Chem 101 Chapter 1- Matter

Chem 101 Chapter 1- Matter - The Basic Tools of Chemistry...

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Unformatted text preview: The Basic Tools of Chemistry Platinum resistance thermometer. This device measures temperatures over a range from about Ϫ259 °C to ϩ962 °C. 10 How Hot Is It? “It’s so hot outside you could fry an egg on the sidewalk!” This is an expression we heard as children. But what does it mean to say that something is hot? We would say it has a high temperature—but what is temperature and how is it measured? Temperature and heat are related but different concepts. Although we will discuss the difference in more detail in Chapter 6, for the moment it is easy to think of them this way: Temperature determines the direction of heat transfer. That is, heat transfers from something at a higher temperature to something at a lower temperature. If you touch your finger to a hot match, heat is transferred to your finger, and you decide the match is hot. Early scientists learned that gases, liquids, and solids expand when heated. In his investigations of heat, Galileo Galilei (1564–1642) invented the “thermoscope,” a simple device that depended on the expansion of a liquid in a tube with increasing temperature. Others developed instruments based on this principle, using liquids such as alcohol and mercury. Among them was Daniel Gabriel Fahrenheit (1686–1736). To create his scale, Fahrenheit initially assigned the freezing point of water as 7.5 °F and body temperature as 22.5 °F. He multiplied these values by 4, and then later adjusted them so that the freezing point of water was 32 °F and body temperature was 96 °F. After Fahrenheit’s death a further revision of the scale established the reference temperatures at their current values, 32 °F for the freezing point of water and 212 °F for the Anders Celsius (1701–1744). boiling point. On the current scale, Swedish astronomer and geognormal body temperature is 98.6 °F. rapher. Archives of the Royal Swedish Academy of Sciences NPL photograph © Crown copyright 1997. Reprinted with permission of the controller of HMSO. 1—Matter and Measurement Chapter Goals Chapter Outline See Chapter Goals Revisited (pages 46 and 47). Test your knowledge of these goals by taking the exam-prep quiz on the General ChemistryNow CD-ROM or website. • • • • 1.1 1.2 Compounds and Molecules 1.4 Identify physical and chemical properties and changes. Physical Properties 1.5 Recognize elements, atoms, compounds, and molecules. Physical and Chemical Changes 1.6 Making Measurements: Precision, Accuracy, and Experimental Error 1.8 • Use metric units and significant figures correctly. • Understand and use the mathematics of chemistry. Units of Measurement 1.7 Apply the kinetic-molecular theory to the properties of matter. Mathematics of Chemistry You might have had your temperature taken with a device that is inserted in your ear. This instrument is essentially a pyrometer. Warm humans emit light, albeit at longer wavelengths than a toaster element. A sensor in the ear thermometer scans the wavelength emitted from the eardrum and reports the temperature. This is a useful measure of body temperature because the eardrum shares blood vessels with the hypothalamus, the area of the brain that regulates body temperature. Charles D. Winters A significant advance in temperature measurement came from Anders Celsius (1701–1744). Celsius was a Swedish geographer and astronomer who constructed the Celsius thermometer, which used liquid mercury in a glass tube. The Celsius thermometer scale originally used 0 as the boiling point of water, and 100 as the freezing point of water—reference points that were reversed after Celsius’s death. His contribution to thermometry was to show experimentally that the freezing point of water is unchanged by atmospheric pressure or the latitude at which the experiment is done. Celsius also showed that, in contrast, the boiling point of water does depend on atmospheric pressure. Both of these observations were important to establishing a standard temperature scale that could be used around the world. In modern science there is an interest in determining low and high temperatures well outside the ranges where alcohol and mercury are liquids. Scientists have created new temperature measuring devices for this purpose. The platinum resistance thermometer, for example, relies on the fact that the electrical resistance of platinum wire changes with temperature in a predictable manner. Such devices are extremely sensitive and can make measurements to within one thousandth of a degree over temperatures ranging from Ϫ259.25 °C to ϩ961.78 °C (the melting point of silver). How do you measure a very high temperature—say, a temperature high enough to boil mercury or melt glass or platinum? From watching the heater element on a stove or in a toaster, you know that heated objects emit light. It turns out that the wavelength of the emitted light can be correlated with temperature. A pyrometer, an optical device, is commonly used for this purpose. Elements and Atoms 1.3 Classify matter. Classifying Matter Infrared thermometer. This device depends on the long wavelength radiation emitted by a warm object. 11 12 Chapter 1 Matter and Measurement magine a tall glass filled with a clear liquid. Sunlight from a nearby window causes the liquid to sparkle, and the glass is cool to the touch. A drink of water would certainly taste good, but should you take a sip? If the glass were sitting in your kitchen you might say yes. But what if this scene occurred in a chemical laboratory? How would you know that the glass held pure water? Or, to pose a more “chemical” question, how would you prove this liquid is water? We usually think of the water we drink as being pure, but this is not strictly true. In some instances material may be suspended in it or bubbles of gases such as oxygen may be visible to the eye. Some tap water has a slight color from dissolved iron. In fact, drinking water is almost always a mixture of substances, some dissolved and some not. As with any mixture, we could ask many questions. What are the components of the mixture—dust particles, bubbles of oxygen, dissolved sodium, calcium, or iron salts—and what are their relative amounts? How can these substances be separated from one another, and how are the properties of one substance changed when it is mixed with another? This chapter begins our discussion of how chemists think about matter. After looking at a way to classify matter, we will turn to some basic ideas about elements, atoms, compounds, and molecules and discover how chemists characterize these building blocks of matter. Finally, we will see how we can use numerical information. I • • • Charles D. Winters • Throughout the chapter this icon introduces a list of resources on the General ChemistryNow CD-ROM or website (http://now that will: help you evaluate your knowledge of the material provide homework problems allow you to take an exam-prep quiz provide a personalized Learning Plan targeting resources that address areas you should study Thinking about matter. Is this a glass of pure water? How can you prove it is? 1.1—Classifying Matter A chemist looks at a glass of drinking water and sees a liquid. This liquid could be the chemical compound water. More likely, the liquid is a homogeneous mixture of water and dissolved substances—that is, a solution. It is also possible the water sample is a heterogeneous mixture, with solids being suspended in the liquid. These descriptions represent some of the ways we can classify matter (Figure 1.1). HETEROGENEOUS MATTER MATTER (may be solid, liquid, or gas) Anything that occupies space and has mass COMPOUNDS Nonuniform composition Elements united in fixed ratios Physically separable into... HOMOGENEOUS MATTER Uniform composition throughout PURE SUBSTANCES Fixed composition; cannot be further purified Physically separable into... Chemically separable into... Combine chemically to form... ELEMENTS Cannot be subdivided by chemical or physical processes SOLUTIONS Homogeneous mixtures; uniform compositions that may vary widely Active Figure 1.1 Classifying matter. See the General ChemistryNow CD-ROM or website to explore an interactive version of this figure accompanied by an exercise. 1.1 Photos: Charles D. Winters Solid Liquid Bromine solid and liquid Classifying Matter Gas Bromine gas and liquid Active Figure 1.2 States of matter—solid, liquid, and gas. Elemental bromine exists in all three states near room temperature. The tiny spheres represent bromine (Br) atoms. In elemental bromine, two Br atoms join to form a Br2 molecule. (See Section 1.3 and Chapter 3.) See the General ChemistryNow CD-ROM or website to explore an interactive version of this figure accompanied by an exercise. States of Matter and Kinetic-Molecular Theory An easily observed property of matter is its state—that is, whether a substance is a solid, liquid, or gas (Figure 1.2). You recognize a solid because it has a rigid shape and a fixed volume that changes little as temperature and pressure change. Like solids, liquids have a fixed volume, but a liquid is fluid—it takes on the shape of its container and has no definite shape of its own. Gases are fluid as well, but the volume of a gas is determined by the size of its container. The volume of a gas varies more than the volume of a liquid with temperature and pressure. At low enough temperatures, virtually all matter is found in the solid state. As the temperature is raised, solids usually melt to form liquids. Eventually, if the temperature is high enough, liquids evaporate to form gases. Volume changes typically accompany changes in state. For a given mass of material, there is usually a small increase in volume on melting—water being a significant exception—and then a large increase in volume occurs upon evaporation. The kinetic-molecular theory of matter helps us interpret the properties of solids, liquids, and gases. According to this theory, all matter consists of extremely tiny particles (atoms, molecules, or ions), which are in constant motion. • In solids these particles are packed closely together, usually in a regular array. The particles vibrate back and forth about their average positions, but seldom does a particle in a solid squeeze past its immediate neighbors to come into contact with a new set of particles. • The atoms or molecules of liquids are arranged randomly rather than in the regular patterns found in solids. Liquids and gases are fluid because the particles are not confined to specific locations and can move past one another. • Under normal conditions, the particles in a gas are far apart. Gas molecules move extremely rapidly because they are not constrained by their neighbors. The molecules of a gas fly about, colliding with one another and with the 13 14 Chapter 1 Matter and Measurement container walls. This random motion allows gas molecules to fill their container, so the volume of the gas sample is the volume of the container. An important aspect of the kinetic-molecular theory is that the higher the temperature, the faster the particles move. The energy of motion of the particles (their kinetic energy) acts to overcome the forces of attraction between particles. A solid melts to form a liquid when the temperature of the solid is raised to the point at which the particles vibrate fast enough and far enough to push one another out of the way and move out of their regularly spaced positions. As the temperature increases even more, the particles move even faster until finally they can escape the clutches of their comrades and enter the gaseous state. Increasing temperature corresponds to faster and faster motions of atoms and molecules, a general rule you will find useful in many future discussions. Matter at the Macroscopic and Particulate Levels The characteristic properties of gases, liquids, and solids just described are observed by the unaided human senses. They are determined using samples of matter large enough to be seen, measured, and handled. Using such samples, we can also determine, for example, what the color of a substance is, whether it dissolves in water, or whether it conducts electricity or reacts with oxygen. Observations and manipulations generally take place in the macroscopic world of chemistry (Figure 1.3). This is the world of experiments and observations. Now let us move to the level of atoms, molecules, and ions—a world of chemistry we cannot see. Take a macroscopic sample of material and divide it, again and again, past the point where the amount of sample can be seen by the naked eye, past the point where it can be seen using an optical microscope. Eventually you reach the level of individual particles that make up all matter, a level that chemists refer to as the submicroscopic or particulate world of atoms and molecules (Figures 1.2 and 1.3). Chemists are interested in the structure of matter at the particulate level. Atoms, molecules, and ions cannot be “seen” in the same way that one views the macroscopic world, but they are no less real to chemists. Chemists imagine what atoms must look like and how they might fit together to form molecules. They create models to represent atoms and molecules (Figures 1.2 and 1.3)—where tiny spheres are used to represent atoms—and then use these models to think about chemistry and to explain the observations they have made about the macroscopic world. It has been said that chemists carry out experiments at the macroscopic level, but they think about chemistry at the particulate level. They then write down their observations as “symbols,” the letters (such as H2O for water or Br2 for bromine molecules) and drawings that signify the elements and compounds involved. This is a useful perspective that will help you as you study chemistry. Indeed, one of our goals is to help you make the connections in your own mind among the symbolic, particulate, and macroscopic worlds of chemistry. Pure Substances Let us think again about a glass of drinking water. How would you tell whether the water is pure (a single substance) or a mixture of substances? Begin by making a few simple observations. Is solid material floating in the liquid? Does the liquid have an odor or an unexpected taste or color? Classifying Matter E 1.1 N Particulate M A G I Photos: Charles D. Winters O B S E R V E I R E P R E Macroscopic S E N T H2O (liquid) 888n H2 O (gas) Symbolic Active Figure 1.3 Levels of matter. We observe chemical and physical processes at the macroscopic level. To understand or illustrate these processes, scientists often try to imagine what has occurred at the particulate atomic and molecular levels and write symbols to represent these observations. A beaker of boiling water can be visualized at the particulate level as rapidly moving H2O molecules. The process is symbolized by indicating that the liquid H2O molecules are becoming H2O molecules in the gaseous state. See the General ChemistryNow CD-ROM or website to explore an interactive version of this figure accompanied by an exercise. Every substance has a set of unique properties by which it can be recognized. Pure water, for example, is colorless, is odorless, and certainly does not contain suspended solids. If you wanted to identify a substance conclusively as water, you would have to examine its properties carefully and compare them against the known properties of pure water. Melting point and boiling point serve the purpose well here. If you could show that the substance melts at 0 °C and boils at 100 °C at atmospheric pressure, you can be certain it is water. No other known substance melts and boils at precisely these temperatures. A second feature of a pure substance is that it cannot be separated into two or more different species by any physical technique such as heating in a Bunsen flame. If it could be separated, our sample would be classified as a mixture. Mixtures: Homogeneous and Heterogeneous A cup of noodle soup is obviously a mixture of solids and liquids (Figure 1.4a). A mixture in which the uneven texture of the material can be detected is called a heterogeneous mixture. Heterogeneous mixtures may appear completely uniform but on closer examination are not. Blood, for example, may not look heterogeneous until you examine it under a microscope and red and white blood cells are revealed (Figure 1.4b). Milk appears smooth in texture to the unaided eye, but magnification 15 Chapter 1 Matter and Measurement ϩ ϩ ϩ Ϫ (a) (b) Ϫ Ϫ a and c, Charles D. Winters; b, Ken Edwards/Science Source/Photo Researchers, Inc. 16 (c) Figure 1.4 Mixtures. (a) A cup of noodle soup is a heterogeneous mixture. (b) A sample of blood may look homogeneous, but examination with an optical microscope shows it is, in fact, a heterogeneous mixture of liquids and suspended particles (blood cells). (c) A homogeneous mixture, here consisting of salt in water. The model shows that salt consists of separate, electrically charged particles (ions) in water, but the particles cannot be seen with an optical microscope. a, Charles D. Winters; b, Littleton, Massachusetts, Spectacle Pond Iron and Manganese Treatment Facility would reveal fat and protein globules within the liquid. In a heterogeneous mixture the properties in one region are different from those in another region. A homogeneous mixture consists of two or more substances in the same phase (Figure 1.4c). No amount of optical magnification will reveal a homogeneous mixture to have different properties in different regions. Homogeneous mixtures are often called solutions. Common examples include air (mostly a mixture of nitrogen and oxygen gases), gasoline (a mixture of carbon- and hydrogen-containing compounds called hydrocarbons), and an unopened soft drink. When a mixture is separated into its pure components, the components are said to be purified (see Figure 1.5). Efforts at separation are often not complete in a sin- (a) (b) Figure 1.5 Purifying water by filtration. (a) A laboratory setup. A beaker full of muddy water is passed through a paper filter, and the mud and dirt are removed. (b) A water treatment plant uses filtration to remove suspended particles from the water. 1.2 17 Elements and Atoms See the General ChemistryNow CD-ROM or website: • Screen 1.5 Mixtures and Pure Substances, for an exercise on identifying pure substances and types of mixtures Charles D. Winters gle step, however, and repetition almost always gives an increasingly pure substance. For example, soil particles can be separated from water by filtration (Figure 1.5). When the mixture is passed through a filter, many of the particles are removed. Repeated filtrations will give water a higher and higher state of purity. This purification process uses a property of the mixture, its clarity, to measure the extent of purification. When a perfectly clear sample of water is obtained, all of the soil particles are assumed to have been removed. • Screen 1.6 Separation of Mixtures, to watch a video on heterogeneous mixtures Homogeneous and heterogeneous mixtures. Which is homogeneous? See Exercise 1.1. Exercise 1.1—Mixtures and Pure Substances ■ Exercise Answers In each chapter of the book you will find a number of Exercises. Their purpose is to help you to check your knowledge of the material in that chapter. Solutions to the Exercises are found in Appendix N. The photo in the margin shows two mixtures. Which is a homogeneous mixture and which is a heterogeneous mixture? 1.2—Elements and Atoms Passing an electric current through water can decompose it to gaseous hydrogen and oxygen (Figure 1.6a). Substances like hydrogen and oxygen that are composed of only one type of atom are classified as elements. Currently 116 elements are known. Of these, only about 90—some of which are illustrated in Figure 1.6—are found in nature. The remainder have been created by scientists. The name and symbol for each element are listed in the tables at the front and back of this book. Carbon (C), sulfur (S), iron (Fe), copper (Cu), silver (Ag), tin (Sn), gold (Au), mercury (Hg), and lead (Pb) were known to the early Greeks and Romans and to the alchemists of ancient China, the Arab world, and medieval Europe. However, many other elements—such as aluminum (Al ), silicon (Si), iodine (I), and helium (He)—were not discovered until the 18th and 19th centuries. Finally, artificial elements—those that do not exist in nature, such as technetium (Tc), plutonium (Pu), and americium (Am)—were made in the 20th century using the techniques of modern physics. Many elements have names and symbols with Latin or Greek origins. Examples include helium (He), named from the Greek word helios meaning “sun,” and lead, whose symbol, Pb, comes from the Latin word for “heavy,” plumbum. More recently discovered elements have been named for their place of discovery or for a person or place of significance. Examples include americium (Am), californium (Cf ), and curium (Cm). The table inside the front cover of this book, in which the symbol and other information for the elements are enclosed in a box, is called the periodic table. We will describe this important tool of chemistry in more detail beginning in Chapter 2. An atom is the smallest particle of an element that retains the characteristic chemical properties of that element. Modern chemistry is based on an understanding and exploration of nature at the atomic level. We will have much more to say about atoms and atomic properties in Chapters 2, 7, and 8, in particular. ■ Writing Element Symbols Notice that only the first letter of an element’s symbol is capitalized. For example, cobalt is Co, not CO. The notation CO represents the chemical compound carbon monoxide. Also note that the element name is not capitalized, except at the beginning of a sentence. ■ Periodic Table See the periodic table at General ChemistryNow. It can be accessed from Screen 1.5 or from the Toolbox. See also the extensive information on the periodic table and the elements at the American Chemical Society website: • • .html Chapter 1 Oxygen—gas Matter and Measurement Hydrogen—gas Mercury—liquid Powdered sulfur—solid Copper wire— solid Iron chips— solid Aluminum— solid Water—liquid (a) (b) Figure 1.6 Elements. (a) Passing an electric current through water produces the elements hydrogen (test tube on the right) and oxygen (test tube on the left). (b) Chemical elements can often be distinguished by their color and their state at room temperature. See the General ChemistryNow CD-ROM or website: • Screen 1.7 Elements and Atoms, and the Periodic Table tool on this screen or in the Toolbox Exercise 1.2—Elements Using the periodic table inside the front cover of this book or on the CD-ROM: (a) Find the names of the elements having the symbols Na, Cl, and Cr. (b) Find the symbols for the elements zinc, nickel, and potassium. 1.3—Compounds and Molecules A pure substance like sugar, salt, or water, which is composed of two or more different elements held together by a chemical bond, is referred to as a chemical compound. Even though only 116 elements are known, there appears to be no limit to the number of compounds that can be made from those elements. More than 20 million compounds are now known, with about a half million added to the list each year. When elements become part of a compound, their original properties, such as their color, hardness, and melting point, are replaced by the characteristic properties of the compound. Consider common table salt (sodium chloride), which is composed of two elements (Figure 1.7): • Sodium is a shiny metal that reacts violently with water. It is composed of sodium atoms tightly packed together. • Chlorine is a light yellow gas that has a distinctive, suffocating odor and is a powerful irritant to lungs and other tissues. The element is composed of Cl2 units in which two chlorine atoms are tightly bound together. Photos: Charles D. Winters 18 1.3 19 Compounds and Molecules Solid sodium, Na Photos: Charles D. Winters + Sodium chloride solid, NaCl Chlorine gas, Cl2 Figure 1.7 Forming a chemical compound. Sodium chloride, commonly known as table salt, can be made by combining sodium metal (Na) and yellow chlorine gas (Cl2). The result is a crystalline solid. • Sodium chloride, or common salt, is a colorless, crystalline solid. Its properties are completely unlike those of the two elements from which it is made (Figure 1.7). Salt is composed of sodium and chlorine bound tightly together. (The meaning of chemical formulas such as NaCl is explored in Sections 3.3 and 3.4.) It is important to distinguish between a mixture of elements and a chemical compound of two or more elements. Pure metallic iron and yellow, powdered sulfur (Figure 1.8a) can be mixed in varying proportions. In the chemical compound iron pyrite (Figure 1.8b), however, there is no variation in composition. Not only does iron pyrite exhibit properties peculiar to itself and different from those of either iron or sulfur, or a mixture of these two elements, but it also has a definite percentage composition by weight (46.55% Fe and 53.45% S). Thus, two major differences Figure 1.8 Mixtures and compounds. Charles D. Winters (a) The substance in the dish is a mixture of iron chips and sulfur. The iron can be removed easily by using a magnet. (b) Iron pyrite is a chemical compound composed of iron and sulfur. It is often found in nature as perfect, golden cubes. (a) (b) 20 Chapter 1 Figure 1.9 Names, formulas, and mod- Matter and Measurement NAME els of some common molecules. Models of molecules appear throughout this book. In such models C atoms are gray, H atoms are white, N atoms are blue, and O atoms are red. FORMULA Water Methane Ammonia Carbon dioxide H2O CH4 NH3 CO2 MODEL exist between mixtures and pure compounds: Compounds have distinctly different characteristics from their parent elements, and they have a definite percentage composition (by mass) of their combining elements. Some compounds—such as table salt, NaCl—are composed of ions, which are electrically charged atoms or groups of atoms [᭤ Chapter 3]. Other compounds— such as water and sugar—consist of molecules, the smallest discrete units that retain the composition and chemical characteristics of the compound. The composition of any compound is represented by its chemical formula. In the formula for water, H2O, for example, the symbol for hydrogen, H, is followed by a subscript “2” indicating that two atoms of hydrogen occur in a single water molecule. The symbol for oxygen appears without a subscript, indicating that one oxygen atom occurs in the molecule. As you shall see throughout this book, molecules can be represented with models that depict their composition and structure. Figure 1.9 illustrates the names, formulas, and models of the structures of a few common molecules. Charles D. Winters 1.4—Physical Properties Figure 1.10 Physical properties. An ice cube and a piece of lead can be differentiated easily by their physical properties (such as density, color, and melting point). You recognize your friends by their physical appearance: their height and weight and the color of their eyes and hair. The same is true of chemical substances. You can tell the difference between an ice cube and a cube of lead of the same size not only because of their appearance (one is clear and colorless, and the other is a lustrous metal ) (Figure 1.10), but also because one is much heavier ( lead) than the other (ice). Properties such as these, which can be observed and measured without changing the composition of a substance, are called physical properties. The chemical elements in Figures 1.6 and 1.7, for example, clearly differ in terms of their color, appearance, and state (solid, liquid, or gas). Physical properties allow us to classify and identify substances. Table 1.1 lists a few physical properties of matter that chemists commonly use. Exercise 1.3—Physical Properties Identify as many physical properties in Table 1.1 as you can for the following common substances: (a) iron, (b) water, (c) table salt (chemical name is sodium chloride), and (d) oxygen. Density Density, the ratio of the mass of an object to its volume, is a physical property useful for identifying substances. Density ϭ mass volume (1.1) 1.4 Table 1.1 21 Physical Properties Some Physical Properties Property Using the Property to Distinguish Substances Color Is the substance colored or colorless? What is the color and what is its intensity? State of matter Is it a solid, liquid, or gas? If it is a solid, what is the shape of the particles? Melting point At what temperature does a solid melt? Boiling point At what temperature does a liquid boil? Density What is the substance’s density (mass per unit volume)? Solubility What mass of substance can dissolve in a given volume of water or other solvent? Electric conductivity Does the substance conduct electricity? Malleability How easily can a solid be deformed? How easily can a solid be drawn into a wire? Viscosity How easily will a liquid flow? Your brain unconsciously uses the density of an object you want to pick up by estimating volume visually and preparing your muscles to lift the expected mass. For example, you can readily tell the difference between an ice cube and a cube of lead of identical size (Figure 1.10). Lead has a high density, 11.35 g/cm3 (11.35 grams per cubic centimeter), whereas the density of ice is slightly less than 0.917 g/cm3. An ice cube with a volume of 16.0 cm3 has a mass of 14.7 g, whereas a cube of lead with the same volume has a mass of 182 g. Density relates the mass and volume of a substance. If any two of three quantities—mass, volume, and density—are known for a sample of matter, the third can be calculated. For example, the mass of an object is the product of its density and volume. Mass 1g2 ϭ volume ϫ density ϭ volume 1cm3 2 ϫ mass 1g2 volume 1cm3 2 Charles D. Winters Ductility Density, mass, and volume. What is the mass of 32 mL of mercury? You can use this approach to find the mass of 32 cm3 [or 32 mL (milliliters)] of mercury in the graduated cylinder in the photo. A handbook of information for chemistry lists the density of mercury as 13.534 g/cm3 (at 20 °C). Mass 1g2 ϭ 32 cm3 ϫ 13.534 g 1 cm3 ϭ 430 g Be sure to notice that the units of cm3 cancel to leave the answer in units of g as required. See the General ChemistryNow CD-ROM or website: • Screen 1.10 Density, for two step-by-step tutorials on determining density and volume ■ Dimensional Analysis The approach to problem solving used in this book is often called dimensional analysis. The essence of this approach is to change one number (A) into another (B) using a conversion factor so that the units of A are changed to the desired unit. See Section 1.8. 22 Chapter 1 Matter and Measurement Example 1.1—Using Density H HO H C C Problem Ethylene glycol, C2H6O2, is widely used in automobile antifreeze. It has a density of 1.11 g/cm3 (or 1.11 g/mL). What volume of ethylene glycol will have a mass of 1850 g? Strategy You know the density and mass of the sample. Because density is the ratio of the mass of a sample to its volume, volume ϭ (mass)(1/density). OH H H ethylene glycol, C2H6O2 density = 1.11 g/cm3 (or 1.11 g/mL) Solution Volume 1cm3 2 ϭ 1850 g a ■ Units of Density The SI unit of mass is the kilogram and the SI unit of length is the meter. Therefore, the SI unit of density is kg/m3. In chemistry the more commonly used unit is g/cm3. To convert from kg/m3 to g/cm3, divide by 1000. 1 cm3 b ϭ 1670 cm3 1.11 g Comment Here we multiply the mass (in grams) by the conversion factor (1 cm3/1.11 g) so that units of g cancel to leave an answer in the desired unit of cm3. Exercise 1.4—Density The density of dry air is 1.18 ϫ 10Ϫ3 g/cm3 (ϭ 0.00118 g/cm3; see Section 1.8 on using scientific notation). What volume of air, in cubic centimeters, has a mass of 15.5 g? Temperature Dependence of Physical Properties Temperature Dependence of Water Density Temperature (°C) Density of Water (g/cm3) 0 (ice) 0.917 0 (liq water) 0.99984 2 0.99994 4 0.99997 10 0.99970 25 0.99707 100 0.95836 The temperature of a sample of matter often affects the numerical values of its properties. Density is a particularly important example. Although the change in water density with temperature seems small, it affects our environment profoundly. For example, as the water in a lake cools, the density of the water increases, and the denser water sinks (Figure 1.11a). This continues until the water temperature reaches 3.98 °C, the point at which water has its maximum density (0.999973 g/cm3). If the water temperature drops further, the density decreases slightly, and the colder water floats on top of water at 3.98 °C. If water is cooled below about 0 °C, solid ice forms. Water is unique among substances in the universe: Ice is much less dense than water, so it floats on water. Because the density of liquids changes with temperature, it is necessary to report the temperature when you make accurate volume measurements. Laboratory glassware used to make such measurements always specifies the temperature at which it was calibrated (Figure 1.11b). Problem-Solving Tip 1.1 Finding Data All the information you need to solve a problem in this book may not be presented in the problem. For example, we could have left out the value of the density in Example 1.1 and assumed you would (a) recognize that you needed density to convert a mass to a volume and (b) know where to find the information. The Appendices of this book contain a wealth of information, and even more is available on the General ChemistryNow CD-ROM and website. Various handbooks of information are available in most libraries; among the best are the Handbook of Chemistry and Physics (CRC Press) and Lange’s Handbook of Chemistry (McGraw-Hill). The most up-to-date source of data is the National Institute for Standards and Technology ( See also the World Wide Web site Webelements ( Physical and Chemical Changes Photos: Charles D. Winters 1.5 (a) (b) Figure 1.11 Temperature dependence of physical properties. (a) Change in density with temperature. Ice cubes were placed in the right side of the tank and blue dye in the left side. The water beneath the ice is cooler and denser than the surrounding water, so it sinks. The convection current created by this movement of water is traced by the dye movement as the denser, cooler water sinks. (b) Temperature and calibration. Laboratory glassware is calibrated for specific temperatures. This pipet or volumetric flask will contain the specified volume at the indicated temperature. Exercise 1.5—Density and Temperature The density of mercury at 0 °C is 13.595 g/cm3, at 10 °C it is 13.570 g/cm3, and at 20 °C it is 13.546 g/cm3. Estimate the density of mercury at 30 °C. Extensive and Intensive Properties Extensive properties depend on the amount of a substance present. The mass and volume of the samples of elements in Figures 1.2 and 1.6 are extensive properties, for example. In contrast, intensive properties do not depend on the amount of substance. A sample of ice will melt at 0 °C, no matter whether you have an ice cube or an iceberg. Density is also an intensive property. The density of gold, for example, is the same (19.3 g/cm3) whether you have a flake of pure gold or a solid gold ring. 1.5—Physical and Chemical Changes Changes in physical properties are called physical changes. In a physical change the identity of a substance is preserved even though it may have changed its physical state or the gross size and shape of its pieces. An example of a physical change is the melting of a solid. The temperature at which this occurs (the melting point ) is often so characteristic that it can be used to identify the solid (Figure 1.12). A physical property of hydrogen gas (H2) is its low density, so a balloon filled with H2 floats in air (Figure 1.13). Suppose a lighted candle is brought up to the balloon. When the heat causes the skin of the balloon to rupture, the hydrogen combines with the oxygen (O2) in the air, and the heat of the candle sets off a chemical reaction (Figure 1.13), producing water, H2O. This reaction is an example of a chemical change, in which one or more substances (the reactants) are transformed into one or more different substances (the products). 23 24 Chapter 1 Matter and Measurement Figure 1.12 A physical property used to distinguish compounds. Photos: Charles D. Winters Aspirin and naphthalene are both white solids at 25 °C. You can tell them apart by, among other things, a difference in physical properties. At the temperature of boiling water, 100 °C, naphthalene is a liquid (left), whereas aspirin is a solid (right). Naphthalene is a white solid at 25 °C but has a melting point of 80.2 °C. Aspirin is a white solid at 25 °C. It has a melting point of 135 °C. The reaction of H2 with O2 is an example of a chemical property of hydrogen. A chemical property involves a change in the identity of a substance. Here the H atoms of the gaseous H2 molecules have become incorporated into H2O. Similarly, a chemical change occurs when gasoline burns in air in an automobile engine or an old car rusts in the air. Burning of gasoline or rusting of iron are characteristic chemical properties of these substances. A chemical change at the particulate level is illustrated by the reaction of hydrogen and oxygen molecules to form water molecules. 2 H2(gas) ϩ 02(gas) 2 H20(gas) ϩ Reactants Products The representation of the change with chemical formulas is called a chemical equation. It shows that the substances on the left (the reactants) produce the substances on the right (the products). As this equation shows, there are four atoms of H and two atoms of O before and after the reaction, but the molecules before the reaction are different from those after the reaction. Unlike a chemical change, a physical change does not result in a new chemical substance being produced. The substances (atoms, molecules, or ions) present before and after the change are the same, but they might be farther apart in a gas or closer together in a solid (Figure 1.2). Finally, as described more fully in Chapter 6, physical changes and chemical changes are often accompanied by transfer of energy. The reaction of hydrogen and oxygen to give water (Figure 1.13), for example, transfers a tremendous amount of energy (in the form of heat and light ) to its surroundings. See the General ChemistryNow CD-ROM or website: • Screen 1.12 Chemical Changes, for an exercise on identifying physical and chemical changes • Screen 1.13 Chemical Change on the Molecular Scale, to watch a video and view an animation of the molecular changes when chlorine gas and solid phosphorus react 1.6 25 Units of Measurement Figure 1.13 A chemical change—the Photos: Charles D. Winters reaction of hydrogen and oxygen. (a) A balloon filled with molecules of hydrogen gas, and surrounded by molecules of oxygen in the air. (The balloon floats in air because gaseous hydrogen is less dense than air.) (b) When ignited with a burning candle, H2 and O2 react to form water, H2O. (See General ChemistryNow Screen 1.11 Chemical Change, for a video of this reaction.) ؉ O2 (gas) (a) H2 (gas) 2 H2O(g) (b) Exercise 1.6—Chemical Reactions and Physical Changes When camping in the mountains, you boil a pot of water on a campfire. What physical and chemical changes take place in this process? Doing chemistry requires observing chemical reactions and physical changes. Suppose you mix two solutions in the laboratory and see a golden yellow solid form. Because this new solid is denser than water, it drops to the bottom of the test tube (Figure 1.14). The color and appearance of the substances, and whether heat is involved, are qualitative observations. No measurements and numbers were involved. To understand a chemical reaction more completely, chemists usually make quantitative observations. These involve numerical information. For example, if two compounds react with each other, how much product forms? How much heat, if any, is evolved? In chemistry, quantitative measurements of time, mass, volume, and length, among other things, are common. On page 31 you can read about one of the fastest growing areas of science, nanotechnology, which involves the creation and study of matter on the nanometer scale. A nanometer (nm) is equivalent to 1 ϫ 10Ϫ9 m Charles D. Winters 1.6—Units of Measurement Chemical and physical changes. A pot of water has been put on a campfire. What chemical and physical changes are occurring here (Exercise 1.6)? Figure 1.14 Qualitative and quantita- Charles D. Winters tive observations. A new substance is formed by mixing two known substances in solution. Of the substance produced we can make several observations. Qualitative observations: yellow, fluffy solid. Quantitative observations: mass of solid formed. 26 Chapter 1 Matter and Measurement (meter), a common dimension in chemistry and biology. For example, a typical molecule is only about 1 nm across and a bacterium is about 1000 nm in length. The scientific community has chosen a modified version of the metric system as the standard system for recording and reporting measurements. This decimal system, used internationally in science, is called the Système International d’Unités (International System of Units), abbreviated SI. Table 1.2 Some SI Base Units Measured Property Name of Unit Abbreviation Mass kilogram kg Length meter m Time second s Temperature kelvin K Amount of substance mole mol Electric current ampere A All SI units are derived from base units, some of which are listed in Table 1.2. Larger and smaller quantities are expressed by using appropriate prefixes with the base unit (Table 1.3). See the General ChemistryNow CD-ROM or website: • Screen 1.16 The Metric System, for a step-by-step tutorial on converting metric units Temperature Scales Three temperature scales are commonly used: the Fahrenheit, Celsius, and Kelvin scales (Figure 1.15). The Fahrenheit scale is used in the United States to report everyday temperatures, but most other countries use the Celsius scale. The Celsius scale is generally used worldwide for measurements in the laboratory. When calculations incorporate temperature data, however, kelvin degrees must be used. The Celsius Temperature Scale The size of the Celsius degree is defined by assigning zero as the freezing point of pure water (0 °C) and 100 as its boiling point (100 °C) (page 10). You can readily interconvert Fahrenheit and Celsius temperatures using the equation T1°C2 ϭ 5 °C 3T 1°F2 Ϫ 324 9 °F but it is best to “calibrate” your senses on the Celsius scale. Pure water freezes at 0 °C, a comfortable room temperature is around 20 °C, your body temperature is 37 °C, and the warmest water you could stand to immerse a finger in is probably about 60 °C. 1.6 Fahrenheit Boiling point of water 212° Freezing point of water Kelvin (or absolute) Celsius 100° 180° 100° 32° 27 Units of Measurement 373 100 K 0° 273 Active Figure 1.15 A comparison of Fahrenheit, Celsius, and Kelvin scales. The reference, or starting point, for the Kelvin scale is absolute zero (0 K ϭ Ϫ273.15 °C), which has been shown theoretically and experimentally to be the lowest possible temperature. See General ChemistryNow CD-ROM or website to explore an interactive version of this figure accompanied by an exercise. Table 1.3 Selected Prefixes Used in the Metric System Prefix Abbreviation Meaning Example mega- M 10 (million) 1 megaton ϭ 1 ϫ 10 tons kilo- k 103 (thousand) 1 kilogram (kg) ϭ 1 ϫ 103 g 6 Ϫ1 (tenth) deci- d 10 centi- c 10Ϫ2 (one hundredth) Ϫ3 (one thousandth) milli- m 10 micro- m 10Ϫ6 (one millionth) Ϫ9 nano- n 10 pico- p 10Ϫ12 f Ϫ15 femto- 10 (one billionth) 6 1 decimeter (dm) ϭ 1 ϫ 10Ϫ1 m 1 centimeter (cm) ϭ 1 ϫ 10Ϫ2 m 1 millimeter (mm) ϭ 1 ϫ 10Ϫ3 m 1 micrometer (mm) ϭ 1 ϫ 10Ϫ6 m ■ Common Conversion Factors 1 kg ϭ 1000 g 1 ϫ 109 nm ϭ 1 m 10 mm ϭ 1 cm 100 cm ϭ 10 dm ϭ 1 m 1000 m ϭ 1 km Conversion factors for SI units are given in Appendix C and inside the back cover of this book. 1 nanometer (nm) ϭ 1 ϫ 10Ϫ9 m 1 picometer (pm) ϭ 1 ϫ 10Ϫ12 m 1 femtometer (fm) ϭ 1 ϫ 10Ϫ15 m The Kelvin Temperature Scale William Thomson, known as Lord Kelvin (1824–1907), first suggested the temperature scale that now bears his name. The Kelvin scale uses the same size unit as the Celsius scale, but it assigns zero as the lowest temperature that can be achieved, a point called absolute zero. Many experiments have found that this limiting temperature is Ϫ273.15 °C (Ϫ459.67 °F). Kelvin units and Celsius degrees are the same size. Thus, the freezing point of water is reached at 273.15 K; that is, 0 °C ϭ 273.15 K. ■ Lord Kelvin William Thomson (1824–1907), known as Lord Kelvin, was a professor of natural philosophy at the University in Glasgow, Scotland, from 1846 to 1899. He was best known for his work on heat and work, from which came the concept of the absolute temperature scale. E. F. Smith Collection/Van Pelt Library/University of Pennsylvania. 28 Chapter 1 Matter and Measurement The boiling point of pure water is 373.15 K. Temperatures in Celsius degrees are readily converted to kelvins, and vice versa, using the relation T 1K2 ϭ ■ Temperature Conversions When converting 23.5 °C to kelvins, adding the two numbers gives 296.65. However, the rules of “significant figures” tell us that the sum or difference of two numbers can have no more decimal places than the number with the fewest decimal places. (See page 40.) Thus, we round the answer to 296.7 K, a number with one decimal place. 1K 3T °C ϩ 273.15 °C4 1 °C ˇ (1.2) Thus, a common room temperature of 23.5 °C is T 1K2 ϭ 1K 123.5 °C ϩ 273.15 °C2 ϭ 296.7 K 1 °C Finally, notice that the degree symbol (°) is not used with Kelvin temperatures. The name of the unit on this scale is the kelvin (not capitalized), and such temperatures are designated with a capital K. See the General ChemistryNow CD-ROM or website: • Screen 1.15 Temperature, for a step-by-step tutorial on converting temperatures Exercise 1.7—Temperature Scales Liquid nitrogen boils at 77 K. What is this temperature in Celsius degrees? Length Charles D. Winters The meter is the standard unit of length, but objects observed in chemistry are frequently smaller than 1 meter. Measurements are often reported in units of centimeters or millimeters, and objects on the atomic and molecular scale have dimensions of nanometers (nm; 1 nm ϭ 1.0 ϫ 10Ϫ9 m) or picometers (pm; 1 pm ϭ 1 ϫ 10Ϫ12 m). Your hand, for example, is about 18 cm from the wrist to the fingertips, and the ant in the photo here is about 1 cm long. Using a special microscope—a scanning electron microscope (SEM)—scientists can zoom in on the face of an ant, then to the ant’s eye, and finally to one segment of the eye (Figure 1.16). If we could continue to zoom in on the ant’s eye in Figure 1.16, we would enter the nanoscale molecular world (Figure 1.17). The DNA (deoxyribonucleic acid) in the ant’s eye is a helical coil of atoms many nanometers long. The rungs of the DNA ladder are approximately 0.34 nm apart, and the helix repeats itself about every 3.4 nm. Zooming in even more, we might encounter a water molecule. Here the distance between the two hydrogen atoms on either side of the oxygen atom is 0.152 nm or 152 pm (pm; picometer, 1 pm ϭ 1 ϫ 10Ϫ12 m). Ant. Your hand is about 18 centimeters long from your wrist to your fingertips. The ant here is about 1 cm in length. Example 1.2—Distances on the Molecular Scale Problem The distance between an O atom and an H atom in a water molecule is 95.8 pm. What is this distance in meters (m)? In nanometers (nm)? Units of Measurement 29 Photos courtesy of Charles Rettner of IBM’s Alamaden Research Center. 1.6 (a) (b) (c) Figure 1.16 Dimensions in biology. These photos were done at the IBM Laboratories using a scanning electron microscope (SEM). The subject was a dead ant. (a) The head of the ant is about 600 micrometers (microns, ␮ m) wide. (This is equivalent to 6 ϫ 10Ϫ4 m or 0.6 mm.) (b) The compound eye of the ant. (c) The scientists at IBM used a special probe to write, on one lens of the ant eye, their advice to science students. The word “homework” is about 1.5 micrometers (microns, ␮ m) long. 95.8 pm Strategy You can solve this problem by knowing the conversion factor between the units in the information you are given (picometers) and the desired units (meters or nanometers). (For more about conversion factors and their use in problem solving see Section 1.8.) There is no conversion factor given in Table 1.3 to change nanometers to picometers, but relationships are listed between meters and picometers and between meters and nanometers (Table 1.3). First, we convert picometers to meters, and then we convert meters to nanometers. ϫ m pm ϫ nm m Picometers ¡ Meters ¡ Nanometers Solution Using the appropriate conversion factors (1 pm ϭ 1 ϫ 10Ϫ12 m and 1 nm ϭ 1 ϫ 10Ϫ9 m), we have 95.8 pm ϫ 9.58 ϫ 10Ϫ11 m ϫ 1 ϫ 10Ϫ12 m ϭ 9.58 ϫ 10Ϫ11 m 1 pm 1 nm ϭ 9.58 ϫ 10Ϫ2 nm or 0.0958 nm 1 ϫ 10Ϫ9 m Comment Notice how the units cancel to leave an answer whose unit is that of the numerator of the conversion factor. The process of using units to guide a calculation is called dimensional analysis and is discussed further on pages 41–43. ■ Powers of Ten The book Powers of Ten explores the dimensions of our universe (Philip and Phylis Morrison, Scientific American Books, 1982). See also the following website in which the “powers of ten” is elegantly animated. primer/java/scienceopticsu/powersof10/ 30 Chapter 1 Matter and Measurement Figure 1.17 Dimensions in the molecular world. Objects on the molecular scale are often given in terms of nanometers (1 nm ϭ 1 ϫ 10Ϫ9 m) or picometers (1 pm ϭ 1 ϫ 10Ϫ12 m). An older non-SI unit is the angstrom unit, where 1 Å ϭ 1.0 ϫ 10Ϫ10 m. The distance between turns of the DNA helix is 3.4 nm. 3.4 nm Charles D. Winters O H H 0.152 nm The distance between the two H atoms in a water molecule is 0.152 nm or 152 pm. Exercise 1.8—Interconverting Units of Length The pages of a typical textbook are 25.3 cm long and 21.6 cm wide. What is each dimension in meters? In millimeters? What is the area of a page in square centimeters? In square meters? Exercise 1.9—Using Units of Length and Density A platinum sheet is 2.50 cm square and has a mass of 1.656 g. The density of platinum is 21.45 g/cm3. What is the thickness of the platinum sheet in millimeters? Volume Charles D. Winters Chemists often use glassware such as beakers, flasks, pipets, graduated cylinders, and burets, which are marked in volume units (Figure 1.18). The SI unit of volume is the cubic meter (m3), which is too large for everyday laboratory use. Therefore, chemists usually use the liter, symbolized by L. A cube with sides equal to 10 cm (0.1 m) has a volume of 10 cm ϫ 10 cm ϫ 10 cm ϭ 1000 cm3 (or 0.001 m3). This is defined as 1 liter. Figure 1.18 Some common laboratory glassware. Volumes are marked in units of milliliters (mL). Remember that 1 mL is equivalent to 1 cm3. 1 liter 1L2 ϭ 1000 mL ϭ 1000 cm3 The liter is a convenient unit to use in the laboratory, as is the milliliter (mL). Because there are exactly 1000 mL (ϭ 1000 cm3) in a liter, this means that 1 cm3 ϭ 0.001 L ϭ 1 mL 1.6 Units of Measurement Chemical Perspectives Professor Alex Zettl of the University of California–Berkeley, holding a model of a carbon nanotube. A bundle of carbon nanotubes. Each tube has a diameter of 1.4 nm, and the bundle is 10–20 nm thick. having diameters of only a few nanometers. Carbon nanotubes are at least 100 times stronger than steel, but only one-sixth as dense. In addition, they conduct heat and electricity far better than copper. As a consequence, carbon nanotubes could be used in tiny, physically strong, conducting devices. Recently, carbon nanotubes have been filled with potassium atoms, making them even better electrical conductors. And even more recently, molecular-sized bearings have been made by sliding one nanotube inside another. Melissa A. Hines/Cornell University Lawrence Berkeley Laboratory A nanometer is one billionth of a meter, a dimension in the realm of atoms and molecules—eight oxygen atoms in a row span a distance of about 1 nanometer. Nanotechnology is one of the hottest fields in science today because the building blocks of those materials having nanoscale dimension can have unique properties. Carbon nanotubes are excellent examples of nanomaterials. These lattices of carbon atoms form the walls of tubes Nanomaterials are by no means new. For the last century tire companies have reinforced tires by adding nanosized particles called carbon black to rubber. Atomic force microscopy (AFM) is an important tool in chemistry and physics to observe materials at the nanometer level. A tiny probe, often a whisker of a carbon nanotube, moves over the surface of a substance and interacts with individual molecules. Here you see an AFM image of a silicon surface about 460 nm on a side and 5 nm high. P. Nikolaev, Rice University, Center for Nanoscale Science and Technology. It’s a Nanoworld! 31 An AFM image of nanobumps on a silicon surface. The average spacing between nanobumps is 38 nm, or about 160 silicon atoms. The average nanobump width is 25 nm or 100 silicon atoms. The units milliliter and cubic centimeter (or “cc”) are interchangeable. Therefore, a flask that contains exactly 125 mL has a volume of 125 cm3. Although not widely used in the United States, the cubic decimeter (dm3) is a common unit in the rest of the world. A length of 10 cm is called a decimeter (dm). Because a cube 10 cm on a side defines a volume of 1 liter, a liter is equivalent to a cubic decimeter: 1 L ϭ 1 dm3. Products in Europe and other parts of the world are often sold by the cubic decimeter. The deciliter, dL, which is exactly equivalent to 0.100 L or 100 mL, is widely used in medicine. For example, standards for amounts of environmental contaminants are often set as a certain mass per deciliter. The state of Massachusetts recommends that children with more than 10 micrograms (10 ϫ 10Ϫ6 g) of lead per deciliter of blood undergo further testing for lead poisoning. Example 1.3—Units of Volume Problem A laboratory beaker has a volume of 0.6 L. What is its volume in cubic centimeters (cm3), milliliters (mL), and deciliters? Strategy Use the information in Table 1.3 to interconvert between units, and use dimensional analysis (see The Mathematics of Chemistry, pages 41–43) as a guide. 32 Chapter 1 Matter and Measurement Solution You should multiply 0.6 L by the conversion factor (1000 cm3/L). The units of L cancel to leave an answer with units of cm3. 0.6 L ؒ 1000 cm3 ϭ 600 cm3 1L Because cubic centimeters and milliliters are equivalent, we can also say that the volume of the beaker is 600 mL. The deciliter is 0.100 L or 100 mL. In deciliters, the volume is 600 mL ؒ 1 dL ϭ 6 dL 100 mL Exercise 1.10—Volume (a) A standard wine bottle has a volume of 750 mL. How many liters does this represent? How many deciliters? (b) One U.S. gallon is equivalent to 3.7865 L. How many liters are in a 2.0-quart carton of milk? (There are 4 quarts in a gallon.) How many cubic decimeters? Mass The mass of a body is the fundamental measure of the quantity of matter, and the SI unit of mass is the kilogram (kg). Smaller masses are expressed in grams (g) or milligrams (mg). 1 kg ϭ 1000 g 1 g ϭ 1000 mg ■ Micrograms Very small masses are often given in micrograms. A microgram is 1/1000 of a milligram or one millionth of a gram. Exercise 1.11—Mass (a) A new U.S. quarter has a mass of 5.59 g. Express this mass in kilograms and milligrams. (b) An environmental study of a river found a pesticide present to the extent of 0.02 microgram per liter of water. Express this amount in grams per liter. 1.7—Making Measurements: Precision, ■ Accuracy The National Institute for Standards and Technology (NIST) is the most important resource for the standards used in science. Comparison with the NIST data is the best test of the accuracy of the measurement. See Accuracy, and Experimental Error The precision of a measurement indicates how well several determinations of the same quantity agree. This is illustrated by the results of throwing darts at a target. In Figure 1.19a, the dart thrower was apparently not skillful, and the precision of the dart’s placement on the target is low. In Figures 1.19b and 1.19c, the darts are clustered together, indicating much better consistency on the part of the thrower—that is, greater precision. Accuracy is the agreement of a measurement with the accepted value of the quantity. Figure 1.19c shows that our thrower was accurate as well as precise—the average of all shots is close to the targeted position, the bull’s eye. 33 Making Measurements: Precision, Accuracy, and Experimental Error Charles D. Winters 1.7 (a) Poor precision and poor accuracy (b) Good precision and poor accuracy (c) Good precision and good accuracy Figure 1.19 Precision and accuracy. Figure 1.19b shows that it is possible to be precise without being accurate—the thrower has consistently missed the bull’s eye, although all the darts are clustered precisely around one point on the target. This is analogous to an experiment with some flaw (either in design or in a measuring device) that causes all results to differ from the correct value by the same amount. The precision of a measurement is often expressed in terms of its standard deviation, a technique of data analysis explored in A Closer Look: Standard Deviation. For A Closer Look Standard Deviation Laboratory measurements can be in error for two basic reasons. First, there may be “determinate” errors caused by faulty instruments or human errors such as incorrect record keeping. So-called “indeterminate” errors arise from uncertainties in a measurement where the cause is not known and cannot be controlled by the lab worker. One way to judge the indeterminate error in a result is to calculate the standard deviation. The standard deviation of a series of measurements is equal to the square root of the sum of the squares of the deviations for each measurement divided by the number of measurements. It has a precise statistical significance: 68% of the values collected are expected to be within one standard deviation of the value determined. (This value assumes a large number of measurements is used to calculate the deviation.) Consider a simple example. Suppose you carefully measured the mass of water delivered by a 10-mL pipet. For five attempts at the measurement (shown in the table, column 2), the standard deviation is found as follows: First, the average of the measurements is calculated (here, 9.984). Next, the deviation of each individual measurement from this value is determined (column 3). These values are squared, giving the values in column 4, and the sum of these values is determined. The standard deviation is then Determination Measured Mass, (g) Difference between Average and Measurement (g) 1 9.990 0.006 4 ϫ 10Ϫ 5 2 9.993 0.009 8 ϫ 10Ϫ 5 3 9.973 0.011 12 ϫ 10Ϫ 5 4 9.980 0.004 2 ϫ 10Ϫ 5 5 9.982 0.002 0.4 ϫ 10Ϫ 5 Square of Difference calculated by dividing this number by 5 (the number of determinations) and taking the square root of the result. Average mass ϭ 9.984 g Sum of squares of differences ϭ 26 ϫ 10Ϫ5 Standard deviation ϭ 26 ϫ 10Ϫ5 ϭ Ϯ0.007 5 B Based on this calculation it would be appropriate to represent the measured mass as 9.984 Ϯ 0.007 g. This would tell a reader that if this experiment were repeated, approximately 68% of the values would fall in the range of 9.977 g to 9.991 g. 34 Chapter 1 Matter and Measurement example, suppose a series of measurements led to a distance of 2.965 cm, and the standard deviation was 0.006 cm. Because the uncertainty shows up in the thousandths position, the value should be reported to the nearest thousandth—that is, 2.965 cm. A standard deviation of 0.006 cm means that 68% of the random measurements we make will be within 1 standard deviation—that is, within Ϯ 0.006 cm. If you are measuring a quantity in the laboratory, you may be required to report the error in the result, the difference between your result and the accepted value, Error ϭ experimentally determined value Ϫ accepted value or the percent error. Percent error ϭ error in measurement ϫ 100% accepted value Example 1.4—Precision and Accuracy Problem A coin has an “accepted” diameter of 28.054 mm. In an experiment, two students measure this diameter. Student A makes four measurements of the diameter of a coin using a precision tool called a micrometer. Student B measures the same coin using a simple plastic ruler. The two students report the following results: Student A Student B 28.246 mm 27.9 mm 28.244 28.0 28.246 27.8 28.248 28.1 What is the average diameter and percent error obtained in each case? Which student’s data are more accurate? Which are more precise? Strategy For each set of values we calculate the average of the results and then compare this average with 28.054 mm. Solution The average for each set of data is obtained by summing the four values and dividing by 4. Student A Student B 28.246 mm 27.9 mm 28.244 28.0 28.246 27.8 28.248 28.1 Average ϭ 28.246 Average ϭ 28.0 Student A’s data are all very close to the average value, so they are quite precise. Student B’s data, in contrast, have a wider range and are less precise. However, student A’s result is less 1.8 35 Mathematics of Chemistry accurate than that of student B. The average diameter for student A differs from the “accepted” value by 0.192 mm and has a percent error of 0.684%: Percent error ϭ 28.246 mm Ϫ 28.054 mm ϫ 100% ϭ 0.684% 28.054 mm Student B’s measurement has an error of only about 0.2%. Comment Possible reasons for the error in Students A’s result are incorrect use of the micrometer or a flaw in the instrument. Exercise 1.12—Error, Precision, and Accuracy Two students measured the freezing point of an unknown liquid. Student A used an ordinary laboratory thermometer calibrated in 0.1 °C units. Student B used a thermometer certified by NIST and calibrated in 0.01 °C units. Their results were as follows: Student A: Ϫ0.3 °C; 0.2 °C; 0.0 °C; and Ϫ0.3 °C Student B: 273.13 K; 273.17 K; 273.15 K; 273.19 K Calculate the average value and, knowing that the liquid was water, calculate the percent error for each student. Which student has the more precise values? Which has the smaller error? 1.8—Mathematics of Chemistry At its core, chemistry is a quantitative science. Chemists make measurements of, among other things, size, mass, volume, time, and temperature. Scientists then manipulate that quantitative numerical information to search for relationships among properties and to provide insight into the molecular basis of matter. This section reviews some of the mathematical skills you will need in chemical calculations. It also describes ways to perform calculations and ways to handle quantitative information. The background you should have to be successful includes the following skills: • Ability to express and use numbers in exponential or scientific notation. • Ability to make unit conversions (such as liters to milliliters). • Ability to express quantitative information in an algebraic expression and solve that expression. An example would be to solve the equation a ϭ 1b/x2c for x. • Ability to prepare a graph of numerical information. If the graph produces a straight line, find the slope and equation of the line. Examples and Exercises using some of these skills follow, and some problems involving unit conversions and solving algebraic expressions are included in the Study Questions at the end of this chapter. Exponential or Scientific Notation Lake Otsego in northern New York is also called Glimmerglass, a name suggested by James Fenimore Cooper (1789–1851), the great American author and an early resident of the village now known as Cooperstown. Extensive environmental studies Charles D. Winters • Ability to read information from graphs. Figure 1.20 Lake Otsego. This lake, with a surface area of 2.33 ϫ 107 m2, is located in northern New York. Cooperstown is a village at the base of the lake, where the Susquehanna River originates. To learn more about the environmental biology and chemistry of the lake, go to 36 Chapter 1 Matter and Measurement have been done along this lake (Figure 1.20), and some quantitative information useful to chemists, biologists, and geologists is given in the following table: used in astronomy. The spiral galaxy M-83 is 3.0 ϫ 106 parsecs away and has a diameter of 9.0 ϫ 103 parsecs. The unit used in astronomy, the parsec (pc), is 206265 AU (astronomical units), and 1 AU is 1.496 ϫ 108 km. Therefore, the galaxy is about 9.3 ϫ 1019 km away from Earth. 2.33 ϫ 107 m2 Maximum depth Figure 1.21 Exponential numbers Quantitative Information Area W. Keel, U. Alabama/NASA Lake Otsego Characteristics 505 m Dissolved solids in lake water 2 ϫ 102 mg/L Average rainfall in the lake basin 1.02 ϫ 102 cm/year Average snowfall in the lake basin 198 cm/year All of the data collected are in metric units. However, some data are expressed in fixed notation (505 m, 198 cm/year), whereas other data are expressed in exponential, or scientific, notation (2.33 ϫ 107 m2). Scientific notation is a way of presenting very large or very small numbers in a compact and consistent form that simplifies calculations. Because of its convenience it is widely used in sciences such as chemistry, physics, engineering, and astronomy (Figure 1.21). In scientific notation the number is expressed as a product of two numbers: N ϫ 10n. N is the digit term and is a number between 1 and 9.9999. . . . The second number, 10n, the exponential term, is some integer power of 10. For example, 1234 is written in scientific notation as 1.234 ϫ 103, or 1.234 multiplied by 10 three times: 1.234 ϭ 1.234 ϫ 101 ϫ 101 ϫ 101 ϭ 1.234 ϫ 103 Conversely, a number less than 1, such as 0.01234, is written as 1.234 ϫ 10Ϫ2. This notation tells us that 1.234 should be divided twice by 10 to obtain 0.01234: 0.01234 ϭ 1.234 ϭ 1.234 ϫ 10 Ϫ1 ϫ 10 Ϫ1 ϭ 1.234 ϫ 10 Ϫ2 101 ϫ 101 Some other examples of scientific notation follow: 10000 ϭ 1 ϫ 104 12345 ϭ 1.2345 ϫ 104 3 1000 ϭ 1 ϫ 10 1234.5 ϭ 1.2345 ϫ 103 2 100 ϭ 1 ϫ 10 123.45 ϭ 1.2345 ϫ 102 1 10 ϭ 1 ϫ 10 12.345 ϭ 1.2345 ϫ 101 0 1 ϭ 1 ϫ 10 (any number to the zero power ϭ 1) 1/10 ϭ 1 ϫ 10Ϫ1 0.12 ϭ 1.2 ϫ 10Ϫ1 Ϫ2 1/100 ϭ 1 ϫ 10 0.012 ϭ 1.2 ϫ 10Ϫ2 Ϫ3 1/1000 ϭ 1 ϫ 10 0.0012 ϭ 1.2 ϫ 10Ϫ3 Ϫ4 1/10000 ϭ 1 ϫ 10 0.00012 ϭ 1.2 ϫ 10Ϫ4 ■ Comparing the Earth and a Plant Cell—Powers of Ten Earth ϭ 12,760,000 meters wide ϭ 12.76 million meters ϭ 1.276 ϫ 107 meters Plant cell ϭ 0.00001276 meter wide ϭ 12.76 millionths of a meter ϭ 1.276 ϫ 10Ϫ5 meters When converting a number to scientific notation, notice that the exponent n is positive if the number is greater than 1 and negative if the number is less than 1. The value of n is the number of places by which the decimal is shifted to obtain the number in scientific notation: 1 2 3 4 5. ϭ 1.2345 ϫ 104 (a) Decimal shifted four places to the left. Therefore, n is positive and equal to 4. 1.8 Problem-Solving Tip 1.2 Using Your Calculator You will be performing a number of calculations in general chemistry, most of them using a calculator. Many different types of calculators are available, but this problemsolving tip describes several of the kinds of operations you will need to perform on a typical calculator. Be sure to consult your calculator manual for specific instructions to enter scientific notation and to find powers and roots of numbers. 1. Scientific Notation When entering a number such as 1.23 ϫ 10Ϫ4 into your calculator, you first enter 1.23 and then press a key marked EE or EXP (or something similar). This enters the “ ϫ 10” portion of the notation for you. You then complete the entry by keying in the exponent of the number, Ϫ4. (To change the exponent from ϩ4 to Ϫ4, press the “ϩ/Ϫ” key.) Mathematics of Chemistry A common error made by students is to enter 1.23, press the multiply key (x), and then key in 10 before finishing by pressing EE or EXP followed by Ϫ4. This gives you an entry that is 10 times too large. Try this! Experiment with your calculator so you are sure you are entering data correctly. 2. Powers of Numbers Electronic calculators usually offer two methods of raising a number to a power. To square a number, enter the number and then press the “x2” key. To raise a number to any power, use the “yx” (or similar) key. For example, to raise 1.42 ϫ 102 to the fourth power: 1. Enter 1.42 ϫ 10 . 2 2. Press “yx”. 3. Enter 4 (this should appear on the display). 37 3. Roots of Numbers To take a square root on an electronic calculator, enter the number and then press the “ 2x” key. To find a higher root of a number, such as the fourth root of 5.6 ϫ 10Ϫ10: x 2. Press the “2y ” key. (On many calculators, the sequence you actually use is to press “2ndF” and then “ϭ.” Alternatively, you press “INV” and then “yx ”.) 1. Enter the number. 3. Enter the desired root, 4 in this case. 4. Press “=”. The answer here is 4.8646 ϫ 10Ϫ3. A general procedure for finding any root is to use the “yx” key. For a square root, x is 0.5 (or 1/2), whereas it is 0.3333 (or 1/3) for a cube root, 0.25 (or 1/4) for a fourth root, and so on. 4. Press “ϭ” and 4.0659 ϫ 108 appears on the display. 0.0 0 1 2 ϭ 1.2 ϫ 10Ϫ3 (b) Decimal shifted three places to the right. Therefore, n is negative and equal to 3. If you wish to convert a number in scientific notation to one using fixed notation (that is, not using powers of 10), the procedure is reversed: 6 . 2 7 3 ϫ 102 ϭ 627.3 (a) Decimal point moved two places to the right because n is positive and equal to 2. 0 0 6.273 ϫ 10Ϫ3 ϭ 0.006273 (b) Decimal point shifted three places to the left because n is negative and equal to 3. Two final points should be made concerning scientific notation. First, be aware that calculators and computers often express a number such as 1.23 ϫ 103 as 1.23E3 or 6.45 ϫ 10Ϫ5 as 6.45E-5. Second, some electronic calculators can readily convert numbers in fixed notation to scientific notation. If you have such a calculator, you may be able to do this by pressing the EE or EXP key and then the “ϭ” key (but check your calculator manual to learn how your device operates). In chemistry you will often have to use numbers in exponential notation in mathematical operations. The following five operations are important: 38 Chapter 1 Matter and Measurement • Adding and Subtracting Numbers Expressed in Scientific Notation When adding or subtracting two numbers, first convert them to the same powers of 10. The digit terms are then added or subtracted as appropriate: 11.234 ϫ 10Ϫ3 2 ϩ 15.623 ϫ 10Ϫ2 2 ϭ 10.1234 ϫ 10Ϫ2 2 ϩ 15.623 ϫ 10Ϫ2 2 ϭ 5.746 ϫ 10Ϫ2 • Multiplication of Numbers Expressed in Scientific Notation The digit terms are multiplied in the usual manner, and the exponents are added algebraically. The result is expressed with a digit term with only one nonzero digit to the left of the decimal: 16.0 ϫ 1023 212.0 ϫ 10Ϫ2 2 ϭ 16.0212.02 ϫ 1023Ϫ2 ϭ 12 ϫ 1021 ϭ 1.2 ϫ 1022 • Division of Numbers Expressed in Scientific Notation The digit terms are divided in the usual manner, and the exponents are subtracted algebraically. The quotient is written with one nonzero digit to the left of the decimal in the digit term: 7.60 ϫ 103 7.60 ϫ 103Ϫ2 ϭ 6.18 ϫ 101 2 ϭ 1.23 1.23 ϫ 10 • Powers of Numbers Expressed in Scientific Notation When raising a number in exponential notation to a power, treat the digit term in the usual manner. The exponent is then multiplied by the number indicating the power: 15.28 ϫ 103 2 2 ϭ 15.282 2 ϫ 103ϫ2 ϭ 27.9 ϫ 106 ϭ 2.79 ϫ 107 • Roots of Numbers Expressed in Scientific Notation Unless you use an electronic calculator, the number must first be put into a form in which the exponent is exactly divisible by the root. For example, for a square root, the exponent should be divisible by 2. The root of the digit term is found in the usual way, and the exponent is divided by the desired root: 23.6 ϫ 107 ϭ 236 ϫ 106 ϭ 236 ϫ 2106 ϭ 6.0 ϫ 103 Significant Figures In most experiments several kinds of measurements must be made, and some can be made more precisely than others. It is common sense that a result calculated from experimental data can be no more precise than the least precise piece of information that went into the calculation. This is where the rules for significant figures come in. Significant figures are the digits in a measured quantity that reflect the accuracy of the measurement. When describing standard deviation on page 33, we used the example of a measurement that was known to be 9.984 with an uncertainty of Ϯ0.007 cm. That is, the last number of our measurement, 0.004 cm, was uncertain to some degree. Our measurement is said to have four significant figures, the last of which is uncertain to some extent. 1.8 39 Mathematics of Chemistry Suppose we want to calculate the density of a piece of metal (Figure 1.22). The mass and dimensions were determined by standard laboratory techniques. Most of these numbers have two digits to the right of the decimal, but they have different numbers of significant figures. Measurement Data Collected Significant Figures Mass of metal 13.56 g 2.50 cm 4 13.56 g Length 6.45 cm Width 2.50 cm 3 Thickness 3.1 mm 6.45 cm 3 2 The quantity 3.1 mm has two significant figures. That is, the 3 in 3.1 is exactly known, but the 1 is not. In general, in a number representing a scientific measurement, the last digit to the right is taken to be inexact. Unless stated otherwise, it is common practice to assign an uncertainty of Ϯ1 to the last significant digit. This means the thickness of the metal piece may have been as small as 3.0 mm or as large as 3.2 mm. When the data on the piece of metal are combined to calculate the density, the result will be 2.7 g/cm3, a number with two significant figures. (The complete calculation of the metal density is given on page 41). The reason for this is that a calculated result can be no more precise than the least precise data used, and here the thickness has only two significant figures. When doing calculations using measured quantities, we follow some basic rules so that the results reflect the precision of all the measurements that go into the calculations. The rules used for significant figures in this book are as follows: 3.1 mm Figure 1.22 Data to determine the density of a metal. Rule 1. To determine the number of significant figures in a measurement, read the number from left to right and count all digits, starting with the first digit that is not zero. Number of Significant Figures 1.23 3; all nonzero digits are significant. 0.00123 g 3; the zeros to the left of the 1 (the first significant digit) simply locate the decimal point. To avoid confusion, write numbers of this type in scientific notation; thus, 0.00123 ϭ 1.23 ϫ 10Ϫ 3. 2.040 g 4; when a number is greater than 1, all zeros to the right of the decimal point are significant. 0.02040 g 4; for a number less than 1, only zeros to the right of the first nonzero digit are significant. 100 g 1; in numbers that do not contain a decimal point, “trailing” zeros may or may not be significant. The practice followed in this book is to include a decimal point if the zeros are significant. Thus, 100. is used to represent three significant digits, whereas 100 has only one significant digit. To avoid confusion, an alternative method is to write numbers in scientific notation because all digits are significant when written in scientific notation. Thus, 1.00 ϫ 102 has three significant digits, whereas 1 ϫ 102 has only one significant digit. 100 cm/m Infinite number of significant digits. This is a defined quantity. Defined quantities do not limit the number of significant figures in a calculated result. p ϭ 3.1415926 The value of certain constants such as p is known to a greater number of significant figures than you will ever use in a calculation. Charles D. Winters Example Standard laboratory balance and significant figures. Such balances can determine the mass of an object to the nearest milligram. Thus, an object may have a mass of 13.456 g (13456 mg, five significant figures), 0.123 g (123 mg, three significant figures), or 0.072 g (72 mg, two significant figures). 40 Chapter 1 Matter and Measurement Rule 2. When adding or subtracting numbers, the number of decimal places in the answer is equal to the number of decimal places in the number with the fewest digits after the decimal. 0.12 2 decimal places 2 significant figures ϩ 1.9 1 decimal place 2 significant figures ϩ10.925 3 decimal places 5 significant figures 12.945 3 decimal places The sum should be reported as 12.9, a number with one decimal place, because 1.9 has only one decimal place. ■ To Multiply or to Add? Take the number 4.68. (a) Take the sum of 4.68 ϩ 4.68 ϩ 4.68. The answer is 14.04, a number with four significant figures. (b) Multiply 4.68 times 3. The answer can have only three significant figures (14.0). You should recognize that different outcomes are possible depending on the type of mathematical operation. ■ Who Is Right—You or the Book? If your answer to a problem in this book does not quite agree with the answers in Appendix N or O, the discrepancy may be the result of rounding the answer after each step and then using that rounded answer in the next step. This book follows these conventions: (a) Final answers to numerical problems in this book result from retaining full calculator accuracy throughout the calculation and rounding only at the end. (b) In Example problems, the answer to each step is given to the correct number of significant figures for that step, but the full calculator accuracy is carried to the next step. The number of significant figures in the final answer is dictated by the number of significant figures in the original data. Rule 3. In multiplication or division, the number of significant figures in the answer should be the same as that in the quantity with the fewest significant figures. 0.01208 ϭ 0.512 or, in scientific notation, 5.12 ϫ 10Ϫ1 0.0236 Because 0.0236 has only three significant digits and 0.01208 has four, the answer should have three significant digits. Rule 4. When a number is rounded off, the last digit to be retained is increased by one only if the following digit is 5 or greater. Full Number Number Rounded to Three Significant Digits 12.696 12.7 16.349 16.3 18.35 18.4 18.351 18.4 One last word on significant figures and calculations: When working problems, you should do the calculation with all the digits allowed by your calculator and round off only at the end of the calculation. Rounding off in the middle can introduce errors. See the General ChemistryNow CD-ROM or website: • Screen 1.17 Using Numerical Information, for tutorials on multiplying and dividing with significant figures, raising significant figures to a power, and taking square roots of significant figures Example 1.5—Using Significant Figures Problem An example of a calculation you will do later in the book (Chapter 12) is Volume of gas 1L2 ϭ 10.120210.0820621273.15 ϩ 232 1230/760.02 1.8 41 Mathematics of Chemistry Calculate the final answer to the correct number of significant figures. Strategy Let us first decide on the number of significant figures represented by each number (Rule 1), and then apply Rules 2 and 3. Solution Number Number of Significant Figures Comments 0.120 3 The trailing 0 is significant. See Rule 1. 0.08206 4 The first 0 to the immediate right of the decimal is not significant. See Rule 1. 273.15 ϩ 23 ϭ 296 3 23 has no decimal places, so the sum can have none. See Rule 2. 230/760.0 ϭ 0.30 2 230 has two significant figures because the last zero is not significant. In contrast, there is a decimal point in 760.0, so there are four significant digits. The quotient may have only two significant digits. See Rules 1 and 3. Analysis shows that one of the pieces of information is known to only two significant figures. Therefore, the volume of gas is 9.6 L, a number with two significant figures. Exercise 1.13—Using Significant Figures (a) How many significant figures are indicated by 2.33 ϫ 107, by 50.5, and by 200? (b) What are the sum and the product of 10.26 and 0.063? (c) What is the result of the following calculation? xϭ 1110.7 Ϫ 642 10.056210.002162 Problem Solving by Dimensional Analysis Suppose you want to find the density of a rectangular piece of metal (Figure 1.22) in units of grams per cubic centimeter (g/cm3). Because density is the ratio of mass to volume, you need to measure the mass and determine the volume of the piece. To find the volume of the sample in cubic centimeters, you multiply its length by its width and its thickness. First, however, all the measurements must have the same units, meaning that the thickness must be converted to centimeters. Recognizing that there are 10 mm in 1 cm, we use this relationship to get a thickness of 0.31 cm: 1 cm 3.1 mm ϫ ϭ 0.31 cm 10 mm With all the dimensions in the same unit, the volume and then the density can be calculated: Length ϫ width ϫ thickness ϭ volume 6.45 cm ϫ 2.50 cm ϫ 0.31 cm ϭ 5.0 cm3 Density ϭ 13.56 g 5.0 cm3 ϭ 2.7 g/cm3 ■ Data to Calculate Metal Density (See Figure 1.22) Mass of metal ϭ 13.56 g Length ϭ 6.45 cm Width ϭ 2.50 cm Thickness ϭ 3.1 mm 42 Chapter 1 Matter and Measurement Dimensional analysis (sometimes called the factor-label method) is a general problem-solving approach that uses the dimensions or units of each value to guide you through calculations. This approach was used above to change 3.1 mm to its equivalent in centimeters. We multiplied the number we wished to convert (3.1 mm) by a conversion factor (1 cm/10 mm) to produce the result in the desired unit (0.31 cm). Units are handled like numbers: Because the unit “mm” was in both the numerator and the denominator, dividing one by the other leaves a quotient of 1. The units are said to “cancel out.” Here this leaves the answer in centimeters, the desired unit. A conversion factor expresses the equivalence of a measurement in two different units (1 cm ϵ 10 mm; 1 g ϵ 1000 mg; 12 eggs ϵ 1 dozen; 12 inches ϵ 1 foot ). Because the numerator and the denominator describe the same quantity, the conversion factor is equivalent to the number 1. Therefore, multiplication by this factor does not change the measured quantity, only its units. A conversion factor is always written so that it has the form “new units divided by units of original number.” Number in original unit Quantity to express in new units new unit ϭ new number in new unit original unit Conversion factor Quantity now expressed in new units See the General ChemistryNow CD-ROM or website: • Screen 1.17 Using Numerical Information, for a tutorial on dimensional analysis Example 1.6—Using Conversion Factors — Density in Different Units Problem Oceanographers often express the density of sea water in units of kilograms per cubic meter. If the density of sea water is 1.025 g/cm3 at 15 °C, what is its density in kilograms per cubic meter? Strategy To simplify this problem, break it into three steps. First, change grams to kilograms. Next, convert cubic centimeters to cubic meters. Finally, calculate the density by dividing the mass in kilograms by the volume in cubic meters. Solution First convert the mass in grams to kilograms. 1.025 g ϫ 1 kg ϭ 1.025 ϫ 10Ϫ3 kg 1000 g No conversion factor is available in one of our tables to directly change units of cubic centimeters to cubic meters. You can find one, however, by cubing (raising to the third power) the relation between the meter and the centimeter: 1 cm3 ϫ a 1m 3 1 m3 b ϭ 1 cm3 ϫ a b ϭ 1 ϫ 10Ϫ6 m3 100 cm 1 ϫ 106 cm3 1.8 Mathematics of Chemistry 43 Therefore, the density of sea water is Density ϭ 1.025 ϫ 10Ϫ3 kg 1 ϫ 10Ϫ6 m3 ϭ 1.025 ϫ 103 kg/m3 Exercise 1.14—Using Dimensional Analysis (a) The annual snowfall at Lake Otsego is 198 cm each year. What is this depth in meters? In feet (where 1 foot ϭ 30.48 cm)? (b) The area of Lake Otsego is 2.33 ϫ 107 m2. What is this area in square kilometers? (c) The density of gold is 19,320 kg/m3. What is this density in g/cm3? (d) See Figure 1.21. Show that 9.0 ϫ 103 pc is 2.8 ϫ 1017 km. Graphing In a number of instances in this text, graphs are used when analyzing experimental data with a goal of obtaining a mathematical equation. The procedure used will often result in a straight line, which has the equation y ϭ mx ϩ b In this equation, y is usually referred to as the dependent variable; its value is determined from (that is, is dependent on) the values of x, m, and b. In this equation x is called the independent variable and m is the slope of the line. The parameter b is the y -intercept—that is, the value of y when x ϭ 0. Let us use an example to investigate two things: (a) how to construct a graph from a set of data points, and (b) how to derive an equation for the line generated by the data. A set of data points to be graphed is presented in Figure 1.23. We first mark off each axis in increments of the values of x and y. Here our x-data range from Ϫ2 to 4, so the x-axis is marked off in increments of 1 unit. The y-data range from 0 to 2.5, so we mark off the y-axis in increments of 0.5. Each data set is marked as a circle on the graph. After plotting the points on the graph (round circles), we draw a straight line that comes as close as possible to representing the trend in the data. (Do not connect the dots!) Because there is always some inaccuracy in experimental data, this line may not pass exactly through every point. To identify the specific equation corresponding to our data, we must determine the y-intercept (b) and slope (m) for the equation y ϭ mx ϩ b . The y -intercept is the point at which x ϭ 0. (In Figure 1.23, y ϭ 1.87 when x ϭ 0). The slope is determined by selecting two points on the line (marked with squares in Figure 1.23) and calculating the difference in values of y ( ¢ y ϭ y 2 Ϫ y 1) and x ( ¢ x ϭ x 2 Ϫ x 1). The slope of the line is then the ratio of these differences, m ϭ ¢ y/ ¢ x. Here the slope has the value Ϫ0.525. With the slope and intercept now known, we can write the equation for the line y ϭ Ϫ0.525x ϩ 1.87 and we can use this equation to calculate y -values for points that are not part of our original set of x-y data. For example, when x ϭ 1.50, y ϭ 1.08. ■ Determining the Slope with a Computer Program—Least-Squares Analysis Generally the easiest method of determining the slope and intercept of a straight line (and thus the line’s equation) is to use a program such as Microsoft Excel. These programs perform a “least squares” or “linear regression” analysis and give the best straight line based on the data. (This line is referred to in Excel as a trendline.) The General ChemistryNow CD-ROM also has a useful plotting program that performs this analysis; see the “Plotting Tool” in the menu on any screen. 44 Chapter 1 Matter and Measurement Figure 1.23 Plotting Data. Data for the variable x are plotted along the horizontal axis (abscissa), and data for y are plotted along the vertical axis (ordinate). The slope of the line, m in the equation y ϭ mx ϩ b, is given by ¢ y/ ¢ x. The intercept of the line with the y-axis (when x ϭ 0) is b in the equation. Using Microsoft Excel with these data, and doing a linear regression (or least-squares) analysis, we find y ϭ Ϫ0.525x ϩ 1.87. 3 Experimental data 2.5 2 x 3.35 2.59 1.08 Ϫ1.19 x = 0, y = 1.87 y 0.0565 0.520 1.38 2.45 1.5 1 x = 2.00, y = 0.82 Using the points marked with a square, the slope of the line is: Slope ϭ 0.5 ⌬y 0.82 Ϫ 1.87 ϭ ϭ Ϫ0.525 ⌬x 2.00 Ϫ 0.00 0 –2 –1 0 1 2 3 4 Exercise 1.15—Graphing To find the mass of 50 jelly beans, we weighed several samples of beans. Number of Beans Mass (g) 5 12.82 11 27.14 16 39.30 24 59.04 Plot these data with the number of beans on the horizontal or x-axis, and the mass of beans on the vertical or y-axis. What is the slope of the line? Use your equation of a straight line to calculate the mass of exactly 50 jelly beans. Problem Solving and Chemical Arithmetic Problem-Solving Strategy Some of the calculations in chemistry can be complex. Students frequently find it is helpful to follow a definite plan of attack as illustrated in examples throughout this book. Step 1: Problem. State the problem. Read it carefully. Step 2: Strategy. What key principles are involved? What information is known or not known? What information might be there just to place the question in the context of chemistry? Organize the information to see what is required and to discover the relationships among the data given. Try writing the information down in table form. If it is numerical information, be sure to include units. 1.8 Mathematics of Chemistry One of the greatest difficulties for a student in introductory chemistry is picturing what is being asked for. Try sketching a picture of the situation involved. For example, we sketched a picture of the piece of metal whose density we wanted to calculate, and put the dimensions on the drawing (page 39). Develop a plan. Have you done a problem of this type before? If not, perhaps the problem is really just a combination of several simpler ones you have seen before. Break it down into those simpler components. Try reasoning backward from the units of the answer. What data do you need to find an answer in those units? Step 3: Solution. Execute the plan. Carefully write down each step of the problem, being sure to keep track of the units on numbers. (Do the units cancel to give you the answer in the desired units?) Don’t skip steps. Don’t do anything except the simplest steps in your head. Students often say they got a problem wrong because they “made a stupid mistake.” Your instructor—and book authors—make them, too, and it is usually because they don’t take the time to write down the steps of the problem clearly. Step 4: Comment and Check Answer. As a final check, ask yourself whether the answer is reasonable. Example 1.7—Problem Solving Problem A mineral oil has a density of 0.875 g/cm3. Suppose you spread 0.75 g of this oil over the surface of water in a large dish with an inner diameter of 21.6 cm. How thick is the oil layer? Express the thickness in centimeters. Strategy It is often useful to begin solving such problems by sketching a picture of the situation. 21.6 cm This helps recognize that the solution to the problem is to find the volume of the oil on the water. If we know the volume, then we can find the thickness because Volume of oil layer ϭ (thickness of layer) ϫ (area of oil layer) So, we need two things: (a) the volume of the oil layer and (b) the area of the layer. Solution First calculate the volume of oil. The mass of the oil layer is known, so combining the mass of oil with its density gives the volume of the oil used: 0.75 g ϫ 1 cm3 ϭ 0.86 cm3 0.875 g Next calculate the area of the oil layer. The oil is spread over a circular surface, whose area is given by Area ϭ p ϫ (radius)2 45 46 Chapter 1 Matter and Measurement The radius of the oil layer is one half its diameter (= 21.6 cm) or 10.8 cm, so Area of oil layer ϭ (3.142)(10.8 cm)2 ϭ 366 cm2 With the volume and the area of the oil layer known, the thickness can be calculated. Thickness ϭ Volume 0.86 cm3 ϭ 0.0023 cm ϭ Area 366 cm2 Comment In the volume calculation, the calculator shows 0.857143. . . . The quotient should have two significant figures because 0.75 has two significant figures, so the result of this step is 0.86 cm3. In the area calculation, the calculator shows 366.435. . . . The answer to this step should have three significant figures because 10.8 has three. When these interim results are combined in calculating thickness, however, the final result can have only two significant figures. Premature rounding can lead to errors. Exercise 1.16—Problem Solving A particular paint has a density of 0.914 g/cm3. You need to cover a wall that is 7.6 m long and 2.74 m high with a paint layer 0.13 mm thick. What volume of paint (in liters) is required? What is the mass (in grams) of the paint layer? Chapter Goals Revisited • • See the General ChemistryNow CD-ROM or website to: Assess your understanding with homework questions keyed to each goal Check your readiness for an exam by taking the exam-prep quiz and exploring the resources in the personalized Learning Plan it provides Now that you have studied this chapter, you should ask whether you have met the chapter goals. In particular, you should be able to Classify matter a. Recognize the different states of matter (solids, liquids, and gases) and give their characteristics (Section 1.1). b. Appreciate the difference between pure substances and mixtures and the difference between homogeneous and heterogeneous mixtures (Section 1.1). c. Recognize the importance of representing matter at the macroscopic level and at the particulate level (Section 1.1). Apply the kinetic-molecular theory to the properties of matter a. Understand the basic ideas of the kinetic-molecular theory (Section 1.1). Recognize elements, atoms, compounds, and molecules a. Identify the name or symbol for an element, given its symbol or name (Section 1.2). General ChemistryNow homework: Study Question(s) 2 b. Use the terms atom, element, molecule, and compound correctly (Sections 1.2 and 1.3). Identify physical and chemical properties and changes a. List commonly used physical properties of matter (Section 1.4). General ChemistryNow homework: SQ(s) 8 Key Equations b. Identify several physical properties of common substances (Section 1.4). c. Use density to connect the volume and mass of a substance (Sections 1.4, 1.6 and 1.8). General ChemistryNow homework: SQ(s) 11, 13 d. Explain the difference between chemical and physical changes (Sections 1.4 and 1.5). e. Understand the difference between extensive and intensive properties and give examples of them (Section 1.4). Use metric units correctly a. Convert between temperatures on the Celsius and Kelvin scales (Section 1.6). General ChemistryNow homework: SQ(s) 19 b. Recognize and know how to use the prefixes that modify the sizes of metric units (Section 1.6). Understand and use the mathematics of chemistry a. Use dimensional analysis to carry out unit conversions. Perform other mathematical operations (Section 1.8). General ChemistryNow homework: SQ(s) 24, 39, 40, 43, 45, 51, 83c, 84a, 89, 94b, 95, 96, 97, 104 b. Know the difference between precision and accuracy and how to calculate percent error (Section 1.7). General ChemistryNow homework: SQ(s) 30 c. Understand the use of significant figures (Section 1.8). General ChemistryNow homework: SQ(s) 78, 80 Key Equations Equation 1.1 (page 20) Density is the quotient of the mass of an object divided by its volume. In chemistry the common unit of density is g/cm3. Density ϭ mass volume Equation 1.2 (page 28) The equation allows the conversion between the Kelvin and Celsius temperature scales. T1K2 ϭ 1K 3T 1°C2 ϩ 273.15 °C4 1 °C Equation 1.3 (page 34) The percent error of a measurement is the deviation of the measurement from the accepted value. Percent error ϭ error in measurement ϫ 100% accepted value 47 48 Chapter 1 Matter and Measurement Study Questions ▲ denotes more challenging questions. ■ denotes questions available in the Homework and Goals section of the General ChemistryNow CD-ROM or website. Blue numbered questions have answers in Appendix O and fully worked solutions in the Student Solutions Manual. Structures of many of the compounds used in these questions are found on the General ChemistryNow CD-ROM or website in the Models folder. Assess your understanding of this chapter’s topics with additional quizzing and conceptual questions at Practicing Skills Matter: Elements and Atoms, Compounds and Molecules (See Exercise 1.2.) 1. Give the name of each of the following elements: (a) C (c) Cl (e) Mg (b) K (d) P (f ) Ni 2. ■ Give the name of each of the following elements: (a) Mn (c) Na (e) Xe (b) Cu (d) Br (f ) Fe 3. Give the symbol for each of the following elements: (a) barium (d) lead (b) titanium (e) arsenic (c) chromium (f ) zinc 4. Give the symbol for each of the following elements: (a) silver (d) tin (b) aluminum (e) technetium (c) plutonium (f ) krypton 5. In each of the following pairs, decide which is an element and which is a compound. (a) Na and NaCl (b) sugar and carbon (c) gold and gold chloride 6. In each of the following pairs, decide which is an element and which is a compound. (a) Pt(NH3)2Cl2 and Pt (b) copper or copper(II) oxide (c) silicon or sand Physical and Chemical Properties (See Exercises 1.3 and 1.6.) 7. In each case, decide whether the underlined property is a physical or chemical property. ▲ More challenging ■ In General ChemistryNow (a) The normal color of elemental bromine is orange. (b) Iron turns to rust in the presence of air and water. (c) Hydrogen can explode when ignited in air. (d) The density of titanium metal is 4.5 g/cm3. (e) Tin metal melts at 505 K. (f ) Chlorophyll, a plant pigment, is green. 8. ■ In each case, decide whether the change is a chemical or physical change. (a) A cup of household bleach changes the color of your favorite T-shirt from purple to pink. (b) Water vapor in your exhaled breath condenses in the air on a cold day. (c) Plants use carbon dioxide from the air to make sugar. (d) Butter melts when placed in the sun. 9. Which part of the description of a compound or element refers to its physical properties and which to its chemical properties? (a) The colorless liquid ethanol burns in air. (b) The shiny metal aluminum reacts readily with orange, liquid bromine. 10. Which part of the description of a compound or element refers to its physical properties and which to its chemical properties? (a) Calcium carbonate is a white solid with a density of 2.71 g/cm3. It reacts readily with an acid to produce gaseous carbon dioxide. (b) Gray, powdered zinc metal reacts with purple iodine to give a white compound. Using Density (See Example 1.1. and the General ChemistryNow Screen 1.10.) 11. ■ Ethylene glycol, C2H6O2, is an ingredient of automobile antifreeze. Its density is 1.11 g/cm3 at 20 ° C. If you need exactly 500. mL of this liquid, what mass of the compound, in grams, is required? 12. A piece of silver metal has a mass of 2.365 g. If the density of silver is 10.5 g/cm3, what is the volume of the silver? 13. ■ A chemist needs 2.00 g of a liquid compound with a density of 0.718 g/cm3. What volume of the compound is required? 14. The cup is a volume measure widely used by cooks in the United States. One cup is equivalent to 237 mL. If 1 cup of olive oil has a mass of 205 g, what is the density of the oil (in grams per cubic centimeter)? 15. A sample of unknown metal is placed in a graduated cylinder containing water. The mass of the sample is 37.5 g, and the water levels before and after adding the sample to the cylinder are as shown in the figure. Which metal in the following list is most likely the sample? (d is the density of the metal.) Blue-numbered questions answered in Appendix O 49 Study Questions (a) Mg, d ϭ 1.74 g/cm3 (b) Fe, d ϭ 7.87 g/cm3 (c) Ag, d ϭ 10.5 g/cm3 (d) Al, d ϭ 2.70 g/cm3 (e) Cu, d ϭ 8.96 g/cm3 (f ) Pb, d ϭ 11.3 g/cm3 25 20 15 15 10 10 5 25. A typical laboratory beaker has a volume of 250. mL. What is its volume in cubic centimeters? In liters? In cubic meters? In cubic decimeters? 25 20 24. ■ A compact disk has a diameter of 11.8 cm. What is the surface area of the disk in square centimeters? In square meters? [Area of a circle ϭ (p)(radius)2.] 5 26. Some soft drinks are sold in bottles with a volume of 1.5 L. What is this volume in milliliters? In cubic centimeters? In cubic decimeters? 27. A book has a mass of 2.52 kg. What is this mass in grams? 28. A new U. S. dime has a mass of 2.265 g. What is this mass in kilograms? In milligrams? Accuracy, Precision, and Error (See Example 1.4.) Graduated cylinders with unknown metal (right). 16. Iron pyrite is often called “fool’s gold” because it looks like gold (see page 19). Suppose you have a solid that looks like gold, but you believe it to be fool’s gold. The sample has a mass of 23.5 g. When the sample is lowered into the water in a graduated cylinder (see Study Question 15), the water level rises from 47.5 mL to 52.2 mL. Is the sample fool’s gold (d ϭ 5.00 g/cm3) or “real” gold (d ϭ 19.3 g/cm3)? Temperature Scales (See Exercise 1.7. and the General ChemistryNow Screen 1.15.) 29. You and your lab partner are asked to determine the density of an aluminum bar. The mass is known accurately (to four significant figures). You use a simple metric ruler to determine its size and calculate the results in A. Your partner uses a precision micrometer and obtains the results in B. Method A (g/cm3) Method B (g/cm3) 2.2 2.703 17. Many laboratories use 25 ° C as a standard temperature. What is this temperature in kelvins? 2.3 2.701 2.7 2.705 18. The temperature on the surface of the sun is 5.5 ϫ 103 ° C. What is this temperature in kelvins? 2.4 5.811 19. ■ Make the following temperature conversions: °C K (a) 16 —-— (b) —-— 370 (c) 40 —-— 20. Make the following temperature conversions: °C K (a) —— 77 (b) 63 —— (c) —— 1450 Using Units (See Examples 1.2 and 1.3. and the General ChemistryNow Screen 1.14.) The accepted density of aluminum is 2.702 g/cm3. (a) Calculate the average density for each method. Should all the experimental results be included in your calculations? If not, justify any omissions. (b) Calculate the percent error for each method’s average value. (c) Which method’s average value is more precise? Which method is more accurate? 30. ■ The accepted value of the melting point of pure aspirin is 135 ° C. Trying to verify that value, you obtain the melting points of 134 ° C, 136 ° C, 133 ° C, and 138 ° C in four separate trials. Your partner finds melting points of 138 ° C, 137 ° C, 138 ° C, and 138 ° C. (a) Calculate the average value and percent error for you and your partner. (b) Which of you is more precise? More accurate? 21. A marathon race covers a distance of 42.195 km. What is this distance in meters? In miles? 22. The average lead pencil, new and unused, is 19 cm long. What is its length in millimeters? In meters? 23. A standard U.S. postage stamp is 2.5 cm long and 2.1 cm wide. What is the area of the stamp in square centimeters? In square meters? ▲ More challenging General Questions These questions are not designated as to type or location in the chapter. They may combine several concepts. 31. A piece of turquoise is a blue-green solid, and has a density of 2.65 g/cm3 and a mass of 2.5 g. ■ In General ChemistryNow Blue-numbered questions answered in Appendix O 50 Chapter 1 Matter and Measurement (a) Which of these observations are qualitative and which are quantitative? (b) Which of these observations are extensive and which are intensive? (c) What is the volume of the piece of turquoise? used, where 1 Å ϭ 1 ϫ 10Ϫ10 m. (The angstrom is not an SI unit.) If the distance between the Pt atom and the N atom in the cancer chemotherapy drug cisplatin is 1.97 Å, what is this distance in nanometers? In picometers? 32. Give a physical property and a chemical property for the elements hydrogen, oxygen, iron, and sodium. (The elements listed are selected from examples given in Chapter 1.) H3N NH3 Pt 1.97Å Cl Cl 33. The gemstone called aquamarine is composed of aluminum, silicon, and oxygen. cisplatin Charles D. Winters 38. The separation between carbon atoms in diamond is 0.154 nm. What is their separation in meters? In picometers? 0.154 nm Aquamarine is the bluish crystal. It is surrounded by aluminum foil and crystalline silicon. (a) What are the symbols of the three elements that combine to make the gem aquamarine? (b) Based on the photo, describe some of the physical properties of the elements and the mineral. Are any the same? Are any properties different? 34. Eight observations are listed below. Which of these observations identify chemical properties? (a) Sugar is soluble in water. (b) Water boils at 100 ° C. (c) Ultraviolet light converts O3 (ozone) to O2 (oxygen). (d) Ice is less dense than water. (e) Sodium metal reacts violently with water. (f ) CO2 does not support combustion. (g) Chlorine is a yellow gas. (h) Heat is required to melt ice. 35. Neon, a gaseous element used in neon signs, has a melting point of Ϫ248.6 ° C and a boiling point of Ϫ246.1 ° C. Express these temperatures in kelvins. 36. You can identify a metal by carefully determining its density (d). An unknown piece of metal, with a mass of 2.361 g, is 2.35 cm long, 1.34 cm wide, and 1.05 mm thick. Which of the following is this element? (a) Nickel, d ϭ 8.90 g/cm3 (b) Titanium, d ϭ 4.50 g/cm3 (c) Zinc, d ϭ 7.13 g/cm3 (d) Tin, d ϭ 7.23 g/cm3 37. Molecular distances are usually given in nanometers (1 nm ϭ 1 ϫ 10Ϫ9 m) or in picometers (1 pm ϭ 1 ϫ 10Ϫ12 m). However, the angstrom (Å) is sometimes ▲ More challenging ■ In General ChemistryNow A portion of the diamond structure 39. ■ A red blood cell has a diameter of 7.5 mm (micrometers). What is this dimension in (a) meters, (b) nanometers, and (c) picometers? 40. ■ Which occupies a larger volume, 600. g of water (with a density of 0.995 g/cm3) or 600. g of lead (with a density of 11.34 g/cm3)? 41. The platinum-containing cancer drug cisplatin contains 65.0% platinum. If you have 1.53 g of the compound, what mass of platinum (in grams) is contained in this sample? 42. The solder once used by plumbers to fasten copper pipes together consists of 67% lead and 33% tin. What is the mass of lead in a 250-g block of solder? 43. ■ The anesthetic procaine hydrochloride is often used to deaden pain during dental surgery. The compound is packaged as a 10.% solution (by mass; d ϭ 1.0 g/mL) in water. If your dentist injects 0.50 mL of the solution, what mass of procaine hydrochloride (in milligrams) is injected? 44. A cube of aluminum (density ϭ 2.70 g/cm3) has a mass of 7.6 g. What must be the length of the cube’s edge (in centimeters)? (See General ChemistryNow Screen 1.10, Tutorial 2, Density.) 45. ■ You have a 100.0-mL graduated cylinder containing 50.0 mL of water. You drop a 154-g piece of brass (d ϭ 8.56 g/cm3) into the water. How high does the water rise in the graduated cylinder? Blue-numbered questions answered in Appendix O 51 Study Questions Charles D. Winters Charles D. Winters 49. Small chips of iron are mixed with sand (see the photo). Is this a homogeneous or heterogeneous mixture? Suggest a way to separate the iron from the sand. Chips of iron mixed with sand. (a) A graduated cylinder with 50.0 ml of water. (b) A piece of brass is added to the cylinder. 46. You have a white crystalline solid, known to be one of the potassium compounds listed below. To determine which, you measure the solid’s density. You measure out 18.82 g and transfer it to a graduated cylinder containing kerosene (in which salts will not dissolve). The level of liquid kerosene rises from 8.5 mL to 15.3 mL. Calculate the density of the solid, and identify the compound from the following list. (a) KF, d ϭ 2.48 g/cm3 (c) KBr, d ϭ 2.75 g/cm3 (b) KCl, d ϭ 1.98 g/cm3 (d) KI, d ϭ 3.13 g/cm3 47. A distant acquaintance has offered to sell you a necklace, said to be pure (24-carat ) gold, for $300. You have some doubts, however; perhaps it is gold plated. You decide to run a test. You have a graduated cylinder and a small balance. You partially fill the cylinder with water and immerse the necklace; the height of water rises from 22.5 mL to 26.0 mL. Then you determine the mass to be 67 g. You recall that the density of gold is 19.3 g/cm3, and that no other element has a density near this value. (Silver has a density of 11.5 g/cm3.) The price of gold on the open market is $380 per troy ounce (1 troy ounce ϭ 31.1 g). Is the necklace gold? Explain your conclusion. Is $300 a good price? Conceptual Questions Water, copper, and mercury. 51. ■ Carbon tetrachloride, CCl4, a liquid compound, has a density of 1.58 g/cm3. If you place a piece of a plastic soda bottle (d ϭ 1.37 g/cm3) and a piece of aluminum (d ϭ 2.70 g/cm3) in liquid CCl4, will the plastic and aluminum float or sink? 52. Figure 1.7 shows a piece of table salt and a representation of its internal structure. Which is the macroscopic view and which is the particulate view? How are the macroscopic and particulate views related? 53. ▲ You have a sample of a white crystalline substance from your kitchen. You know that it is either salt or sugar. Although you could decide by taste, suggest another property that you could use to determine the sample’s identity. (Hint: You may use the World Wide Web or a handbook of chemistry in the library to find some pertinent information.) Charles D. Winters 48. The mineral fluorite contains the elements calcium and fluorine. What are the symbols of these elements? How would you describe the shape of the fluorite crystals in the photo? What can this tell us about the arrangement of the atoms inside the crystal? 50. The following photo shows copper balls, immersed in water, floating on top of mercury. What are the liquids and the solids in this photo? Which substance is most dense? Which is least dense? Charles D. Winters (b) (a) The mineral fluorite, calcium fluoride. ▲ More challenging ■ In General ChemistryNow Blue-numbered questions answered in Appendix O 52 Chapter 1 Matter and Measurement 54. Milk in a glass bottle was placed in the freezer compartment of a refrigerator overnight. By morning a column of frozen milk emerged from the bottle. Explain this observation. 57. You can figure out whether a substance floats or sinks if you know its density and the density of the liquid. In which of the liquids listed below will high-density polyethylene (HDPE, a common plastic whose density is 0.97 g/mL) float? (HDPE does not dissolve in these liquids.) Charles D. Winters 56. ▲ The density of pure water is given at various temperatures. T(°C) d(g/cm3) 4 0.99997 15 0.99913 25 0.99707 35 0.99406 ■ In General ChemistryNow 0.9997 0.7893 The alcohol in alcoholic beverages 0.7914 Toxic; gasoline additive to prevent gas line freezing Acetic acid 1.0492 Component of vinegar Glycerol 1.2613 Solvent used in home care products. 58. Hexane (C6H14, d ϭ 0.766 g/cm3), perfluorohexane (C6F14, d ϭ 1.669 g/cm3), and water are immiscible liquids; that is, they do not dissolve in one another. You place 10 mL of each liquid in a graduated cylinder, along with pieces of high-density polyethylene (HDPE, d ϭ 0.97 g/cm3), polyvinyl chloride (PVC, d ϭ 1.36 g/cm3), and Teflon (density ϭ 2.3 g/cm3). None of these common plastics dissolve in these liquids. Describe what you expect to see. 59. Make a drawing, based on the kinetic-molecular theory and the ideas about atoms and molecules presented in this chapter, of the arrangement of particles in each of the cases listed here. For each case draw ten particles of each substance. Your diagram can be two-dimensional. Represent each atom as a circle and distinguish each kind of atom by shading. (a) a sample of solid iron (which consists of iron atoms) (b) a sample of liquid water (which consists of H2O molecules) (c) a sample of water vapor (d) a homogeneous mixture of water vapor and helium gas (which consists of helium atoms) (e) a heterogeneous mixture consisting of liquid water and solid aluminum; show a region of the sample that includes both substances (f ) a sample of brass (which is a homogeneous mixture of copper and zinc) 60. You are given a sample of a silvery metal. What information would you seek to prove that the metal is silver? 61. Suggest a way to determine whether the colorless liquid in a beaker is water. If it is water, does it contain dissolved salt? How could you discover whether salt is dissolved in the water? Suppose your laboratory partner tells you that the density of water at 20 ° C is 0.99910 g/cm3. Is this a reasonable number? Why or why not? ▲ More challenging Toxic; the major component of automobile antifreeze Methanol Gallium metal. 1.1088 Ethanol 55. The element gallium has a melting point of 29.8 ° C. If you held a sample of gallium in your hand, should it melt? Explain briefly. Properties, Uses Water Frozen milk in a glass bottle. Density (g/cm3) Ethylene glycol Charles D. Winters Substance 62. Describe an experimental method that can be used to determine the density of an irregularly shaped piece of metal. Blue-numbered questions answered in Appendix O 53 Study Questions 63. Three liquids of different densities are mixed. Because they are not miscible (do not form a homogeneous solution with one another), they form discrete layers, one on top of the other. Sketch the result of mixing carbon tetrachloride (CCl4, d ϭ 1.58 g/cm3), mercury (d ϭ 13.546 g/cm3), and water (d ϭ 1.00 g/cm3). 64. Diabetes can alter the density of urine, so urine density can be used as a diagnostic tool. People with diabetes may excrete too much sugar or too much water. What do you predict will happen to the density of urine under each of these conditions? (Hint: Water containing dissolved sugar has a higher density than pure water.) 69. Many foods are fortified with vitamins and minerals. For example, some breakfast cereals have elemental iron added. Iron chips are used instead of iron compounds because the compounds can be converted by the oxygen in air to a form of iron that is not biochemically useful. Iron chips, in contrast, are converted to useful iron compounds in the gut, and the iron can then be absorbed. Outline a method by which you could remove the iron (as iron chips) from a box of cereal and determine the mass of iron in a given mass of cereal. (See General ChemistryNow Screens 1.1 and 1.18 Chemical Puzzler.) Charles D. Winters 65. The following photo shows the element potassium reacting with water to form the element hydrogen, a gas, and a solution of the compound potassium hydroxide. Some breakfast cereals contain iron in the form of elemental iron. Charles D. Winters 70. Describe what occurs when a hot object comes in contact with a cooler object. (See General ChemistryNow Screen 1.15 Temperature.) Potassium reacting with water to produce hydrogen gas and potassium hydroxide. (a) What states of matter are involved in the reaction? (b) Is the observed change chemical or physical? (c) What are the reactants in this reaction and what are the products? (d) What qualitative observations can be made concerning this reaction? 71. Study the animation of the conversion of P4 and Cl2 molecules to PCl3 molecules on General ChemistryNow CD-ROM or website Screen 1.13 Chemical Change on the Molecular Scale. (a) What are the reactants in this chemical change? What are the products? (b) Describe how the structures of the reactant molecules differ from the structures of the product molecules. 72. The photo below shows elemental iodine dissolving in ethanol to give a solution. Is this a physical or a chemical change? 66. A copper-colored metal is found to conduct an electric current. Can you say with certainty that it is copper? Why or why not? Suggest additional information that could provide unequivocal confirmation that the metal is copper. 68. Four balloons (each with a volume of 10 L and a mass of 1.00 g) are filled with a different gas: Helium, d ϭ 0.164 g/L Neon, d ϭ 0.825 g/L Argon, d ϭ 1.633 g/L Krypton, d ϭ 4.425 g/L If the density of dry air is 1.12 g/L, which balloon or balloons float in air? ▲ More challenging Charles D. Winters 67. What experiment can you use to: (a) Separate salt from water? (b) Separate iron filings from small pieces of lead? (c) Separate elemental sulfur from sugar? Elemental iodine dissolving in ethanol. (See General ChemistryNow Screen 1.9 Exercise, Physical Properties of Matter.) ■ In General ChemistryNow Blue-numbered questions answered in Appendix O 54 Chapter 1 Matter and Measurement Mathematics of Chemistry These questions provide an additional review of the mathematical skill used in general chemistry as presented in Section 1.8. sis using a computer program) and then write the equation for the resulting straight line. What is the slope of the line? What is the concentration when the absorbance is 0.635? Concentration (M) Absorbance Exponential Notation 0.00 0.00 73. Express the following numbers in exponential or scientific notation. (a) 0.054 (b) 5462 (c) 0.000792 1.029 ϫ 10Ϫ 3 0.257 2.058 ϫ 10Ϫ 3 0.518 3.087 ϫ 10Ϫ 3 0.771 Ϫ3 1.021 74. Express the following numbers in fixed notation (e.g., 123 ϫ 102 ϭ 123). (a) 1.62 ϫ 103 (b) 2.57 ϫ 10Ϫ4 (c) 6.32 ϫ 10Ϫ2 75. Carry out the following operations. Provide the answer with the correct number of significant figures. (a) (1.52)(6.21 ϫ 10Ϫ3) (b) (6.21 ϫ 103) Ϫ (5.23 ϫ 102) (c) (6.21 ϫ 103) Ϭ (5.23 ϫ 102) 76. Carry out the following operations. Provide the answer with the correct number of significant figures. (a) (6.25 ϫ 102)3 (b) 22.35 ϫ 10Ϫ3 (c) 12.35 ϫ 10Ϫ3 2 1/3 Significant Figures (See Exercise 1.13.) 77. Give the number of significant figures in each of the following numbers: (a) 0.0123 g (c) 1.6402 g (b) 3.40 ϫ 103 mL (d) 1.020 L 78. ■ Give the number of significant figures in each of the following numbers: (a) 0.00546 g (c) 2.300 ϫ 10Ϫ4 g (b) 1600 mL (d) 2.34 ϫ 109 atoms 4.116 ϫ 10 82. To determine the average mass of a popcorn kernel you collect the following data: Number of kernels 5 0.836 12 2.162 35 5.801 Plot the data with number of kernels on the x-axis and mass on the y-axis. Draw the best straight line using the points on the graph (or do a least-squares or linear regression analysis using a computer program) and then write the equation for the resulting straight line. What is the slope of the line? What does the slope of the line signify about the mass of a popcorn kernel? What is the mass of 50 popcorn kernels? How many kernels are there in a handful of popcorn (20.88 g)? 83. Using the graph below: (a) What is the value of x when y ϭ 4.0? (b) What is the value of y when x ϭ 0.30? (c) ■ What are the slope and the y- intercept of the line? (d) What is the value of y when x ϭ 1.0? 8.00 79. Carry out the following calculation, and report the answer with the correct number of significant figures. 7.779 d 55.85 80. ■ Carry out the following calculation, and report the answer to the correct number of significant figures. 11.682 c 23.56 Ϫ 2.3 d 1.248 ϫ 103 81. You are asked to calibrate a spectrophotometer in the laboratory and collect the following data. Plot the data with concentration on the x-axis and absorbance on the y -axis. Draw the best straight line using the points on the graph (or do a least-squares or linear regression analy■ In General ChemistryNow 6.00 5.00 4.00 3.00 Graphing (See Exercise 1.15. Use the plotting program on the General ChemistryNow CD-ROM or website or Microsoft Excel.) ▲ More challenging 7.00 y values 10.05462116.00002 c Mass (g) 2.00 1.00 0 Blue-numbered questions answered in Appendix O 0 0.10 0.20 0.30 x values 0.40 0.50 55 Study Questions 84. ■ Use the graph below to answer the following questions. 25.00 20.00 91. An ancient gold coin is 2.2 cm in diameter and 3.0 mm thick. It is a cylinder for which volume ϭ (p)(radius)2(thickness). If the density of gold is 19.3 g/cm3, what is the mass of the coin in grams? y values 15.00 92. Copper has a density of 8.96 g/cm3. An ingot of copper with a mass of 57 kg (126 lb) is drawn into wire with a diameter of 9.50 mm. What length of wire (in meters) can be produced? [Volume of wire ϭ (p)(radius)2( length)]. 10.00 5.00 0 (a) What is the volume of this cube in cubic nanometers? In cubic centimeters? (b) The density of NaCl is 2.17 g/cm3. What is the mass of this smallest repeating unit (“unit cell”)? (c) Each repeating unit is composed of four NaCl “molecules.” What is the mass of one NaCl molecule? 0 1.00 2.00 3.00 4.00 5.00 x values (a) Derive the equation for the straight line, y ϭ mx ϩ b. (b) What is the value of y when x ϭ 6.0? Using Equations 85. Solve the following equation for the unknown value, C. (0.502)(123) ϭ (750.)C 86. Solve the following equation for the unknown value, n. (2.34)(15.6) ϭ n(0.0821)(273) 87. Solve the following equation for the unknown value, T. (4.184)(244)(T Ϫ 292.0) ϩ (0.449)(88.5)(T Ϫ 369.0) ϭ 0 88. Solve the following equation for the unknown value, n. 1 1 Ϫ246.0 ϭ 1312 c 2 Ϫ 2 d 2 n Problem Solving 89. ■ Diamond has a density of 3.513 g/cm3. The mass of diamonds is often measured in “carats,” where 1 carat equals 0.200 g. What is the volume (in cubic centimeters) of a 1.50-carat diamond? 90. ▲ The smallest repeating unit of a crystal of common salt is a cube (called a unit cell) with an edge length of 0.563 nm. 93. ▲ In July 1983, an Air Canada Boeing 767 ran out of fuel over central Canada on a trip from Montreal to Edmonton. (The plane glided safely to a landing at an abandoned airstrip.) The pilots knew that 22,300 kg of fuel were required for the trip, and they knew that 7682 L of fuel were already in the tank. The ground crew added 4916 L of fuel, which was only about one fifth of what was required. The crew members used a factor of 1.77 for the fuel density—the problem is that 1.77 has units of pounds per liter and not kilograms per liter! What is the fuel density in units of kg/L? What mass of fuel should have been loaded? (1 lb ϭ 453.6 g.) 94. When you heat popcorn, it pops because it loses water explosively. Assume a kernel of corn, with a mass of 0.125 g, has a mass of only 0.106 g after popping. (a) What percentage of its mass did the kernel lose on popping? (b) ■ Popcorn is sold by the pound in the United States. Using 0.125 g as the average mass of a popcorn kernel, how many kernels are there in a pound of popcorn? (1 lb ϭ 453.6 g.) 95. ▲ The aluminum in a package containing 75 ft2 of kitchen foil weighs approximately 12 ounces. Aluminum has a density of 2.70 g/cm3. What is the approximate thickness of the aluminum foil in millimeters? (1 oz ϭ 28.4 g.) 96. ▲ The fluoridation of city water supplies has been practiced in the United States for several decades. It is done by continuously adding sodium fluoride to water as it comes from a reservoir. Assume you live in a medium-sized city of 150,000 people and that 660 L (170 gal ) of water is consumed per person per day. What mass of sodium fluoride (in kilograms) must be added to the water supply each year (365 days) to have the required fluoride concentration of 1 ppm (part per million)—that is, 1 kilogram of fluoride per 1 million kilograms of water? (Sodium fluoride is 45.0% fluoride, and water has a density of 1.00 g/cm3.) 97. ■ ▲ About two centuries ago, Benjamin Franklin showed that 1 teaspoon of oil would cover about 0.5 acre of still water. If you know that 1.0 ϫ 104 m2 ϭ 2.47 acres, and that there is approximately 5 cm3 in a teaspoon, what is the thickness of the layer of oil? How might this thickness be related to the sizes of molecules? 0.563 nm sodium chloride, NaCl ▲ More challenging ■ In General ChemistryNow Blue-numbered questions answered in Appendix O 56 Chapter 1 Matter and Measurement 98. ▲ Automobile batteries are filled with an aqueous solution of sulfuric acid. What is the mass of the acid (in grams) in 500. mL of the battery acid solution if the density of the solution is 1.285 g/cm3 and if the solution is 38.08% sulfuric acid by mass? 99. A piece of copper has a mass of 0.546 g. Show how to set up an expression to find the volume of this piece of copper in units of liters. (Copper density ϭ 8.96 g/cm3.) (See General ChemistryNow Screen 1.17 Tutorial 1, Using Numerical Information.) 100. Evaluate the value of x in the following expressions: (a) x ϭ [(9.345 ϫ 10Ϫ4)(6.23 ϫ 106)]3 (b) x ϭ 211.23 ϫ 10Ϫ2 2 14.5 ϫ 105 2 3 (c) x ϭ 2 11.23 ϫ 10Ϫ2 2 14.5 ϫ 105 2 Show the answers to the correct number of significant figures. (See General ChemistryNow CD-ROM or website Screen 1.17 Tutorial 4, Using Numerical Information.) 101. A 26-meter tall statue of Buddha in Tibet is covered with 279 kg of gold. If the gold was applied to a thickness of 0.0015 mm, what surface area is covered (in square meters)? (Gold density ϭ 19.3 g/cm3.) 102. At 25 ° C the density of water is 0.997 g/cm3, whereas the density of ice at Ϫ10 ° C is 0.917 g/cm3. (a) If a soft-drink can (volume ϭ 250. mL) is filled completely with pure water at 25 °C and then frozen at Ϫ10 ° C, what volume does the solid occupy? (b) Can the ice be contained within the can? 103. Suppose your bedroom is 18 ft long, 15 ft wide, and the distance from floor to ceiling is 8 ft, 6 in. You need to know the volume of the room in metric units for some scientific calculations. (a) What is the room’s volume in cubic meters? In liters? (b) What is the mass of air in the room in kilograms? In pounds? (Assume the density of air is 1.2 g/L and that the room is empty of furniture.) 104. ■ A spherical steel ball has a mass of 3.475 g and a diameter of 9.40 mm. What is the density of the steel? [The volume of a sphere ϭ (4/3)pr 3 where r ϭ radius.] 105. ▲ The substances listed below are clear liquids. You are asked to identify an unknown liquid that is known to be one of these liquids. You pipette a 3.50-mL sample into a beaker. The empty beaker had a mass of 12.20 g, and the beaker plus the liquid weighed 16.08 g. Substance Known Density at 25 ° C (g/cm3) Ethylene glycol 1.1088 (the major component of antifreeze) Water 0.9997 Ethanol 0.7893 (the alcohol in alcoholic beverages) Acetic acid 1.0492 (the active component of vinegar) Glycerol 1.2613 (a solvent, used in home care products) (a) Calculate the density and identify the unknown. ▲ More challenging ■ In General ChemistryNow (b) If you were able to measure the volume to only two significant figures (that is, 3.5 mL, not 3.50 mL), will the results be sufficiently accurate to identify the unknown? Explain. 106. ▲ You have an irregularly shaped chunk of an unknown metal. To identify it, you determine its density and then compare this value with known values that you look up in the chemistry library. The mass of the metal is 74.122 g. Because of the irregular shape, you measure the volume by submerging the metal in water in a graduated cylinder. When you do this, the water level in the cylinder rises from 28.2 mL to 36.7 mL. (a) What is the density of the metal? (Use the correct number of significant figures in your answer.) (b) The unknown is one of the seven metals listed below. Is it possible to identify the metal based on the density you have calculated? Explain. Metal Density (g/cm3) Metal Density (g/cm3) zinc 7.13 nickel 8.90 iron 7.87 copper 8.96 cadmium 8.65 silver 10.50 cobalt 8.90 107. ▲ A 7.50 ϫ 102-mL sample of an unknown gas has a mass of 0.9360 g. (a) What is the density of the gas? Express your answer in units of g/L. (b) Nine gases and their densities are listed below. Compare the experimentally determined density with these values. Can you determine the identity of the gas based on the experimentally determined density? (c) A more accurate measure of volume is made next, and the volume of this sample of gas is found to be 7.496 ϫ 102 mL. Using a more accurate density calculated using this value, can you now determine the identity of the gas? Gas Density (g/L) Gas Density (g/L) B2H6 1.2345 C2H4 1.2516 CH2O 1.3396 CO 1.2497 Dry air 1.2920 C2H6 1.3416 N2 1.2498 NO 1.2949 O2 1.4276 108. ▲ The density of a single, small crystal can be determined by the flotation method. This method is based on the idea that if a crystal and a liquid have precisely the same density, the crystal will hang suspended in the liquid. A crystal that is more dense will sink; one that is less dense will float. If the crystal neither sinks nor floats, then the density of the crystal equals the density of the liquid. Generally, mixtures of liquids are used to get the proper density. Chlorocarbons and bromocarbons (see Blue-numbered questions answered in Appendix O 57 Study Questions the list below) are often the liquids of choice. If the two liquids are similar, then volumes are usually additive and the density of the mixture relates directly to composition. (An example: 1.0 mL of CHCl3, d ϭ 1.4832 g/mL, and 1.0 mL of CCl4, d ϭ 1.5940 g/mL, when mixed, give 2.0 mL of a mixture with a density of 1.5386 g/mL. The density of the mixture is the average of the values of the two individual components.) The problem: A small crystal of silicon, germanium, tin, or lead (Group 4A in the periodic table) will hang suspended in a mixture made of 61.18% (by volume) CHBr3 and 38.82% (by volume) CHCl3. Calculate the density and identify the element. (You will have to look up the values of the density of the elements in a manual such as the The Handbook of Chemistry and Physics in the library or in a World Wide Web site such as WebElements at, Liquid Density (g/mL) Liquid Density (g/mL) CH2Cl2 1.3266 CH2Br2 2.4970 CHCl3 1.4832 CHBr3 2.8899 CCl4 1.5940 CBr4 109. ▲ Suppose you have a cylindrical glass tube with a thin capillary opening, and you wish to determine the diameter of the capillary. You can do this experimentally by weighing a piece of the tubing before and after filling a portion of the capillary with mercury. Using the following information, calculate the diameter of the capillary. Mass of tube before adding mercury ϭ 3.263 g Mass of tube after adding mercury ϭ 3.416 g Length of capillary filled with mercury ϭ 16.75 mm Density of mercury ϭ 13.546 g/cm3 Volume of cylindrical capillary filled with mercury ϭ (p)(radius)2(length) Do you need a live tutor for homework problems? Access vMentor at General ChemistryNow at for one-on-one tutoring from a chemistry expert 2.9609 ▲ More challenging ■ In General ChemistryNow Blue-numbered questions answered in Appendix O ...
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This note was uploaded on 02/18/2009 for the course CHEM 101 taught by Professor Williamson during the Fall '08 term at Texas A&M.

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