come from a series of nuclear reactions, the overall result of which is the transformation of four hydrogen atoms into one helium atom. How much energy is released in the formation of one helium atom? (Include the annihilation energy of the two positrons formed in the nuclear reactions with two electrons.) Ans. 26.8 MeV 21.36. It is proposed to use the nuclear fusion reaction 2:H -+ :H + :H + energy to produce industrial electric power. If the output is to be 50 MW and the energy of the above reaction is used with 30 percent efficiency, how many grams of deuterium fuel will be needed per day? Ans. 149 g/day 21.37. A pure radiochemical preparation was observed to disintegra1,e at the rate of 4280 counts per minute at 1:35 p.m. At 4: 55 p.m. of the same day, the disintegration rate of the sample was only 1070 counts per minute. The disintegration rate is proportional to the number of radioactive atoms in the sample. What is the half-life of the material? Ans. 100 min as 21.38. An atomic battery for pocket watches has been developed which uses the beta particles from 147Pm the primary energy source. The half-life of '47Pm is 2.62 years. How long would it take for the rate of beta emission in the battery to be reduced to 10 percent of its initial value. A m . 8.7 years 21.39. A set of piston rings weighing 120 g was irradiated with neutrons in a nuclear reactor, converting some of the cobalt present in the steel to 6oCo.Irradiation continued until the total 6oCoactivity was 360 mCi. The rings were inserted in an automobile engine which was operated for 24 h under average conditions, following which 0.27 pCi of 6oCo activity was found in the oil filter. Calculate the rate of wear of the piston rings in mg/y based on the assumption that all the eroded metal was captured in the oil filter. Ans. 33 mg/y 21.40. A charcoal sample taken from a fire pit in an archeologist's excavation of a rock shelter was believed to have been formed when early occupants of the shelter burned wood for cooking. A 100-mg sample of pure carbon from the charcoal was found in 1979 to have a disintegratilon rate of 0.25 count per minute. In what millennium did the tree grow from which the archeological sample was taken? Use the data from Problem 21.11. Ans. 15 millennia ago 21.41. All naturally occurring rubidium cores contain 87Sr, resulting from the beta decay of 87Rb. In naturally occurring rubidium, 278 of every 1 000 rubidium atoms are 87Rb.A mineral containing 0.85% rubidium was analyzed and found to contain 0.0089/o strontium. Assumling that all of the strontium originated by radioactive decay of 87Rb,estimate the age of the mineral. 87Rbhas a half-life of 4.9 x 10" years. Ans. 2.6 x 109 years 21.42. Transuranium elements were originally believed not to occur in nature because of their relatively short halflives. Then 244pU was reported in a natural ore. The half-life of 244pU is 8.0 x 107 years. If this element is stabler than any of its radioactive predecessors and thus has not been produced in this ore in significant 364 NUCLEAR PROCESSES [CHAP. 21 amounts since the ore was deposited, what fraction of the original 244Pucontent would still be present if the ore is assumed to be 5 x 109 years old. A ~ 1~ - l ~ 0 . 21.43. Prior to the use of nuclear weapons, the specific activity of 14Cin soluble ocean carbonates was found to be 16 disintegrations per minute per gram of carbon. The amount of carbon in these carbonates has been estimated as 4.5 x 10l6 kg. How many megacuries of I4C did the ocean carbonates contain? Ans. 320 MCi 21.44. Refer to Problems 21.13 and 21.1 1. Calculate the mass of pure I4C needed to provide 27 pCi of beta activity. Ans. 6.1 x 10-6 grams 21.45. If the limit of a particular detection system is 0.002 disintegration per second for a 1 g sample, what would be the maximum half-life that this system could detect on a 1-g sample of a nuclide whose mass number is around 200? ~ n s .3 x 1 0 ' ~ years 21.46. The activity of 30 pg of 247Cm 2.8 nCi. Calculate the disintegration rate constant and the half-life of 247Cm. is Ans. 1.42 x lO-I5 s-l, 1.55 x 107 years 21.47. How much heat would be developed per hour from a 1-Ci I4C source if all the energy of the P--decay were imprisoned? Ans. 3 J/h 21.48. 32Phas a 14.3 day half-life and emits an energetic p- particle, and so is useful for nutritional studies in plants and animals. A 0.10 pCi sample of 32Pin the form of a soluble phosphate was added to a hydroponic bath containing a young tomato plant. Exactly three weeks later the entire plant was rinsed, dried, and macerated. A 1/100 aliquot was placed in a total activity counter. In a 1.00-min period an average of 625 counts was recorded. How efficient was the growing tomato plant in utilizing the phosphorus? Ans. 78% efficient 21.49. A chemist had a solution which contained various sodium salts and she wanted to determine the total sodium content. She added a quantity of HC1, which she judged to be in excess of the sodium ion equivalent, then added exactly 0.422 9 g of pure NaCl, which had a total 22Naactivity of 22 1 10 counts/min. (22Nais a positron emitter with a half-life of 2.6 years.) The solution was thoroughly mixed, then evaporated until crystals of NaCl were obtained. These were filtered off and repeatedly recrystallized until a small batch of pure NaCl was obtained with an activity of 483 counts/min per gram. What was the mass of sodium ion in the original solution? Ans. 17.8 g Appendix A Exponents A. The following is a partial list of powers of 10. 100 = 1 10' = 10 102 = 10 x 10 = 100 103 = 10 x 10 x 10 = 1 ooo 104 = 10 x 10 x 10 x 10 = ioooo 105 = 10 x 10 x 10 x 10 x 10 = 100000 i o 6 = OX OX i o X i o x IOX 10- 1 = - = o 1 10 .1 1 16)- 2 =-=- 1 = 0.01 102 100 1 1 10-3 = -= -= 0.001 103 1000 1 10-~=-=-104 1 - 0.000 1 ioooo i o = ioooooo In the expression 105, the base is 10 and the exponent is 5. B. In multiplication, exponents of like bases are added. (1) 0 3 x as = a3+5= a ' (2) 102 x 103 = 102+3= 105 (3) 10 x 10 = 10"' = 102 (4) 107 1 0 - ~= 1 0 ~ =~ - 104 (5) (4 x 104)(2 x 10--6)= 8 + 104-6 = 8 x 10-2 (6) (2 x 105)(3 x 10--2)= 6 x 105-2 = 6 x 103 C In division, exponents of like bases are subtracted. . D. Any number may be expressed as an integral power of 10, or as the product of two numbers one of which is an integral power of 10 (e.g. 300 = 3 x 102). (1) (2) (3) (4) 22400 = 2.24 x 104 7200000 = 7.2 x 106 454 = 4.54 x 102 0.454 = 4.54 x 10-' (5) 0.0454 = 4.54 x 1Cl-2 (6) 0.00006 = 6 x lO-' (7) 0.003 06 = 3.06 x 10-3 (8) 0.000 000 5 = 5 x 10-7 Moving the decimal point one place to the right is equivalent to multiplying a number by 10; moving the decimal point two places to the right is equivalent to multiplying by 100, and so on. Whenever the decimal point is moved to the right by n places, compensation can be achieved by dividing at the same time by 10n; the value of the number remains unchanged. Thus 3.25 0.032 5 = -= 3.25 x 10-2 1o2 365 366 EXPONENTS [APP. A Moving the decimal point one place to the left is equivalent to dividing by 10. Whenever the decimal point is moved to the left n places, compensation can be achieved by multiplying at the same time by Ion; the value of the number remains unchanged. For example, 7296 = 72.96 x 102 = 7.296 x 103 E. An expression with an exponent of zero is equal to 1. (1) a = 1 (2) 10' = 1 (3) (3 x 10)' = 1 (4) 7 x 10' = 7 (5) 8.2 x 10' = 8.2 ' F. A factor may be transferred from the numerator to the denominator of a fraction, or vice versa, by changing the sign of the exponent. 1 (1) 10-4 =104 5 (2) 5 x 10-3 =103 (3) 10-2=7 x 102 7 (4) - 5aV2= -a2 5 G. The meaning of the fractional exponent is illustrated by the following. (1) 102/3= (2) lO3l2= &$ (3) 101/2= (4) 43/2 = fi= J64= 8 I. To extract the square root of a power of 10, divide the exponent by 2. If the exponent is an odd number it should be increased or decreased by 1, and the coefficient adjusted accordingly. To extract the cube root of a power of 10, adjust so that the exponent is divisible by 3; then divide the exponent by 3 . The coefficients are treated independently. (1) (2) d%%%6 d = s= d?x &? = 3 x 102 or 300 x x = 6 x 10' or 60 J3.6X103= J36X102 = d % &ZiF = (3) (4) (5) J4.9X10-5 = J49X10-6 = d @ d s = 7 x lOP3 or 0.007 10 or 50 = 2 x 103 or 2000 Bx -;/1.25x105 = ViZTii7 = YEG x -;/103 = 5 x J. Multiplication and division of numbers expressed as powers of ten. 8000 x 2500 = (8 x 103)(2.5x 103) = 20 x 106 = 2 x 107 or 20000000 0.0078 __ 7.8 x 10-3 = 6.5 x 10-5 or 0.000065 120 1.2 x 102 (4 ) (4 10-~)(5 1 0 ~ = ~ 1 0 - ~ ) ( 5x~ io8) = 4 x 52 x 1 0 - ~ + ~100 x 105 = 1 x 107 = (6000000)(0.00004)2 - (6 x 106)(4x 10-5)4 - 6 x 44 106 x 10-20 X x (8 x 102)2(2 10-4)3 - g2 X 23 104 X 10-l2 (800)2(0.0002)3 - 6 x 256 64 x 8 X- 106-20 104-12 = 10-l~ 10-8 = 3 x 10-6 ~ APP. A] EXPONENTS 367 (6) ( d m ) ( d s ) ( m= ) d G z ) ( w ) ( e ) ( = (2 x 10-3)(9 x 10')(4 x 10-2) 10-4 == 7.2 x 10-3 or 0.0072 = 72 x (7) ( & G x P ) ( Y r n ) ( V T i G z F ) ( ; / 6 4 . 1 o - ' ) ( V 5 z ) ( ~ ) = = (4 x 10-')(3 x 10')(6 x 10-*) = 72 x 10-2.or 0.72 Appendix B Significant Figures INTRODUCTION The numerical value of every observed measurement is an approximation. No physical measurement, such as mass, length, time, volume, velocity, is ever absolutely correct. The accuracy (reliability) of every measurement is limited by the reliability of the measuring instrument, which is never absolutely reliable. Consider that the length of an object is recorded as 15.7 cm. By convention, this means that the length was measured to the nearest tenth of a centimeter and that its exact value lies between 15.65 and 15.75 cm. If this measurement were exact to the nearest hundredth of a centimeter, it would have been recorded as 15.70 cm. The value 15.7 cm represents three signijicantfigures (1, 5, 7), while 15.70 cm represents four signzjicant figures (1, 5, 7, 0). A significant figure is one which is known to be reasonably reliable. Similarly, a recorded mass of 3.406 2 g, observed with an analytical balance, means that the mass of the object was determined to the nearest tenth of a milligram and represents five significant figures (3,4, 0, 6, 2), the last figure (2) being reasonably correct and guaranteeing the certainty of the preceding four figures. A 50-mL buret has markings 0.1 ml apart, and the hundredths of a milliliter are estimated. A recorded volume of 41.83 cm3 represents four significant figures. The last figure (3), being estimated, may be in error by one or two digits in either direction. The preceding three figures (4, 1, 8) are completely certain. In elementary measurements in chemistry and physics, the last digit is an estimated figure and is considered as a significant figure. A recorded volume of 28 mL represents two significant figures (2, 8). If this same volume were written as 0.028 L, it would still contain only two significant figures. Zeros appearing as the first figures of a number are not significant, since they merely locate the decimal point. However, the values 0.028 0 L and 0.280 L represent three significant figures (2, 8, and the last zero); the value 1.028 L represents four significant figures (1, 0, 2, 8); and the value 1.0280 L represents five significant figures (1, 0, 2, 8, 0). Similarly, the value 19.00 for the atomic mass of fluorine contains four significant figures. The statement that a body of ore weighs 9800 lb does not indicate definitely the accuracy of the weighing. The last two zeros may have been used merely to locate the decimal point. If it was weighed to the nearest hundred pounds, the weight contains only two significant figures and may be written exponentially as 9.8 x 103 lb. If weighed to the nearest ten pounds it may be written as 9.80 x 103 lb, which indicates that the value is accurate to three significant figures. Since the zero in this case is not needed to locate the decimal point, it must be a significant figure. If the object was weighed to the nearest pound, the weight could be written as 9.800 x 103 lb (four significant figures). Likewise, the statement that the velocity of light is 186000 mi/s is accurate to three significant figures, since this value is accurate only to the nearest thousand miles per second; to avoid confusion, it may be written as 1.86 x 105 mi/s. (Normally the decimal point is placed after the first significant figure.) EXACT NUMBERS Some numerical values are exact to as many significant figures as necessary, by definition. Included in this category are the numerical equivalents of prefixes used in unit definition. For example, 1 cm = 0.01 m by definition, and the units conversion factor, 1.0 x 1OP2 m/cm, is exact to an infinite number of significant figures. 368 APP. B] SIGNIFICANT FIGURES 369 Other numerical values are exact by definition. For example, the atomic mass scale was established by fixing the mass of one atom of 12C as 12.0000 U. As many more zeros could be added as desired. Other examples include the definition of the inch (1 in = 2.540 0 cm) and the calorie (1 cal = 4.184 00 J). ROUNDING OFF A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. When the first digit dropped is less than 5, the last digit retained should remain unchanged; when it is greater than 5, 1 is added to the last digit retained. When it is exactly 5, 1 is added to the last digit retained if that digit is odd. Thus the quantity 51.75 g may be rounded off to 51.8 g; 51.65 g to 51.6 g; 51.85 g to 51.8 g. When more than one digit is to be dropped, rounding off should be done in a block, not one digit at a time. ADDITION AND SUBTRACTION The answer should be rounded off after adding or subtracting, so as to retain digits only as far as the first column containing estimated figures. (Remember that the last significant figure is estimated.) Examples Add the following quantities expressed in grams. 25.340 5.465 0.322 31.127 g (Ans.) (2) (1) 58.0 0.003 8 ___- (3) 4.20 1.652 3 0.015 5.867 3 = 5.87 (4) 415.5 3.64 0.238 419.378 0.000 01 58.003 8 1 = 58.0 g (Ans.) g (Ans.) =419.4 g (AILS.) An alternative procedure is to round off the individual numbers before performing the arithmetic operation, retaining only as many columns to the right of the decimal as would give a digit in every item to be added or subtracted. Examples (2), (3), and (4) above would be done as follows: (2) 58.0 0.0 0.0 58.0 g (3) 4.20 1.65 0.02 ___ 5.87 g (4) 415.5 3.6 0.2 419.3 g answer. The last place, however, is Note that the answer to (4) differs by one in the last place from the 1~-evious known to have some uncertainty in it. MULTIPLICATION AND DIVISION The answer should be rounded off to contain only as many significant figures as are contained in the least exact factor. For example, when multiplying 7.485 x 8.61, or when dividing 0.1642 t 1.52, the answer should be given in three significant figures. This rule is an approximation to a more exact statement that the fractional or percentage error of a product or quotient cannot be any less than the fractional or percentage error of any one factor. For this reason, numbers whose first significant figure is 1 (or occasionally 2) must contain an additional significant figure to have a given fractional error in comparison with a number beginning with 8 or 9. Consider the division 9.84 -= 1.06 9.3 370 SIGNIFICANT FIGURES [APP. B By the approximate rule, the answer should be 1.1 (two significant figures). However, a difference of 1 in the last place of 9.3 (9.3 f 0.1) results in an error of about 1 percent, while a difference of 1 in the last place of 1.1 (1.1 f 0.1) yields an error of roughly 10 percent. Thus the answer 1.1 is of much lower percentage accuracy than 9.3. Hence in this case the answer should be 1.06, since a difference of 1 in the last place of the least exact factor used in the calculation (9.3) yields a percentage of error about the same (about 1 percent) as a difference of 1 in the last place of 1.06 (1.06 f 0.01). Similarly, 0.92 x 1.13 = 1.04. In nearly all practical chemical calculations, a precision of only two to four significant figures is required. Therefore the student need not perform multiplications and divisions manually. Even if an electronic calculator is not available, an inexpensive 10-in slide rule is accurate to three significant figures, and a table of 4-place logarithms is accurate to four significant figures. Since not all electronic calculators are alike, detailed instructions cannot be given here. Read your instruction manual. You should purchase a calculator which, in addition to +, -, x, and -+ functions, provides at least the following: scientific notation (powers of ten); logarithms and antilogarithms (inverse logarithms) both natural and common (base ten); and exponentials (y"). If it has these functions, it will probably have reciprocals (1/x), squares, square roots, and trigonometric functions as well. The use of the arithmetic functions is fairly obvious, but you should use powers of ten except in trivial cases. To enter "96500" for instance, consider it 9.65 x 104 and enter 9.65 EE4. (On most calculators "EE4" means x 104).The calculator keeps track of the decimal point and provides an answer between one and ten times the appropriate power of ten. It will usually display many more figures than are significant, and you will have to round off the final result. If at least one factor was entered as a power of ten, the power-of-ten style will prevail in the display, and you need not fear running "off scale", nor will any significant figures disappear off scale. Appendix C Logarithms 1NTRODUCTION Common Logarithms The logarithm (log, in abbreviated form) of a positive number is the exponent, or power, of a given base that is required to produce that number. For example, since 1000 = 103, 100 = 102, 10 = 10*, 1 = loo, then the logarithms of 1 000, 100, 10, 1, to the base 10 are respectively 3, 2, 1, 0. It is obvious that 1 0 ' . will give some number greater than 10 (which is 10') but smaller than 100 (102). Actually, ~ ~ ~ ~ 101.5377 - 34.49; hence log 34.49 = 1.537 7. The system of logarithms whose base is 10 (called the common or Briggsiun system) had been widely used for making numerical computations (multiplication and division) before the advent of electronic calculators and computers. The techniques will be presented later in this appendix and numerical tables are provided in Appendix D. Students who have calculators will not need these techniques but should learn them for the sake of improving their understanding of logarithms and as a backup method of calculation. Natural Logarithms The logarithmic function that occurs commonly in physics and chemistry as part of the solution to certain differential equations has as its base not the number 10 but the transcendental number e = 2.718 28 . . . . To differentiate between the common and the natural or Nupierian logarithms, a more explicit notation could be used: log,, N = x and log, N == y , where 10" = N and ey = N . In this book, and in many chemistry and physics books, the notation Log N is used to indicate the logarithm to the base 10, and In N to indicate the natural logarithm to the base e. Although it is possible to construct logarithm tables for the base e, or for any other base for that matter, there is a simple relationship between any two base systems such that a number found by the use of the common logarithmic table can easily be converted to the logarithm to any other base. Specifically, for the interconversion of common and natural logarithms, it can be shown that In N = 2.303 log N Other Properties of Logarithms Negative numbers do not have logarithms. The logarithm of a number greater than 1 is positive; of a number less than 1 it is negative. The logarithm of 0 is infinitely negative. Given the value of the logarithm of a number of a certain base, the process of finding the number is called taking the antilogarithm. Taking the antilogarithm is synonymous with raising the base to a power equal to the logarithm. Logarithms themselves have no units. Also in principle one: cannot take the logarithm of a number with units, but in fact one often does so when using relationships for which the units have been designated. For example, pH = - log[H+] provided that [)-I+]is in units of mol/L. (1) OBTAINING LOGS ON AN ELECTRONIC CALCULATOR Enter the number, either in direct or exponential notation. Be sure it has a positive sign. Press LOG for the common log or LN for the natural log and the result will be displayed. To find the antilog of a value, enter it, adjust its sign to plus or minus as required, and press first INV (inverse), then LOG or 37 1 372 LOGARITHMS [APP. C LN, depending whether the value is a common log or natural log. Some calculators have the function 10" and ex in place of INV LOG and INV LN, respectively. BASIC PRINCIPLES OF LOGARITHMS A. Since logarithms are exponents, all properties of exponents are also properties of logarithms. The logarithm of the product of two numbers is the sum of their logarithms. logab = loga + logb log ( 5 280 x 48) = log 5 280 + log48 B. The logarithm of the quotient of two numbers is the logarithm of the numerator minus the logarithm of the denominator. a log -= log 536 - log 24.5 log-=loga-logb 536 24.5 b The log of the nth power of a number is n times the log of the number. log an = n log a This rule can be used to derive ( I ) . If x = logN C. log (4.28)3= 3 log 4.28 that is, if 10" = N then from Basic Principle of Logarithms C above, applied now to the base e, 1nN = In 10" = x In 10 = (logN)(ln 10) The numerical value of In 10 is 2.302 585.. . ; so to four significant figures, In N = 2.303 log N D. The log of the nth root of a number is the log of the number, divided by n. 1 1 1 n logd%=-log32 2 10gm=~log792 logG=-loga This is a special case of C, since the nth root of a number is the number raised to the (l/n)th power. USING COMMON LOGARITHMS FOR CALCULATION Definition of Terms The number before the decimal point is the characteristic of the log, and the decimal fraction part is the rnantissa of the log. In the example, log 34.49 = 1.5377, the characteristic is 1 and the mantissa is -5377. The mantissa of the log of a number is found in tables, printed without the decimal point. Each mantissa in the tables is understood to have a decimal point preceding it, and the mantissa is always considered positive. The characteristic is determined by inspection from the number itself according to the following rules: (1) For a number greater than 1, the characteristic is positive and is one less than the number of digits before the decimal point. For example: Number Characteristic 5297 348 900 34.8 60 4.764 3 3 2 1 2 1 0 0 APP. C] LOGARITHMS 373 (2) For a positive number less than I, the characteristic is negative and its magnitude is one more than the number of zeros immediately following the decimal point. The negative sign of the characteristic is written in one of three ways: (a) above the characteristic, as 1, 2, and so on; (b) as 9. -10,8, -10, and so on; (c) in front of the number in the usual way, as -1, -2, and so on. Thus the characteristic of the logarithm of 0.348 5 is i , 9 . - 10, or -1; of the logarithm of 0.051 3, it is 2,8. - 10, or -2. (This discussion does not apply to the common logarithms of numbers less than one which are read off an electronic calculator. The difference, which is always negative, appears directly in the display). Using the Common Logarithm Tables Suppose it is required to find the complete log of the number 728. In the table of logarithms in Appendix D glance down the N column to 72, then horizontally to the right to column 8 and note the entry 8621 which is the required mantissa. Since the characteristic is 2, log 728 = 2.862 1. (This means that 728 = 102.g621 1. The mantissa for log 72.8, for log 7.28, for log 0.728, for log 0.072 8, etc., is .862 1, but the characteristics differ. Thus: log728 = 2.862 1 log72.8 = 1.862 1 log7.28 = 0.862 1 log0.728 = 1.862 1 or 9.862 1 - 10 or log0.0728 = 2.862 1 or 8.862 1 - 10 or log0.00728 = 3.862 1 or 7.862 1 - 10 or - 1 + .862 1 - 2 + .862 1 - 3 + .862 1 - To find log 46.38: Glance down the N column to 46, then horizontally to column 3 and note the mantissa 6656. Moving farther to the right along the same line., the figure 7 is found under column 8 of Proportional Parts. The required mantissa is .665 6 + .OOO 7 = .666 3. Since the characteristic is 1, 10g46.38 = 1.666 3. The mantissa for log 4 638, for log 463.8, for log 46.38, etc., is .666 3, but the characteristics differ, as in the example above. Exercises. Find the logarithms of the following numbers. + sign must be written before the mantissa because the mantissa is always positive. Note that if a negative characteristic is written by the third method given above, -3 for example, a (1) 454 (2) 5280 (3) 96500 (4) 30.48 (5) 1.057 0.621 0.9463 0.0353 0.0022 (10) 0.0002645 (6) (7) (8) (9) Ans. (1) 2.657 1 (2) 3.7226 (3) 4.9845 (4) 1.4840 (5) 0.024 1 9.793 1 - 10 9.9760 - 10 (8) 2.5478 or 8.5478 - 10 (9) 3.3424 or 7.3424- 10 (10) 4.4224 or 6.4224 - 10 (6) 1.793 1 or (7) 1.9760 or or -1 or -1 or -2 or -3 or -4 + .793 1 + .9760 + S478 + .3424 + .4224 Sometimes the log of a number must be used in an algebraic equation, such as y = 7.5logx, or in graphs. If x is greater than 1, log x is positive and there is no special problem. If x is between zero and 1, however, logx is negative. This negative log, according to the above rules, is written as the sum of a positive mantissa and a negative characteristic. For algebraic manipulations it is preferable to treat log x as a single number with a definite sign, either positive or negative. For such a purpose, 2.748 6 would be written as - 1.2514, obtained by adding -2 and +.748 6 algebraically. Exercises. Write the logarithms of the following numbers as quantit:ies suitable for substitution in an algebraic equation. (1) 0.275 (2) 0.000394 (3) 0.0149 Ans. (1) -0.5607 (2) -3.4045 (3) -1.8268 Antilogarithms The antilogarithm is the number corresponding to a given logarithm. "The antilog of 3" means "the number whose log is 3"; that number is obviously 1000. In general, antilog N = 10N. 374 LOGARITHMS [APP. C Suppose it is required to find the antilog of 2.6747, i.e., the number whose log is 2.6747. The characteristic is 2 and the mantissa is .6747. Using the table of Logarithms in Appendix D, locate the mantissa 6739 in the table, corresponding to the digits 472. This is 8 less than the desired 6747, corresponding in the proportional parts table to a 9 for the fourth digit, and giving the result 4729. Since the characteristic is 2, there are three digits to the left of the decimal point. Hence 472.9 is the required number. It should be understood that the antilog of 1.6747 is 47.29; the antilog of 0.674 7 is 4.729; the antilog of 9.674 7 - 10 is 0.472 9; etc. On the other hand, the antilog of - 1.674 7 must be rewritten as antilog of 2.325 3, or 8.325 3 - 10, or -2 + .325 3, before the tables may be used, because only positive mantissas are found in the table. Exercises. Find the numbers corresponding to the following logarithms. (1) 3.1568 (2) 1.6934 (3) 5.6934 (6) 0.9142 (4) 2.5000 (5) 2.0436 (7) 0.0008 (8) 9.7507 - 10 or i.7507 or (9) 8.0034 - 10 or 2.0034 or (10) 7.2006 - 10 or 3.2006 or (11) -0.2436 (12) -3.7629 + .750 7 - 2 + .0034 - 3 + .2006 -1 Ans. (1) 1435 (2) 49.37 (7) 1.002 (8) 0.5632 (3) 493700 (9) 0.01008 (4) 316.2 (10) 0.001 587 (5) 110.6 (11) 0.5707 (6) 8.208 (12) 0.000 1726 Logarithms of Numbers Expressed in Scientific Notation + log4.50 = 7 + 0.653 2 = 7.653 2 log (4.50 x 10-4) = log4.50 + log 10-4 = -4 + l0g4.50 = -4 + 0.653 2 = 4.653 2 = 6.653 2 - 10 log (4.50 x 107) = log4.50 + log 107 = 7 In general, if a number is expressed as a product of two factors, the first being a number between 1 and 10, and the second being an integral power of ten, the logarithm has as its mantissa the logarithm of the first factor and as its characteristic the exponent of ten in the second factor. Exercises. Find the logarithms of the following numbers. (1) 3.75 x 10' (3) 6.6 x 10-27 (2) 6.02 1 0 ~ ~ 0.75 104 (4) (5) 60.3 x 10F8 (6) 2.09 x IO-l5 Ans. (1) 2.5740 (2) 23.7796 (3) 37.819 5 = 3.819 5 - 30 = -27 + .819 5 (4) 3.875 1 + .7803 (6) 15.320 1 = 5.320 1 - 20 = -15 + .320 1 (5) 7.7803 = 3.7803 - 10 = -7 Conversely, an antilogarithm can be expressed directly in scientific notation. The power of 10 is the characteristic; the coefficient of the power of 10 is the antilog of the mantissa, with the decimal point in the antilog following the first digit. antilog 3.8420 = (antilog 3420) x (antilog 3) = 6.95 x 103 antilog 3.842 0 = (antilog .842 0) x [antilog(-3)] = 6.95 x lOP3 antilog - 3.8420 = antilog 4.1580 = (antilog .1580) x [antilog(-4)] = 1.439 x 10-4 Exercises. Write the antilogarithms of the following in scientific notation. (1) 10.4769 (2) i9.2044 (3) 5.0403 (4) 4.140 2 - 20 (5) 7.621 6 (6) -8.276 3 Ans. (1) 2.998 x 10" (2) 1.602 x 10-19 (3) 1.097 x 10' (4) 1.381 x (5) 4.184 x 107 (6) 5.292 x 10-9 APP. C] LOGARITHMS 375 Illustrations of the Use of Logarithms 1. Find the value of 487 x 2.45 x 0.0387. Let x = 487 x 2.45 x 0.038 7. log x = log 487 + log 2.45 log 0.038 7 = 1.6644 x = antilog 1.6644 = 46.17 or 46.2 (to three significant figures) log 48 7 log 2.45 log0.038 7 = 2.6875 = = logx = + or log 136.3 = 2.134 5 0.3892 8.587 7 - 10 (add) 11.6644 - 10 1.6644 2. Find x = 136.3 65.38 log x = log 136.3 - log 65.38 = 0.3 19 1 x = antilog0.319 1 = 2.085 ~ log 65.38 = 1.815 4 (subtract) lolgx = 0.319 1 3. Find x = 1 log 1 = 0 = 10.0000 - 10 22.4 log x = log 1 - log 22.4 = 8.649 8 - 10 log 22.4 = 1.3502 (subtract) x = antilog 8.649 8 - 10 = 0.044 56 or 0.044 6 logx = 8.649 8 - 10 (three significant figures) -, Adding or subtracting 10.0000 - 10,20.000 0 - 20, etc., to or from any logarithm does not change its value. 4. 17.5 x 1.92 Find x = 0.283 x 0.031 4 logx= (log 17.5 +log 1.92)(log 0.283 -I- log 0.03 1 4) log 17.5 = 1.2430 l0g0.283 = 9.451 8 - 10 log 1.92 = 0.283 3 (add) log0.0314 = 8.496 9 - 10 (add) 17.9487 - 20 or 7.948 7 - 10 1.5263 or 11.5263 - 10 = 11.526 3 - 10 log 17.5 + log 1.92 log0.283 + log0.0314 = 7.948 7 - 10 (subtract) logx = 3.5776 x = antilog 3.577 6 = 378 1 or 3.78 x 103 Find x = (6.138)3. l o g x = 3(10g6.138) = 3(0.788 1) = 2.3643 x = antilog 2.364 3 = 23 1.4 5. 6. Find x = d ' or (7514)1'2. logx =4(10g7514) =&(3.8758)= 1.9379 x = antilog 1.9379 = 86.68 7 . Find x = or (0.0592)'/3. l o g x = flog0.0592 = i(8.7723 - 10) = t(28.7723 - 30) = 9.5908 - 10 x = antilog 9.590 8 - 10 = 0.389 8 or 0.390 (three significant figures) Jm' 8. Find x = 10gx=&(310g152)= & ( 3 ~ 2 . 1 8 1 8 ) = $ ( 6 . 5 4 5 4 ) = 3 . . 2 7 2 7 x = antilog 3.272 7 = 1 874 or 1.87 x 103 (three significant figures) 9. Find (6.8 x 10-4)3 or (6.8)3 x 10-l2. 10g(6.8)~ 3(10g6.8) 3(0.832 5 ) = 2.497 5 = (6.8)3 = antilog2.497 5 = 314.4 or 3.14 x 102 Then (6.8 x 10-4)3 = 3.14 x 102 x 10-l2 = 3.14 x 10-" = I 43. (three significant figures) 376 LOGARITHMS [APP. C 10. Find 48.31 x 10-" or d83.1 x 10-'* or d m x 10-6. = (log 83.1) = (1.919 6) = 0.959 8 log d m = antilog0.959 8 = 9.1 16 or 9.12 Then d8.31 x 10-" = 9.12 x 10-6 4 4 11. Find x = 971*665 log x = 1.665(10g97) = 1.665(1.986 8) = 3.309 x = antilog 3.309 = 2.04 x 103 log 1.665 = .2214 log 1.987 = .298 3 log(1.665 x 1.987) = ,5197 1.665 x 1.987 = 3.309 Note that the logarithm of a logarithm must be taken if logarithm tables are used to perform this type of calculation. 12. Find x = ln28.25 x = 2.303 10g28.25 = 2.303 x 1.4510 log x = log 2.303 log 1.451 0 = .524 0 x = antilog 0.5240 = 3.342 + 10g2.303 = .362 3 log 1.4510 = .1617 (add) = .5240 logx 10g57.9 = 1.7627 l0g2.303 = .3623 , log4.512 = .6544 (add) 2.779 4 13. Find x = 57.91n (3.25 x 104). x = 57.9 x 2.303 log (3.25 x 104) = 57.9 x 2.303 x 4.512 log x = log 57.9 + log 2.303 + log 4.5 12 = 2.7794 x = antilog 2.779 4 = 602 Appendix D Four-Place Common Logarithms N 10 11 12 13 14 15 16 17 18 19 O 1 2 0086 0492 0864 1206 1523 1818 209 5 2355 2601 2833 3054 3263 3464 3655 3838 4014 4183 4346 4502 4654 3 4 1 5 6 7 8 !? Proportional Parts 1 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 8 8 7 6 6 6 5 5 5 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 12 11 10 10 9 8 8 7 7 7 6 6 6 6 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 3 4 17 15 14 13 12 11 11 10 9 9 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 5 21 19 17 16 15 14 13 12 12 11 11 10 10 9 9 9 8 8 8 7 7 7 7 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 5 6 25 23 21 19 18 17 16 15 14 13 13 12 12 11 11 7 29 26 24 23 21 20 18 17 16 16 15 14 14 13 12 8 33 30 28 26 24 22 21 20 19 18 17 16 15 15 14 9 37 34 31 29 27 25 24 22 21 20 19 18 17 17 16 m 0043 0414 0453 0792 0828 1139 1173 1461 1492 1761 2041 2304 2553 2788 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 6021 6128 6232 6335 6435 6532 6628 6721 6812 6902 6990 7076 7160 7243 7324 1790 2068 2330 2577 2810 3032 3243 3444 3636 3820 3997 4166 4330 4487 4639 4786 4928 5065 5198 5328 5453 5575 5694 5809 5922 603 1 6138 6243 6345 6444 6542 6637 6730 682 1 691 1 6998 7084 7168 725 1 7332 0128 0170 053 1 0569 0899 0934 1239 1271 1553 1584 1847 2122 2380 2625 2856 3075 3284 3483 3674 3856 1875 2148 2405 2648 2878 3096 3304 3502 3692 3874 0212 0253 0294 0607 0645 0682 0969 1004 1038 1303 1335 1367 1614 1644 1673 1903 2175 2430 2672 2900 3118 3324 3522 3711 3892 4065 4232 4393 4548 4698 1931 2201 2455 2695 2923 3139 3345 3541 3729 3909 4082 4249 4409 4564 4713 1959 2227 2480 2718 2945 3160 3365 3 560 3747 3927 4099 4265 4425 4579 4728 0334 0374 0719 0755 1072 1106 1399 1430 1703 1732 1987 2253 2504 2742 2967 3181 3385 3579 3766 3945 4116 4281 4440 4594 4742 2014 2279 2529 2765 29139 3201 3404 3598 3784 3962 4133 4298 4456 4609 4757 20 21 22 23 24 25 26 27 28 29 403 1 4048 4200 4216 4362 4378 4518 4533 4669 4683 4829 4969 5105 5237 5366 5490 5611 5729 5843 5955 6064 6170 6274 6375 6474 6571 6665 6758 6848 6937 7024 7110 7193 7275 7356 10 12 14 15 10 11 13 15 9 1 1 1 3 1 4 9 11 1 2 1 4 9 1 0 1 2 1 3 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 4800 4814 4942 4955 5079 5092 5211 5224 5340 5353 5465 5587 5705 582 1 5933 6042 6149 6253 6355 6454 5478 5599 5717 5832 5944 6053 6160 6263 6365 6464 4843 4857 487 1 4886 4900 4983 4997 5011 5024 5038 5119 5132 5145 5159 5172 5250 5263 5276 5289 5302 5378 539 1 5403 5416 5428 5502 5623 5740 5855 5966 6075 6180 6284 6385 6484 6580 6675 6767 6857 6946 7033 7118 7202 7284 7364 5514 5635 5752 5866 5977 6085 6191 6294 6395 6493 6590 6684 6776 6866 6955 7042 7126 7210 7292 7372 5527 5647 5763 5877 5988 6096 620 1 6304 6405 6503 6599 6693 6785 6875 6964 7050 7135 7218 7300 7380 5539 5658 5775 5888 5999 6107 6212 6314 6415 6513 6609 6702 6794 6884 6972 7059 7143 7226 7308 7388 5551 5670 5786 5899 6010 6117 6222 6325 6425 652.2 6618 6712 6803 6893 698'1 706'7 7152 7235 7316 7396 9 1 0 1 1 1 8 1 0 1 1 1 8 9 1 1 1 8 9 1 0 1 8 9 1 0 1 7 7 7 7 7 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 6 3 2 2 2 1 9 1 0 1 1 8 1 0 1 1 8 9 1 0 8 9 1 0 8 9 1 0 8 7 7 7 7 7 7 6 6 6 6 6 6 6 6 7 9 1 0 8 9 8 9 8 9 8 9 8 7 7 7 7 7 7 7 6 6 8 9 8 8 8 8 8 8 7 7 7 9 655 1 656 1 6646 6656 6739 6749 6830 6839 6920 6928 7007 7093 7177 7259 7 340 7016 7101 7185 7267 7348 N I 0 1 2 3 4 5 6 7 8 9 377 378 FOUR-PLACE COMMON LOGARITHMS [APP. D ~~ ~ N 56 57 58 59 60 61 62 63 64 55 0 7404 7482 7559 7634 7709 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 845 1 8513 8573 8633 8692 8751 8808 8865 8921 8976 903 1 9085 9138 9191 9243 9294 9345 9395 9445 9494 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 1 7412 7490 7566 7642 7716 7789 7860 793 1 8000 8069 8136 8202 8267 833 1 8395 8457 8519 8579 8639 8698 8756 8814 8871 8927 8982 9036 9090 9 143 9196 9248 9299 9350 9400 9450 9499 9547 9595 9643 9689 9736 9782 9827 9872 9917 996 1 2 7419 7497 7574 7649 7723 7796 7868 7938 8007 8075 8142 8209 8274 8338 8401 8463 8525 8585 8645 8704 8762 8820 8876 8932 8987 9042 9096 9149 9201 9253 9304 9355 9405 9455 9504 9552 9600 9647 9694 9741 9786 9832 9877 992 1 9965 3 7427 7505 7582 7657 773 1 7803 7875 7945 8014 8082 8149 8215 8280 8344 8407 8470 853 1 8591 865 1 8710 8768 8825 8882 8938 8993 9047 9101 9154 9206 9258 9309 9360 9410 9460 9509 9557 9605 9652 9699 9745 979 1 9836 988 1 9926 9969 4 7435 7513 7589 7664 7738 78 10 7882 7952 8021 8089 8156 8222 8287 835 1 8414 8476 8537 8597 8657 8716 8774 8831 8887 8943 8998 9053 9106 9159 9212 9263 9315 9365 9415 9465 9513 9562 9609 9657 9703 9750 9795 984 1 9886 9930 9974 / 5 7443 7520 7597 7672 7745 7818 7889 7959 8028 8096 8162 8228 8293 8357 8420 8482 8543 8603 8663 8722 8779 8837 8893 8949 9004 9058 9112 9165 921 7 9269 9320 9370 9420 9469 9518 9566 9614 966 1 9708 9754 9800 9845 9890 9934 9978 6 745 1 7528 7604 7679 7752 7825 3896 7966 8035 8102 8169 8235 8299 8363 8426 8488 8549 8609 8669 8727 8785 8842 8899 8954 9009 9063 9117 9170 9222 9274 9325 9375 9425 9474 9523 957 1 9619 9666 971 3 9759 9805 9850 9894 9939 9983 7 7459 7536 7612 7686 7760 7832 7903 7973 8041 8109 8176 8241 8306 8370 8432 8494 8555 8615 8675 8733 8791 8848 8904 8960 901 5 9069 9122 9175 9227 9279 9330 9380 9430 9479 9528 9576 9624 967 1 9717 9763 9809 9854 9899 9943 9987 8 7466 7543 7619 7694 7767 7839 7910 7980 8048 8116 8182 8248 8312 8376 8439 8500 8561 862 1 8681 8739 8797 8854 8910 8965 9020 9074 9128 9180 9232 9284 9335 9385 9435 9484 9533 958 1 9628 9675 9722 9768 98 14 9859 9903 9948 999 1 9 7474 7551 7627 770 1 7774 7846 7917 798 7 8055 8122 8189 8254 8319 8382 8445 8506 8567 8627 8686 8745 8802 8859 8915 897 1 9025 9079 9133 9186 9238 9289 9340 9390 9440 9489 9538 9586 9633 9680 9727 9773 9818 9863 9908 9952 9996 ~ -~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Proportional Parts 2 3 4 5 6 7 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 7 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 8 9 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 9 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 N 0 1 2 3 4 5 6 7 8 9 Absolute temperature, 3, 10 Absolute zero of temperature, 3 Absorption of light, 107, 115 Acids and bases, 265 Arrhenius concept, 265 Bronsted-Lowry concept, 265, 272 carboxylic, 229, 23 1 catalysis, 348 conjugate pair, 265, 272 equivalent mass, 189, 196 indicators, 268, 286 ionization, 265, 274 Lewis concept, 266, 274 neutralization, 206 strong and weak, 265, 272 titration, 270, 286 water ionization, 266, 279 Actinides, 112 Activated state, 340 Activity, 247 Activity coefficient, 247 Alcohol, 229, 230 Aldehyde, 229, 231 Aliphatic compounds, 228 Alkanes, 227, 230 Alkenes, 228, 230 Alkynes, 229 Alpha decay, 355 Alpha particle, 354, 355 Amide, 229, 231 Amine, 229, 231 Ampere (unit), 315, 322 Amphiprotic substance, 266 Angstrom (unit), 2, 7 Angular momentum, 108 quantum number, 109 Annihilation of electron and positron, 355 Antibonding orbitals, 129, 144 Aromatic compounds, 228 Arrhenius concept of acids and bases, 265 Asymmetric carbon atom, 228, 239 Atmosphere (unit), 2, 7 Atmospheric pressure, 61, 64 Atomic mass, 17, 19 table, 385 Atomic mass unit, 17, 165 Atomic number, 17, 354 Aufbau (see Buildup principle) Avogadro's hypothesis, 77 Avogadro's number, 18 ' Balancing of equations (Cont.): oxidation-reduction, 177, 179 Band theory of metallic bonding, 138 Bar (unit), 2, 61 as pressure standard, 61, 95 Beta decay, 355 Beta particle, 354 Bimolecular reactions, 339 Binding energy, nuclear, 353, 356 Biochemistry, 232 Bipyramid, trigonal, 147, 148 Body-centered cubic lattice, 161 Bohr orbit, 107, 117 Bohr theory, 107, 117 Boiling-point elevation, 2 16, 219 Bond angles, 132, 146 Bondenergy, 124, 143, 155 Bond length., 132, 146 Bond order, 130, 144 Bonding in metals, 138 Bonding orbitals, 129, 144 Boyle's law, 62, 65 Bravais lattice, 161 British thermal unit, 79 Bronsted-Lowry concept, 265, 272 Buffer solutions, 268, 286 Buildup principle, for atoms, 112 for diatomic molecules, 138 Calorie, 79 kilocalorie, 79 nutritionist's, 79 Calorimetry, 93, 97 Carbohydrate, 233 Catalysts, 250, 337, 348 Cells, galvanic, 316, 325 Cellulose, 233 Celsius scale, 3, 10 Cesium chloride structure, 167 Charge of electron, 108, 353 Charles' law, 62, 65 Chemical equations, balancing of, 47, 177 calculations from, 45, 48, 185, 197 ionic notation, 176 Chemical equilibrium, 246 effect of concentration, 250, 253 effect of pressure, 249, 253 effect of temperature, 248, 253 Chiral carbon atom, 228, 239 Close packing, 162, 165 Collision frequency, 80 Combustion, heat of, 98, 100 Common-ion effect, 275, 278 Complex ions (see Coordination complexes) 379 Back-bonding in complexes, 158 Balancing of equations, 47 nuclear, 354, 356 380 INDEX Composition calculations, 27, 32 Concentration of a solution, 189, 192 Conjugate acid and base, 265, 272 Constant heat summation (see Hess' law) Conversion factors, 5 Conversion of units, 4, 5 Coordination complexes: absorption of light, 149 dissociation constant, 299, 301 isomerism, 135, 151 magnetic properties, 135, 149 molecular orbitals, 134 nomenclature, 133 optical activity, 137, 151 spin, 135, 149 stability constant, 299, 301 Coordination number, 162, 165 Coulomb (unit), 298, 315 Couples, 317, 325 Covalence, 124, 140 coordinate, 133 Covalent bonding, 124, 164 Covalent radii, 112, 119 table, 132 Crystal forces, 161, 168 Crystals, 161 covalent, 164, 168, 169 ionic, 164 molecular, 163, 168 Cubic lattices, 161 body-centered, 161 cesium chloride structure, 166, 167 face-centered, 161, 165 sodium chloride structure, 170, 171 Curie (unit), 355 Current, electric, 315, 322 Dalton (unit), 17 Dalton's law, 63, 69 Dating by radioactivity, 359 De Broglie, 109 Debye (unit), 155 Delocalization of electrons, 124 Density, 8 ideal gases, 63, 69, 82 units, 2, 8 Deuteron, 354 Dew point, 264 Diamagnetism, 114 Diffusion of gases, 81 Dilution problems, 191, 198 Dimensional analysis, 4 Dimensions, 1 Dipole moment, 126, 144 Dipole-dipole attractions, 164 Disproportionation, 326 Dissociation constant: acids and bases, 265, 274 Dissociation constant (Cont.): coordination complexes, 299, 301 Distribution, law of, 218, 233 Efflorescence, 259 Effusion of gases, 81, 87 Einstein equation, 353 Electric current, 315, 322 Electrical units, 315, 322 Electrochemistry, 3 15 Electrode potential, 316, 325 table, 318 Electrode reactions in electrolysis, 321, 327 Electrolysis, electrode reactions, 321, 327 Faraday's laws, 3 15, 323 Electron, 353, 354 charge, 108 configurations, 112 delocalization, 124 mass, 108 repulsion, 147 sea (in metals), 138, 139 spin, 109 Electron affinity, 114, 119 Electron volt, 113, 114, 364 Electronegativity, 126 Empirical formula, 19, 27, 30 End point, 270 Endothermic process, 94 Energy, of activation, 339, 346 binding, 353, 356 electrical, 3 15, 322 heat, 93 internal, 93 light, 107, 115 mass equivalent, 353, 356 Enthalpy, 93, 94, 96 of condensation, 99 of formation, 95, 97, 100; table, 96 of fusion, 94, 99 of neutralization, 101 of reaction, 95, 97, 100 standard, 95, 100 of sublimation, 96 of vaporization, 95, 99 Entropy, 243, 250 Equations: balancing, 47 calculations from, 45, 48, 78, 83, 185, 205 nuclear, 354, 356 oxidation-reductioq, 177, 179 thermochemical, 95, 100 Equilibrium (see Chemical equilibrium) Equilibrium constant, 247, 254 acids and bases, 265, 274 dissociation of complexes, 299 gases, 247, 254 hydrolysis, 268, 28 1 INDEX 38 1 Equilibrium constant (Cont.): ionization of solvent, 273 ionization of water, 266, 279 solubility product, 299, 302 stability of complexes, 299, 301 Equivalent (eq), unit, 174, 189, 293 Equivalent mass: acids and bases, 189, 196 electrode processes, 3 16, 323 oxidation-reduction, 190, 197 Ester, 229, 231 Estimation, 5 Ether, 229, 231 Excess reactant, 46, 51 Exothermic process, 94 Exponents, 365 Extensive quantities, 94 Extraction, 218, 223 Face-centered cubic lattice, 161, 165 Factor-label method, 4, 49 Fahrenheit scale, 3, 10 Faraday constant, 3 15 Faraday's laws, 315, 323 Fat, 233 First law of thermodynamics, 243 First-order reactions, 338 Force, units of, 2 Formal charge, 126, 141 Formation, enthalpy of, 95, 97, 100; table, 96 free energy of, 245, 252; table, 245 Formula, 18 empirical, 19, 27, 30 from precise nuclidic mass, 28, 37 ionic, 139 molecular, 30, 82 and oxidation state, 179 structural, 125, 140 Formula unit, 19 Fractional order, 338, 348 Free energy, 243 and cell potential, 319, 327 of formation, 252; table, 245 of oxidation-reduction, 319, 327 of reaction, 246, 252 Freezing-point lowering, 2 15, 2 18 Frequency factor, 338 Frequency of light, 107, 115 Functional groups, 229, 235 Fundamental particles, 353 Fusion, heat of, 94, 99 Galvanic cell, 316, 325 Gamma ray, 354 Gas constant, 78, 87 Gas laws, 61, 64, 77, 81 Gas volumes, 61, 64, 78, 81 Gases, density, 63, 69, 82 diffusion and effusion, 80, 81, 83, 87 ideal or perfect, 62, 64, 77 molecular masses, 77, 81 partial pressure, 63, 69, 85 solubility in liquids, 218, 222 stoichiornetry, 78 Gay-Lussac's law, 63, 66 Geometrical isomers, 137, 150 Gibbs free energy, 243 Graham's law, 81, 87 Ground state, 127 Half-cell potentials, 3 17, 325 Half-life, 338, 342, 355, 359 Halide, organic, 229, 230 Heat, 93 of combustion, 98 of condensation, 99 of fusion, 94, 99 of reaction, 95, 97, 100 specific, 93, 97, 98 of sublimation, 96 units, 79 of vaporization, 95, 99 Heat capacity, 93, 97, 98 Henry's law, 218, 222 Hertz (unit), 107 Hess' law, 95, 97 Hexagonal unit cell, 162, 168 Holes, octahedral and tetrahedral, 163, 166 Hund's rule, 114, 118 Hybrid orbitals, 127, 130, 131, 132 Hybridization, 127, 132, 134 Hydrates, 31 Hydrocarbons, 227 Hydrogen bonding, 163, 169 Hydrogen-like orbitals, 11 1 Hydrolysis, 268, 28 1 Ideal gas, 62, 68, 77 Ideal gas law, 77 Ideal solution, 21 5 Inch, 2 Indicator, 268, 286 Internal energy, 93 Ion-electron, method for redox equations, 177, 178, 179 Ionic radii, 112, 119; table, 164 Ionization, primary, 270 secondary, 270 Ionization constant of acids and bases, 265, 274 Ionization energy, 113, 117 Ionization of water, 266, 279 Ions, complex (see Coordination complexes) Irreversible process, 243 Isomerism, 135, 150, 228 geometrical, 137, 150, 228, 234, 238 382 INDEX Isomerism (Cont.): optical, 228, 234, 239 structural, 136, 228, 234, 237 Isotopes, 17, 19 Joule (unit), 79 K-electron capture, 355 Kelvin temperature, 3, 10 Ketone, 229, 231 Kilocalorie, 79 Kilogram, 1 Kilowatt, 322 Kilowatt-hour, 322 Kinetic theory of gases, 79, 87 Kinetics, 337 Lanthanides, 1 12 Lattice, 161 body-centered, 161 face-centered, 161, 165 hexagonal, 162, 168 simple cubic, 161 Law, of Boyle, 62, 65 of Charles, 62, 65 of constant heat summation, 95 of Dalton, 68, 69 of distribution, 218, 223 of Faraday, 315, 323 first, of thermodynamics, 243 of Gay-Lussac, 63, 66 of Graham, 83 of Henry, 218, 222 of Hess, 95 of multiple proportions, 23 periodic, 111 of Raoult, 215, 220 of relativity, 116 second, of thermodyamics, 243 third, of thermodynamics, 244 Le Chatelier's principle, 248 Length, units of, 1, 2, 5, 6 Lewis acids and bases, 266, 274 Lewis formulas, 125, 140 Ligand, 133 Ligand field, 135, 149 Light, absorption, 107 emission, 107 energy, 107, 115 frequency and wavelength, 107, 1 15 particle character, 107 wave character, 107 Limiting reactant, 46, 51 Lipid, 233 Liquids, intermolecular forces in, 164, 170 Liter, 2, 6 Logarithms, 371 common, table of, 377 Logarithms (Cont.): natural, 371 London forces, 164 Magnetic properties: diamagnetism, 1 14 paramagnetism, 1 14 Magnetic quantum number, 109 Main Group elements, 112 Manometer, 6 1 Mass, electronic, 108, 353 energy equivalent of, 353, 354, 356 nuclidic, 17; table, 18, 386 units of, 1, 2, 5, 6 Mass number, 17, 354 molecular, 19 Mass spectrometer, 28 Maxwell-Boltzmann distribution, 79 Measurements, 1 Mechanism of reactions, 339, 346 Metallic bonding, 138 Meter (unit), 1 Microcurie, 355 Millicurie, 355 Milliequivalent, 196, 205 Molality, 190, 195 Molar concentration, 189, 194, 205 Molar mass, 19, 37 of gases, 77, 81 Molar volume, 77 Mole, 18, 19 Mole fraction, 190, 196 of gases, 85, 86 Mole method, 49 Mole percent, 191 Molecular orbitals, 128, 134, 144 Molecular mass, 19 of gases, 77, 81 Multicenter bonds, 130 Multiple proportions, law of, 23 Nanometer, 115 Natural logarithms, 338, 371 Nernst equation, 320, 327 Neutralization, 206, 270 enthalpy of, 101 Neutron, 17, 353 Newton (unit), 2 Noble gases, 112, 118 Nodal plane, 11 1 Nodal surface, 110 Nomenclature, coordination compounds, 133 ionic compounds, 124, 139 organic compounds, 227, 233 Nonbonding orbitals, 129 Nonstoichiometric factors, 28 Normality, 189, 196, 205 Nuclear equations, 354, 356 INDEX 383 Nucleic acid, 233 Nucleon, 17, 353, 355 Nucleus, 17, 353 Nuclide, 17, 358 Nuclidic mass, 17, 357; table, 18, 386 Nuclidic molecular mass, 28, 37 Octahedral hole, 163 Octahedron, 134, 151 Octet rule, 124, 140 Olefins, 227 Optical isomers, 137, 151, 228, 234 Orbital, 109 angular dependence, 111 antibonding, 129 bonding, 129 hybrid, 127 molecular, 128, 134, 145 nonbonding, 129 Order of reaction, 337, 341 Organic compounds, 227 Osmotic pressure, 216, 217, 221 Oxidation-reduction, 175 balancing equations, 177, 179 cell potential, 317, 319, 325 equivalent mass, 190, 197 free energy, 319, 327 spontaneous reactions, 3 19, 326 Oxidation state, 175, 179 method of balancing equations, 178, 180 Paramagnetism, 114, 144 Partial pressure, 63, 69, 86 equilibrium constant, 247 Pascal (unit), 2 Pauli exclusion principle, 111 Percent ionic character, 144 Percentage composition and formula, 27, 30, 32 Perfect gas (see Ideal gas) Periodic table, 112 chart, inside back cover Permittivity, 108 Perovskite structure, 168 pH, 267, 279 Photoelectric effect, 1 15 Photon, 107, 115 Pi-bonds, 130 pK, 267 Planck's constant, 107 pOH, 267 Polybasic acids, 270, 283 Polymer, 232, 236, 239 Polyprotic acids, 270, 283 Positron, 353 emission, 355, 357 Potential, half-cell and cell, 316, 317, 319, 325 nonstandard, 3 19 standard, 316, 325; table, 318 Potential difference, 3 15 Pound, 2, 5 Power, electrical, 315 Pre-exponential factor, 339 Pressure, 61 equilibrium constant, 248 gages, 61 of gases, 61 osmotic, 216, 217, 221 partial, 63, 69, 85 units of, 2, 7, 61 vapor pressure of solution, 215, 220 vapor pressure of solvent, 63, 70 Primary ionization, 270 Principal quantum number, 109 Protein, 233 Proton, 17, 353 Pyramid, square, 148 Q,energy of nuclear reaction, 354, 357 Q, ion product, 300 Q, reaction quotient, 247 Quadratic equation, formula, 256 Quantitative factor, 28 Quantum, 107, 116 Quantum mechanics, 109 Quantum number, 109 Radii: covalent, 112, 119; table, 132 ionic, 112, 1 19; table, 164 van der Waals, 112 Radioactive dating, 359 Radioactive tracer, 360 Radioactivity, 355 Radiochemistry, 355 Raoult's law, 215, 220 Rate constants, 337, 341 and equilibrium constants, 345 radioactive decay, 359 Reaction mechanism, 339, 346 Reaction order, 337, 338, 341 Reaction rate, 337, 338 Reaction types, 46 . Reactions of gases, 83 Reduced m(assof nucleus, 108, 117 Reducing agent, equivalent mass, 190, 197 Reduction (see Oxidation-reduction) Resonance, 125, 131, 141 Reversible process, 243 Rydberg constant, 108, 117 Salt bridge, 316, 317 Schrodinger equation, 109 Screening, electronic, 112 Second law of thermodynamics, 243 384 INDEX Second-order reactions, 337, 341 Secondary ionization, 270 Shapes of molecules, 131, 144, 146 Shell, electron, 109 SI units, 1 prefixes, 2 Significant figures, 368 Size, of atoms and ions, 112 Skeleton equation, 47 Sodium chloride structure, 170, 171 Solubility, of gases, 218, 222 of salts, 300, 302 Solubility product, 299, 302 and standard potential, 329 Solute, 189 Solutions: buffer, 268, 286 concentrations of, 189, 192 of gases, 218, 222 ideal, 215 osmotic pressure, 216, 217, 221 of precipitates, 300 standard, 189, 205 vapor pressure, 215, 220 Solvent, 189 Specific heat, 93, 97 Spectrochemical series (of ligands), 135 Spectrum, 107 hydrogen, 107, 108 Speed of gas molecules, 79, 83, 84 average, 79 most probable, 80 root-mean-square, 80, 87 Spin, high and low, 135, 149 Spontaneous process, 243, 3 19, 326 Square pyramid, 148 Stability constant, 299, 301 and standard potential, 329 Stability of oxidation states, 326 Standard conditions (S.T.P.), 62 Standard electrode potential (see Standard potential) Standard enthalpy, 95; table, 96 Standard entropy, table, 245 Standard free energy of formation, table, 245 Standard potential, 316, 325 and solubility product, 329 and stability constant, 329 table, 318 Standard solutions, 189, 205 Standard state, 95, 245 of dissolved ions, 95 Starch, 233 State of a system, 95 State functions, 243 Stoichiometry, 25 Structural formulas, 125, 140 Structural isomers, 136, 150 Sublimation, heat of, 96 Sugar, 233 Symbol of an element, 18 Temperature, 3, 10 absolute, 3, 10 Celsius, 3, 10 Fahrenheit, 3, 10 Kelvin, 3, 10 Termolecular reactions, 339 Tetrahedral hole, 154 Tetrahedron, 127, 146 Thermodynamics, first law, 243 second law, 243 third law, 244 Titration, 270 Ton, metric, 5, 7, 8 Torr, 2, 61 Transition elements, 112 Trigonal bipyramid, 147, 148 Unimolecular reactions, 339 Unit cell, 161 Unit factor method, 4 Units, 1-10 electrical, 3 15, 322 heat, 79 multiples and submultiples, 2 Universal gas constant, 78, 85 Valence-bond representation, 124 Valence electrons, 112 Van der Waals forces, 163, 164 Van der Waals radius, 112 Vapor pressure, 63, 70 lowering of, 21 5, 220 Vaporization, heat of, 95, 99 Volt, 315, 322 Volume, units of, 1, 6 VSEPR (Valence Shell Electron Pair Repulsion), 132 Water, density, 7 ionization, 266, 279 specific heat, 93 triple point, 3 vapor pressure, 63, 70 Watt (unit), 315, 322 Wave number, 107 Wavelength, 107, 115 of electron beam, 109, 116 Zinc blende structure, 171 Table of Atomic Masses (1995) Scaled to the relative atomic mass A, (I2C)=12 Name Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Bohrium Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Curium Dubnium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Symbol Ac A1 Am Sb Ar As At Ba Bk Be Bi Bh B Hr Cd Ca Cf (3 Ce CS Atomic Number 89 13 95 51 18 33 85 56 97 4 83 107 5 35 48 20 98 6 58 55 17 24 27 29 96 105 66 99 68 63 100 9 87 64 31 32 79 72 Atomic Mass (227) 26.981 54 (243) 121.760 39.948 74.921 60 (2 10) 137.327 (247) 9.012 18 208.980 38 (264) 10.811 79.904 112.411 40.078 (25 1 ) 12.010 7 140.1 16 132.905 45 35.452 7 51.996 1 58.933 20 63.546 (247) (262) 162.50 (252) 167.26 15 1.964 (257) 18.998 40 (223) 157.25 69.723 72.6 1 196.966 55 178.49 385 - Name Hassium Helium Holmium Hydrogen Indium Iodine Iridium Iron Symbol Hs He Ho H In I Ir Fe Kr La Lr Pb Li Lu Mg Mn Mt Md Hg MO Nd Ne NP Ni Nb N No OS Atomic Number 108 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 109 101 80 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 Atomic Mass (265) 4.002 60 164.930 32 1.007 94 114.818 126.904 47 192.217 55.845 83.80 138.905 5 (262) 207.2 6.94 1 174.967 24.305 0 54.938 05 (268) (258) 200.59 95.94 144.24 20.179 7 (237) 58.693 4 92.906 38 14.006 74 (259) 190.23 15.999 4 106.42 30.973 76 195.078 (244) (209) 39.098 3 140.907 65 (145) 231.035 88 C1 Cr CO C U Cm Db D Y ES Er Eu Fm F Fr Gd Ga Ge Au Hf Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Meitnerium Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium 0 Pd P Pt Pu P O K Pr Pm Pa 386 TABLE OF ATOMIC MASSES ~~ ~ Name Radium Radon Rhenium Rhodium Rubidium Ruthenium Rutherfordium Samarium Scandium Seaborgium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Symbol Ra Rn Re Rh Rb Ru Rf Sm sc sg Se Si Ag Na Sr S Ta Atomic Number 88 86 75 45 37 44 104 62 21 106 34 14 47 11 38 16 73 Atomic Mass (226) (222) 186.207 102.905 50 85.467 8 101.07 (26 1) 150.36 44.955 91 (263) 78.96 28.085 5 107.868 2 22.989 77 87.62 32.066 180.947 9 Name Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten (Wolfram) Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Symbol Tc Te Tb T1 Th Tm Sn Ti W U V Xe Yb Y Zn Zr Atomic Number 43 52 65 81 90 69 50 22 74 92 23 54 70 39 30 40 Atomic Mass (98) 127.60 158.925 34 204.383 3 232.038 1 168.934 21 118.710 47.867 183.84 238.028 9 50.941 5 131.29 173.04 88.905 85 65.39 9 1.224 A value in parentheses for an element without any stable nuclides is the atomic mass number of the isotope of that element of longest known half-life. Nuclidic Masses of Selected Radionuclides Name Actinium Americium Astatine Berkelium Bohrium Californium Curium Dubnium Einsteinium Fermium Francium Hassium Lawrencium Meitnerium Symbol Ac Am At Bk Bh Cf Cm Db Es Fm Fr Hs Lr Mt Atomic Number 89 95 85 97 107 98 96 105 99 100 87 108 103 109 Nuclidic Mass 227.027 7 243.0614 209.987 1 247.070 3 264.12 251.079 6 247.070 3 262.1 14 4 252.083 0 257.095 1 223.019 7 265. I30 6 262.1 10 --- Name Mendelevium Neptunium Nobelium Plutonium Polonium Promethium Protactinium Radium Radon Rutherfordium Seaborgium Technetium Thorium Uranium Md NP No Pu P O Pm Pa Ra Rn Rf sg Tc Th U Atomic Number 101 93 102 94 84 61 91 88 86 104 106 43 90 92 Nuclidic Mass 258.098 4 237.048 2 259.101 1 244.064 2 208.982 4 144.912 7 231.035 9 226.025 4 222.017 6 261.108 9 263.1 18 6 97.907 2 232.038 0 238.050 8 The listed mass is that for the longest-lived isotope of the listed element. ...