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Analysis Dimensional Learning Goal: To understand how to use dimensional analysis to solve problems. Dimensional analysis is a useful tool for solving problems that involve unit conversions. By carefully tracking units and conversion factors, you can avoid many of the errors commonly encountered in chemistry problems. Dimensional analysis can also help you work through problems when you are not sure where to begin. Dimensional analysis involves multiplying a given value by a conversion factor. This results in a value in the new units. The conversion factor can be the ratio of any two quantities, as long as the ratio is equal to one. For example, dimensional analysis could be used to determine the number of nickels in 3432.35 To begin, write down the starting value, 3432.35 . Next, convert dollars to cents. This conversion involves a simple conversion factor: . Note that the "dollar" unit should appear on the bottom of this conversion factor, since "dollars" appears on the top of the starting value. Finally, convert cents to nickels. This conversion also involves a simple conversion factor: This time "cents" should be on the bottom of the conversion factor, since it was on top of the previous conversion factor. Combining these expressions gives us . . . This can also be written as a fraction: Finally, cancel units. Since dollars are divided by dollars, and cents are divided by cents, both of these units can be canceled. Multiplying through gives the final result: You wash dishes for a chemistry laboratory to make extra money for laundry. You earn 6 each shift lasts 75 . Your laundry requires 6 . , and Part A How many shifts must you work if you wish to wash 10 of laundry? Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Find the number of quarters require d Hint not displaye d Hint A.3 Find the number of dollars require d Hint not displaye d Scientific Notation Scientific notation can be used to express very large or very small numbers more easily. For example, In MasteringChemistry, you may enter answers in scientific notation in the following format: 2.998*10^8 5.09*10^-6 Part A Convert the number 0.000127 to scientific notation, then enter the answer using the Masterin gChemis try format. Express your answer numerica lly. ANSWER: 1 . 2 7 1 0 4 C o r r e c t Entering scientific notation on a calculator It is important, especially for exams, that you be able to enter scientific notation in your calculator. All calculators are different, but look for a button such as E, EE, or EXP. If this button exists on your calculator, it means "times ten to the" and so you never actually type the number 10. For example, would be entered on a calculator as 2.998 E 8 For entering negative exponents, some calculators require the sign to be entered after the exponent. For example, might be entered on a calculator as 5.09 E -6 or as 5.09 E 6Also note that many calculators have a "change sign" button (+/-) that should be used for exponents instead of the button that indicates subtraction. When a number is written as , it means and therefore on a calculator would be entered as 1 E 3 Finally, you may need to put your calculator into scientific or exponential mode. This can often be done by pressing 2nd SCI on your calculator after computing an answer. Part B Use your calculato r to determin e the answer to the followin g calculati on: Express the answer in Masterin gChemis try format using three significa nt figures. Hint B.1 Buttons to use on a calculat or Hint not displaye d Express your answer numerica lly using three significa nt figures. ANSWER: Significant Figures in Calculations Learning Goal: To learn how to round an answer to the correct number of significant figures. When we report a measurement in science, we are careful to report only digits known with certainty, plus a final digit that is recognized to be uncertain. We call this set of digits significant figures. Often, these numbers are used in calculations. When we use a calculator, the calculator does not automatically account for significant figures, so we have to do so ourselves. There are some basic rules for handling significant figures in calculations as shown in the figure outlined to the right. Multiplication and division and When multiplying or dividing, the final answer is rounded to the same number of significant figures as the measured number with the fewest significant figures. Addition and subtraction When adding or subtracting, the final answer is rounded to the same number of digits to the right of the decimal point as the measured number with the fewest digits to the right of the decimal point. Part A Compute . Round the answer appropri ately. Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Determi ne the number of significa nt figures in the final answer Hint not displaye d Express your answer numerica lly using the proper number of significa nt figures. ANSWER: The Scientific Method Learning Goal: To understand how the scientific method can be used to search for explanations of nature. The scientific method is a procedure used to search for explanations of nature. The scientific method consists of making observations, formulating hypotheses, designing and carrying out experiments, and repeating this cycle. Observations can be either quantitative or qualitative. Quantitative observations are measurements consisting of both numbers and units, such as the observation that ice melts at . In contrast, qualitative observations are observations that do not rely on numbers or units, such as the observation that water is clear. A hypothesis is a tentative explanation of the observations. The hypothesis is not necessarily correct, but it puts the scientist's understanding of the observations into a form that can be tested through experimentation. Experiments are then performed to test the validity of the hypothesis. Experiments are observations preferably made under conditions in which the variable of interest is clearly distinguishable from any others. If the experiment shows that the hypothesis is incorrect, the hypothesis can be modified, and further experiments can be carried out to test the modified hypothesis. This cycle is repeated, continually refining the hypothesis. If a large set of observations follow a reproducible pattern, this pattern can be summarized in a lawa verbal or mathematical generalization of a phenomenon. For example, over the years people observed that every morning the sun rises in the east, and every night the sun sets in the west. These observations can be described in a law stating, "The sun always rises in the east and sets in the west." After a great deal of refinement, a hypothesis can lead to a theory. A theory is an explanation of why something happens. For example, Newton's theory of gravitation explains why objects tend to fall toward the Earth (as well as explaining the interactions between the Earth and the other planets, etc). However, theories can still be further refined or even replaced. Einstein's theory of general relativity was able to better explain certain astronomical observations related to gravity, and therefore it replaced Newton's theory of gravitation (although Newton's theory still holds true under most everyday conditions). Similarly, the geocentric theory (that the Earth is the center of the universe) was replaced by the heliocentric theory (that the Earth revolves around the sun) based on further observations and testing of predictions. Note that a scientific theory is not the same as the popular definition of a theorynamely, a "guess" or "speculation." Instead, a theory is an explanation that can hold up against repeated experimentation. It may not be perfect, but it is the best explanation possible based on available evidence. In the course of a conversation, you observe that three of your friends like horror movies. Horror movies happen to be your favorite type of movie as well. You also know that all of these friends were born in the same week that you were, even in the same year. An astrology-loving friend hypothesizes that people born in that week like horror movies more than other genres of movies. You decide to use the scientific method to test this hypothesis. Part A What should you do next? ANSWER: Com e up with a theo ry to expl ain why mov ie pref eren ce is relat ed to birth wee k. Perf orm expe rime nts to test your hypo thesi s. Prop ose seve ral alter nativ e hypo thes es. Refi ne Using Conversion Factors Chemistry is an experimental science in which a consistent system of measurements is used. Scientists throughout the world use the International System of Units (Systme International or SI), which is based on the metric system. However, the metric system in not used on an everyday basis in the United States. It is often necessary to convert between the English system, which is used in the United States, and the SI system. One problem with the SI system is the size of the units sometimes turns out to be inconveniently large or small. Therefore it is often necessary to make conversions within the SI system. For this reason, SI units are mte a odified through the use of prefixes when they refer to either larger or smaller quantities. amount of money to spend on gas. nded daily amount of dietary fat (based on a 2000 calories a day diet) is about 67 (0.067 ). Chemical Properties and Physical Properties Substances have both chemical properties and physical properties. Part A Which of the followin g propertie s of iron are chemical propertie s? Hint A.1 Chemic al properti es definitio n Hint not displaye d Hint A.2 Physical properti es definitio n Hint not displaye d Check all that apply. ANSWER: It is shin y. It rusts in Convert between Metric and English Units Keeping track of units during conversions can get confusing. Therefore, a good strategy is to use dimensional analysis. In dimensional analysis, unit factors are used to cancel out unwanted units and leave the desired units. Consider the following example: How many football fields could fit on a 1 relevant conversion factors: stretch of land? Here are the Then Part A The speed of light is . How fast is this in miles per hour ( )? Use the conversi on . Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Find a factor to convert meters to miles Hint not displaye d Hint A.3 Find a factor to convert seconds to minutes Counting Significant Figures When we report a measurement in science, we are careful to report only digits known with certainty, plus a final digit that is recognized to be uncertain. It is important to know how to count significant figures correctly, including in numbers containing zeros. Number Signific ant figures Explanat ion 23.56 4 All nonzero digits are significa nt. 23.560 5 Trailing zeros are significa nt when a decimal point is shown. 1000 1 Trailing zeros are not consider ed significa nt when no decimal point is shown. 1000. 4 Showing the decimal point indicates that the precedin g zeros are significa nt. 500.01 5 Zeros Density of a Regular Solid By definition, Typical units of density are or . Part A The density of iron is 7.87 . What is the mass of a 3.00 4. 00 1. 00 block of iron? Hint A.1 Calculat e the volume of iron Hint not displaye d Express your answer in grams. ANSWER: m a s s = 9 4 . 4 C o r r e c Measurement Consider two graduated cylinders in units of milliliters. that measure Part A How many milliliter s of liquid does the smaller graduate d cylinder contain? Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Determi ne the differen ce between marking s on the smaller cylinder Hint not displaye d Hint A.3 How to determi ne the number of significa nt figures Hint not Rounding It will be helpful to remember two guidelines about significant figures. 1. 2. After multiplication or division, the answer cannot have more significant figures than any of the original numbers. After addition or subtraction, the answer cannot have more digits to the right of the decimal point than any of the original numbers. After determining the appropriate number of significant figures, round off your final answer. 1. 2. 3. If the first digit you drop is greater than 5, add 1 to the last digit you keep. You are rounding up. If the first digit you drop is less than 5, do nothing to the digits you keep. You are rounding down. If the digit you drop is 5, and there are no following digits, round down. If there are digits following the 5, round up. Part A A bakery measure d the mass of a whole cake as 0.870 .A customer bought one slice of the cake and measure d the mass of the slice as . 0.1151 What is the mass of the remainin g cake, without the one slice? Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Calculat e the unroun ded mass of remaini ng cake Hint not displaye d Classification of Matter All matter can be classified as either a mixture or a pure substance. Mixtures can be further classified as heterogeneous or homogeneous. Pure substances can be further classified as elements or compounds. Part A In the followin g images, the white spheres represent hydroge n atoms and the red spheres represent oxygen atoms. Classify each sample as an element, a compoun d, or a mixture. Hint A.1 Define element s, compou nds, and mixture s Hint not displaye d Drag each item to the appropri ate bin. ANSWER: View Correct Density Determination The accompanying sketches suggest four observations made on a small block of plastic material. Express the density numerica lly in grams per cubic centimet er. ANSWER: 1 . 1 6 C o r r e c t Since the block is 40.0 submerg ed, the density of the block is 40.0 of the density of the liquid. Alternate data Laboratory measurements always contain some error. Depending on the type of equipment used, that error can be quite substantial. One way to validate our density value from Part B is to do a completely different experiment to measure of the density of the block, then compare the two measurements. Part C Based on the first and fourth sketches , what is the density of the block? Hint C.1 How to approac h the problem Hint not displaye d Hint C.2 Calculat e the mass of the displace d water Hint not displaye d Intensive Properties and Extensive Properties Physical properties can be classified as either intensive properties or extensive properties. An intensive property is independent of the amount of the sample. An extensive property varies with the amount of the sample. Part A Classify the followin g as intensive or extensiv e propertie s of gold. Hint A.1 Analyze color, mass, volume, and melting point Hint not displaye d Hint A.2 Analyze density Hint not displaye d Drag the appropri ate items to their respectiv e bins. ANSWER: View Correct Mixtures, Solutions, Pure Chemical Substances, Compounds, and Elements Matter can be classified into several categories. All substances are either pure chemical substances or mixtures. Mixtures can be separated into their component parts through physical means, whereas pure chemical substances require chemical or nuclear reactions to separate them. Pure chemical substances can be further divided into elements or compounds. Mixtures can be either heterogeneous or homogeneous. Homogeneous mixtures are also called solutions. Part A A sample of tap water consists of water, fluoride ions, calcium ions, and several other dissolve d ions. The ions are evenly distribut ed througho ut the water. Which term or terms could be used to describe this sample of tap water? Hint A.1 Define the terms Hint not displaye d Check all that apply. ANSWER: mixt ure Physical Change or Chemical Reaction A chemical reaction (also known as a chemical change) produces substances that are chemically different from the starting materials. An example of a chemical reaction is the formation of water from hydrogen and oxygen gas. In a physical change, a substance changes its physical appearance but not its chemical identity. An example of physical change is the formation of liquid water from solid water, a familiar process called melting. Physically, liquid water looks very different from solid water (ice) but the chemical identity, water, is the same for both. ANSWER: cond ensa tion of etha nol evap orati on of etha nol form ation of etha nol from gluc ose by yeas t form ation of a carb on depo sit insid e the flask burn ing natu ral gas form ation of carb on diox ide gas from Temperature Scales Temperature can be measured using three different scales: Celsius, Fahrenheit, and Kelvin. The Celsius degree ( ) is the common unit of temperature, but most people in the United States use the Fahrenheit degree ( ). The kelvin ( ) is used in science. You can convert from one temperature scale to another by using the conversions The boiling and freezing points of water are constants. Water boils at 212 , 100 , or 373.15 . Although these temperatures are equivalent, the scales used to measure them are different. Water freezes at 32 , 0 , or 273.15 . Note that these conversions are based on the fact that water will boil or freeze at a certain temperature regardless of the scale used to measure it. The average normal human body temperature is 98.6 . Part A What is normal body temperat ure in degrees Celsius? Hint A.1 Identify the best formula Hint not displaye d Hint A.2 Determi ne the order of operatio ns for the equatio n Hint not displaye d Express your answer numerica lly in degrees Celsius. ANSWER: 3 7 . 0 C o Density Calculations Density is an intensive physical property that relates the mass of an object to its volume. Density, which is simply the mass of an object divided by its volume, is expressed in the SI derived unit for a liquid or for a solid. Most substances expand or contract when heated or cooled, so the density values for a substance are temperature dependent.. Part A A particula r brand of gasoline has a density of 0.737 at 25 . How many grams of this gasoline would fill a 12.1 tank? Hint A.1 How to approac h the problem Hint not displaye d Hint A.2 Convert gallons to millilite rs Hint not displaye d Hint A.3 Find an equatio n for mass Measurements: Accuracy and Precision Accuracy describes the correlation between the measured value and the accepted value. The accuracy of a measurement, or set of measurements, can be expressed in terms of error: The larger the error is, the less accurate is the measurement. Precision describes the reproducibility of a measurement. To evaluate the precision of a set of measurements, start by finding the deviation of each individual measurement in the set from the average of all the measurements in the set: Note that deviation is always positive because the vertical lines in the formula represent absolute value. The average of all the deviations in the set is called the average deviation. The larger the average deviation is, the less precise is the data set. In a laboratory experiment, the density of a concentrated sugar solution was determined by measuring the volume of the solution and corresponding mass. Three students each made a set of measurements using a different balance and used the results to calculate density. Their results are summarized here. Density ( Trial Set 1 Set 2 Set 3 a 1.380 1.347 1.418 b 1.400 1.479 1.499 c 1.374 1.401 1.301 Part A ) The accepted density of the sugar solution is 1.388 . Based on the average value of each data set, rank the data sets in decreasing order of accuracy. Hint A.1 Choose the quantity that estimate s accurac y Hint not displaye d Hint A.2 Evaluat e the accurac y of Set 1 Hint not displaye d Problem 1.6 Part A What is the length of the pencil in the figure if the scale reads in centimet ers? ANSWER: = 7 . 5 C o r r e c t Problem 1.8 Part A How many significa nt figures should be reported for the volume of the metal bar shown below ? Express your answer as an integer. ANSWER: 2 C o r r e c t Part B If the Problem 1.14 Give the chemical symbol or name for each of the following elements, as appropriate. Part A carbon Express your answer as a chemical symbol. Register to View Answero r r e c t Part B nitrogen Express your answer as a chemical symbol. Register to View Answero r r e c t Problem 1.24 Use appropriate metric prefixes to write the following measurements without use of exponents. Part A ANSWER: 6.35 63.5 6.35 63.5 635 63.5 C o r r e c t Part B ANSWER: Problem 1.36 Indicate the number of significant figures in each of the following measured quantities. Part A 3.774 Express your answer as an integer. ANSWER: 4 C o r r e c t Part B 205 Express your answer as an integer. ANSWER: 3 C o r r e c t Problem 1.64 The U.S. quarter has a mass of 5.67 and is approximately 1.55 thick. Part A How many quarters would have to be stacked to reach 575 , the height of the Washing ton Monume nt? Express your answer using three significa nt figures. ANSWER: 1 . 1 3 1 0 5 C o r r e c t Part B Problem 1.10 Draw a logic map indicating the steps you would take to convert miles per hour to kilometers per second. Part A Write down the conversi on factor for each step, as done in the diagram on page 26 in the textbook. ANSWER: My Answer: Problem 1.22 A beaker contains a clear, colorless liquid. Part A If it is water, how could you determin e whether it containe d dissolve d table salt? Do not taste it! ANSWER: My Answer: Problem 1.81 In 2006, Professor Galen Suppes, from the University of Missouri-Columbia, was awarded a Presidential Green Challenge Award for his system of converting glycerin, , a by-product of biodiesel production, to propylene glycol, . Propylene glycol produced in this way will be cheap enough to replace the more toxic ethylene glycol that is the primary ingredient in automobile antifreeze. Part A If 50.0 of propylen e glycol has a mass of 51.80 , what is its density? Express your answer using three significa nt figures. ANSWER: = 1 . 0 4 C o r r e c t Part B To obtain the same antifreez e Problem 1.74 A coin dealer offers to sell you an ancient gold coin that is 2.7 in diameter and 2.5 in thickness. Part A The density of gold is 19.3 . How much should the coin weigh if it is pure gold? Express your answer using two significa nt figures. ANSWER: = 2 8 C o r r e c t Part B If gold sells for $640 per troy ounce, how Problem 1.65 In the United States, water used for irrigation is measured in acre-feet. An acre-foot of water covers an acre to a depth of exactly 1 . An acre is 4840 households for 1.00 yr. . An acre-foot is enough water to supply two typical Part A If desalinat ed water costs 2150 per acrefoot, how much does desalinat ed water cost per liter? ANSWER: 1 . 7 4 1 0 3 C o r r e c t Part B How much would it cost one househol d per day if it were Problem 1.49 The density of air at ordinary atmospheric pressure and 25 Part A What is the mass, in kilogram s, of the air in a room that measures 12.0 12.5 10 ? Express your answer using two significa nt figures. ANSWER: is 1.19 . = 5 1 C o r r e c t Problem 1.47 Part A How many liters of wine can be held in a wine barrel whose capacity is 33 ? Express your answer using two significa nt figures. ANSWER: = 1 . 2 1 0 2 C o r r e c t Part B The Problem 1.45 Perform the following conversions. Part A 6.00 days to , Express your answer using three significa nt figures. ANSWER: 6 . 0 0 d a y s = 5 . 1 8 1 0 5 C o r r e c t Part B Problem 1.43 Perform the following conversions. Part A 8.1102 to Express your answer using two significa nt figures. ANSWER: 8 . 1 1 0 2 = 8 1 C o r r e c t Part B 5.0108 to Express your Problem 1.40 Carry out the following operations, and express the answer with the appropriate number of significant figures. Part A , ANSWER: = 2 . 9 1 0 3 C o r r e c t Part B ANSWER: = 8 . 3 7 7 1 0 7 Problem 1.32 A cylindrical rod formed from silicon is 18.2 2.33 . long and has a mass of 2.20 . The density of silicon is Part A What is the diameter of the cylinder? (The volume of a cylinder is given by , where is the radius, and is its length.) ANSWER: = 8 . 1 3 C o r r e c t Problem 1.26 Part A The temperat ure on a warm summer day is 83 . What is the temperat ure in ? Express your answer using two significa nt figures. ANSWER: 8 3 = 2 8 C o r r e c t Part B Many scientific Problem 1.80 You are assigned the task of separating a desired granular material, with a density of 3.62 , from an undesired granular material that has a density of 2.04 . You want to do this by shaking the mixture in a liquid in which the heavier material will fall to the bottom and the lighter material will float. A solid will float on any liquid that is more dense. Using the Internet or a handbook of chemistry, find the densities of the following substances: Part A carbon tetrachlo ride ANSWER: = 1 . 5 9 C o r r e c t Part B hexane ANSWER: = 0 . 6 6 0 C o r r e c t Problem 1.34 Part A Indicate which of the followin g are exact numbers. Drag the items into the appropri ate bins. ANSWER: View Correct Problem 1.20 A match is lit and held under a cold piece of metal. The following observations are made. (a) The match burns. (b) The metal gets warmer. (c) Water condenses on the metal. (d) Soot (carbon) is deposited on the metal. Part A Which of these occurren ces are due to physical changes, and which are due to chemical changes? Drag the items into the appropri ate bins. ANSWER: View Correct ... View Full Document

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