3.3. Key Questions. What happens when a measurement is multiplied by a

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1 LESSON 3.3 Key Objectives EXPLAIN what happens when a measurement is multiplied by a conversion factor DESCRIBE the kinds of problems that can be easily solved using dimensional analysis. Additional Resources Reading and Study Workbook, Lesson 3.3 Core Teaching Resources, Lesson 3.3 Review Engage CHEMISTRY & YOU Have students examine the photograph that opens the section. Ask Has anyone ever noticed a chart or table in a bank or in the newspaper relating the values of foreign currency to the U.S. dollar? (yes or no) that these are tables are called currency conversion tables. These are the daily values that allow people to relate one currency to another. Ask How could you know for certain which amount of money is worth more 75 euros or 75 British pounds? (Convert these values to a familiar currency U.S. dollars.) Activate Prior Knowledge Provide various types of measuring devices to the class. Tell students to select one device and record the length of their shoe. Then ask them to create a chart and record the length of their shoe in inches, feet, yards, millimeters, centimeters, and meters. As a class, discuss how students determined the measurements. Begin a list of conversions on the word wall. National Science Education Standards A Solving Conversion Problems Key Questions What happens when a measurement is multiplied by a conversion factor? What kinds of problems can you solve using dimensional analysis? Vocabulary 84 Focus on ELL CHEMISTRY & YOU Q: How can you convert U.S. dollars to euros? Perhaps you have traveled to another country or are planning to do so. If so, you know (or will soon discover) that different countries have different currencies. As a tourist, exchanging money is an important part of having a good trip. After all, you often must use cash to pay for meals, transportation, and souvenirs. Because each country s currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is essential. Conversion problems are readily solved by a problem-solving approach called dimensional analysis. Conversion Factors What happens when a measurement is multiplied by a conversion factor? If you think about any number of everyday situations, you will realize that a quantity can usually be expressed in several different ways. For example, consider the monetary amount $1. 1 dollar 4 quarters 10 dimes 20 nickels 100 pennies These are all expressions, or measurements, of the same amount of money. The same thing is true of scientific quantities. For example, consider a distance that measures exactly 1 meter. 1 meter 10 decimeters 100 centimeters 1000 millimeters These are different ways to express the same length. Whenever two measurements are equivalent, a ratio of the two meaurements will equal 1, or unity. For example, you can divide both sides of the equation 1 m 100 cm by 1 m or by 100 cm. 1m 100 cm 1m 1 m 1 or 1 m 100 cm 100 cm 100 cm 1 The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors. A conversion factor is a ratio of equivalent measurements. The measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom). The conversion factors shown above are read one hundred centimeters per meter and one meter per hundred centimeters. 1 CONTENT AND LANGUAGE Present academic vocabulary for this lesson, such as analyze, calculate, and evaluate. Pronounce the words and have students repeat them. Provide explanations, examples, and visuals of each word so students are aware of what tasks lie ahead when they encounter these words. 2 FRONTLOAD THE LESSON Preview the conversion factors and Figure 3.12 in the text, and point out that conversion factors are fractions that contain both numbers and units of measurement. Briefly review the basics of working with fractions, including terminology and equivalent fractions. 84 Chapter 3 Lesson 3 3 COMPREHENSIBLE INPUT When writing problems on the board, assign a different color to each unique unit of measurement in the problem. This will assist students in understanding that a unit in the numerator can only be cancelled by an equivalent unit in the denominator.

2 1 meter Smaller number Larger number 1m = 10 1 m 100 cm Figure 3.12 illustrates another way to look at the relationships in a conversion factor. Notice that the smaller number is part of the measurement with the larger unit. That is, a meter is physically larger than a centimeter. The larger number is part of the measurement with the smaller unit. Conversion factors are useful in solving problems in which a given measurement must be expressed in some other unit of measure. When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same. For example, even though the numbers in the measurements 1 g and 10 dg (decigrams) differ, both measurements represent the same mass. In addition, conversion factors within a system of measurement are defined quantities or exact quantities. Therefore, they have an unlimited number of significant figures and do not affect the rounding of a calculated answer. Here are some additional examples of pairs of conversion factors written from equivalent measurements. The relationship between grams and kilograms is 1000 g 1 kg. The conversion factors are 1000 g 1 kg and 1 kg 1000 g Figure 3.13 shows a scale that can be used to measure mass in grams or kilograms. If you read the scale in terms of grams, you can convert the mass to kilograms by multiplying by the conversion factor 1 kg/1000 g. The relationship between nanometers and meters is given by the equation 10 9 nm 1 m. The possible conversion factors are 10 9 nm 1 m and 1 m 10 9 nm Common volumetric units used in chemistry include the liter and the microliter. The relationship 1 L 10 6 L yields the following conversion factors: 1 L 10 6 L and 10 6 L 1 L Based on what you have learned about metric prefixes, you should easily be able to write conversion factors that relate equivalent metric quantities. 100 centimeters Larger unit Monetary Exchange Rates Smaller unit Figure 3.12 Conversion Factor The two parts of a conversion factor, the numerator and the denominator, are equal. See conversion factors animated online. KINETIC A R T Figure 3.13 Measuring Mass This scale is calibrated to measure mass to the nearest 20 g. Interpret Photos What is the scale showing in grams? In kilograms? K Scientific Measurement 85 The conversion of chemical units is similar to the exchange of currency. Americans who travel outside the United States must exchange U.S. dollars for foreign currency at a given rate of exchange. These exchange rates vary from day to day. The daily exchange rates affect all international monetary transactions. Each time one type of money is exchanged for another, the current exchange rate serves as a conversion factor. International currency traders keep track of exchange rates 24 hours a day through a linked computer network. Foundations for Reading BUILD VOCABULARY Have students write definitions of conversion factor and dimensional analysis in their own words. (Sample answers: a ratio of equivalent measurements used to convert a quantity from one unit to another; a technique of problem-solving that uses the units that are part of a measurement to solve the problem.) Conversion Factors USE VISUALS Have students inspect Figure Emphasize that a conversion factor relates two equivalent measurements. Ask What two parts does every measurement have? (a number and a unit) Point out that if this is so, then every conversion factor must contain two numbers and two units so that one number and its unit equal another number and its unit. Explore Class Activity PURPOSE Students will use dimensional analysis to convert common units. MATERIALS copies of a recipe, lists of equivalents and conversions among the following measurements: teaspoon, tablespoon,1/4 cup,1/2 cup, and 1 cup (These lists are found in most cookbooks.) PROCEDURE Distribute the recipe and the conversion list to pairs of students. that they must rewrite the recipe so that it can feed six times the number of serving sizes suggested by the recipe. Point out that it would be tedious to have to measure out a particular ingredient (pick out one) in teaspoons or tablespoons six times, so students must rewrite the recipe in appropriately larger units. After students have rewritten the recipe, have student pairs exchange and compare recipes. EXPECTED OUTCOME Students should use the conversion lists to write simple conversion factors, such as 3 teaspoons/1 tablespoon, and then rewrite the recipe using larger measurements. Answers FIGURE 3.13 The scale is showing 220 grams, or 0.22 kg. Scientific Measurement 85 LESSON 3.3

3 LESSON 3.3 Dimensional Analysis START A CONVERSATION that dimensional analysis is an extremely powerful problem-solving tool. Learning this method requires extra effort on the part of students. They must often use multiple conversion factors. The extra effort can be justified because the proper manipulation of the units assures accurate manipulation of the numbers. Emphasize that students should use dimensional analysis as a tool for solving all of the problems they encounter in chemistry. Their first question about any quantity should be, What are the units of this quantity? By comparing the units of various quantities in a problem, students can discover whether they need to perform any unit conversions before proceeding. Mary worked 43,200 seconds this week. How many hours did Mary work this week? (12 hours) Sample Problem 3.9 The first conversion factor is based on 1 h 60 min. The unit hours must be in the denominator so that the known unit will cancel. The second conversion factor is based on 1 min 60 s. The unit minutes must be in the denominator so that the desired units (seconds) will be in your answer. Dimensional Analysis Using Dimensional Analysis How many seconds are in a workday that lasts exactly eight hours? Analyze List the knowns and the unknown. To convert time in hours to time in seconds, you ll need two conversion factors. First you must convert hours to minutes: h min. Then you must convert minutes to seconds: min s. Identify the proper conversion factors based on the relationships 1 h 60 min and 1 min 60 s. What kinds of problems can you solve using dimensional analysis? Some problems are best solved using algebra. For example, converting a kelvin temperature to Celsius can be done by using the equation C K 273. Many problems in chemistry are conveniently solved using dimensional analysis. Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements. The best way to explain this technique is to use it to solve an everyday situation, as in Sample Problem 3.9. As you read Sample Problem 3.10, you might see how the same problem could be solved algebraically but is more easily solved using dimensional analysis. In either case, you should choose the problem-solving method that works best for you. Try to be flexible in your approach to problem solving, as no single method is best for solving every type of problem. 60 min 1 h 60 s 1 min time worked 8 h 1 hour 60 min 1 minute 60 s seconds worked? s Before you do the actual arithmetic, it s a good idea to make sure that the units cancel and that the numerator and denominator of each conversion factor are equal to each other. Multiply the time worked by the conversion factors. 60 min 8 h 60 s 28,000 s s 1 h 1 min Evaluate Does the result make sense? The answer has the desired unit (s). Since the second is a small unit of time, you should expect a large number of seconds in 8 hours. The answer is exact since the given measurement and each of the conversion factors is exact. 36. How many minutes are there in exactly one week? 37. How many seconds are in an exactly 40-hour work week? Chapter 3 Lesson 3 Foundations for Math CONVERSION FACTORS Point out that a conversion factor is a ratio of equivalent measurements that is, the numerator should contain both a number and a unit, and the denominator should contain both a number and a unit. For example, the relationship 60 minutes = 1 hour yields the following conversion factors: 60 minutes 1 hour and 1 hour 60 minutes Knowing the unit of the given quantity and the desired unit of the answer dictates which conversion factor to use. To convert from hours to minutes, the denominator of the conversion factor must contain hours in order for the known unit to cancel, so you would use 60 min/1 h. Note that Sample Problem 3.9 involves two conversions (from hours to minutes, and from minutes to seconds), each requiring its own conversion factor. 60 minutes In Sample Problem 3.9, note that two such expressions are used, and 1 hour 60 seconds. This reflects the fact that for each equivalent relationship there are two 1 minute possible conversion factors that can be written.

4 M CHEM C T U T O R Sample Problem 3.10 Using Dimensional Analysis The directions for an experiment ask each student to measure 1.84 g of copper (Cu) wire. The only copper wire available is a spool with a mass of 50.0 g. How many students can do the experiment before the copper runs out? From the known mass of copper, use the appropriate conversion factor to calculate the number of students who can do the experiment. The desired conversion is mass of copper number of students. The experiment calls for 1.84 grams of copper per student. Based on this relationship, you can write two conversion factors. Because the desired unit for the answer is students, use the second conversion factor. Multiply the mass of copper by the conversion factor g Cu 1 student mass of copper available 50.0 g Cu Each student needs 1.84 grams of copper. number of students? 1 student and 1.84 g Cu Note that because students cannot be fractional, the answer is rounded down to a whole number. 1 student 50.0 g Cu students 27 students 1.84 g Cu ADDRESS MISCONCEPTIONS Students may think that because a conversion factor equals unity, it does not matter whether the conversion factor or its reciprocal is used in a calculation. Remind students that a given pair of equivalent measurements yields two different conversion factors, only one of which can be used to obtain the correct conversion. An experiment requires that each student use 8 grams of baking soda. If a full container of baking soda holds 340 grams, how many students can do the experiment? (42 students) LESSON 3.3 Evaluate Does the result make sense? The unit of the answer (students) is the one desired. You can make an approximate calculation using the following conversion factor. 1 student 2 g Cu Multiplying the above conversion factor by 50 g Cu gives the approximate answer of 25 students, which is close to the calculated answer. 38. An experiment requires that each student use an 8.5-cm length of magnesium ribbon. How many students can do the experiment if there is a 570-cm length of magnesium ribbon available? Here s a tip: The equalities needed to write a particular conversion factor may be given in the problem. In other cases, you ll need to know or look up the necessary equalities. 39. A 1.00-degree increase on the Celsius scale is equivalent to a 1.80-degree increase on the Fahrenheit scale. If a temperature increases by 48.0 C, what is the corresponding temperature increase in F? 40. An atom of gold has a mass of g. How many atoms of gold are in 5.00 g of gold? Scientific Measurement 87 Foundations for Math ROUNDING to students that some problems require a more logical rounding technique than using significant figures or decimal places. For example, money is typically rounded to the hundredths place because our money system includes pennies. The number of buses needed to transport a certain number of students on a field trip would be rounded UP to the next whole bus to make sure all students are accommodated. Have students think of other examples where rounding would not follow the conventional rules. In Sample Problem 3.10, logically there is no such thing as student, so the result of the calculation is rounded to a whole number. that the number is rounded DOWN (unlike in the bus example above) because there is not enough copper for another student to do the experiment. Answers min s students ºF atoms Scientific Measurement 87

5 LESSON 3.3 START A CONVERSATION that measurements are often made using one unit and then converted into a related unit before being used in calculations. For example, students might measure volume in liters or milliliters in the laboratory, but express it as cubic centimeters in a calculation. to the students that conversions are done using conversion factors. Emphasize that these conversion factors are ratios of equivalent physical quantities, such as 1 ml/1 cm 3. MAKE A CONNECTION Reassure students that they are more familiar with conversions than they may realize. Point out examples of everyday conversions, such as converting money from cents to dollars and converting time from minutes to hours. Start out by giving them practice with everyday examples. Ask A chicken needs to be cooked 20 minutes for each pound it weighs. How long should the chicken be cooked if it weighs 4.5 pounds? (4.5 lb 20 min/lb = 90 min; 90 min 1 h/60 min = 1.5 h. Most students will automatically relate 90 minutes to 1.5 hours. This may help them become comfortable with the process.) If students are having difficulty with conversion factors, provide a more tactile environment for students to discover these relationships. Divide the class in half and have each group challenge the other to write the conversion factor given two related units. Remind them that each conversion factor can appear in two forms depending on which value they put in the numerator. CHEMISTRY & YOU Sample answer (the conversion factor will vary with the exchange rate): $50 ( 1/$1.360) = Make the following conversions. a L to cubic centimeters ( cm 3 ) b mg to grams ( g) c m to micrometers ( μm) d cm to millimeters ( mm) CHEMISTRY & YOU Q: Look up the exchange rate between U.S. dollars and euros on the Internet. Write a conversion factor that allows you to convert from U.S. dollars to euros. How many euros could you buy with $50? Sample Problem 3.11 Use the relationship 1 g 10 dg to write the correct conversion factor. Multiply the known mass by the conversion factor. 88 Simple Unit Conversions In chemistry, as in everyday life, you often need to express a measurement in a unit different from the one given or measured initially. Dimensional analysis is a powerful tool for solving conversion problems in which a measurement with one unit is changed to an equivalent measurement with another unit. Sample Problems 3.11 and 3.12 walk you through how to solve simple conversion problems using dimensional analysis. Converting Between Metric Units Express 750 dg in grams. (Refer to Table 3.2 if you need to refresh your memory of metric prefixes.) The desired conversion is decigrams grams. Multiply the given mass by the proper conversion factor. 41. Using tables from this chapter, convert the following: a km to meters b. 4.6 mg to grams c g to centigrams 1 g 10 dg 1 g 750 dg 75 g 10 dg mass 750 dg 1 g 10 dg Evaluate Does the result make sense? Because the unit gram represents a larger mass than the unit decigram, it makes sense that the number of grams is less than the given number of decigrams. The answer has the correct unit (dg) and the correct number of significant figures. Foundations for Math 42. Convert the following: 3 a. 15 cm to liters b g to kilograms c. 6.7 s to milliseconds d g to micrograms mass? g Note that the known unit (dg) is in the denominator and the unknown unit (g) is in the numerator. MORE ON METRIC CONVERSIONS to students that it is possible to convert through the basic metric unit in order to quickly go from very large units to very small units or vice versa. This requires more than one conversion factor in the dimensional analysis calculation, each of which involves the basic metric unit. Demonstrate this example: How many centigrams are there in 0.8 kilograms? 0.8 kg g 100 cg ,000 cg 1 kg 1 g Point out that the desired unit (cg) is placed in the numerator of the last conversion factor. In Sample Problem 3.11, point out that the initial unit is placed in the denominator of the conversion factor and the desired unit is placed in the numerator. 88 Chapter 3 Lesson 3

6 M CHEM C T U T O R Sample Problem 3.12 Using Density as a Conversion Factor What is the volume of a pure silver coin that has a mass of 14 g? The density of silver (Ag) is 10.5 g/cm 3. You need to convert the mass of the coin into a corresponding volume. The density gives you the following relationship between volume and mass: 1 cm 3 Ag 10.5 g Ag. Multiply the given mass by the proper conversion factor to yield an answer in cm 3. Use the relationship 1 cm 3 Ag 10.5 g Ag to write the correct conversion factor. Multiply the mass of the coin by the conversion factor. 43. Use dimensional analysis and the given densities to make the following conversions: 3 a g of boron to cm of boron. The density of boron is 2.34 g/cm 3. 3 b g of mercury to cm of mercury. The density of mercury is 13.5 g/cm 3. 1 cm 3 Ag 10.5 g Ag mass 14 g density of silver 10.5 g/cm 3 1 cm 14 g Ag Ag cm 3 Ag 10.5 g Ag Evaluate Does the result make sense? Because a mass of 10.5 g of silver has a volume of 1 cm 3, it makes sense that 14.0 g of silver should have a volume slightly larger than 1 cm 3. The answer has two significant figures because the given mass has two significant figures. volume of coin cm 3 Notice that the known unit (g) is in the denominator and the unknown unit (cm 3 ) is in the numerator. 45. What is the mass, in grams, of a sample of cough syrup that has a volume of 50.0 cm 3? The density of cough syrup is g/cm 3. Explore Class Activity PURPOSE Students will use dimensional analysis to convert between English and metric units. MATERIALS Internet access or copies of media guides containing vital statistics, such as heights and weights, of players on a sports team (These guides are available from local sports franchises.) PROCEDURE Distribute the media guides and assign each group of students a set of players. Ask the group to convert heights and weights into heights and masses expressed in meters and kilograms, respectively. Have students document their approach, including dimensional analysis expressions, conversion factors, and calculations. EXPECTED OUTCOMES Students should use conversion factors, such as 2.54 cm/1 inch and 454 g/1 lb, to convert their measurements. The density of mineral oil is g/cm 3. What is the mass in grams of a sample of mineral oil that has a volume of 2.50 cm 3? (2.09 grams) LESSON Rework the preceding problems by applying the following equation: mass Density volume Density can be used to write two conversion factors. To figure out which one you need, consider the units of your given quantity and the units needed in your answer. Scientific Measurement 89 Differentiated Instruction ELL ENGLISH LANGUAGE LEARNERS Provide as much class time as possible for students to work on problem assignments in cooperative learning groups of varying proficiencies. Encourage students with high proficiency to assist students with lower proficiencies by reading the problem out loud. LPR LESS PROFICIENT READERS Students may benefit from a reminder that certain key words and phrases in each word problem indicate the unknown quantity and its units. Some of these phrases are: how much, what is, how long, determine, and find. L3 ADVANCED STUDENTS Direct students attention to Tables 3.3, 3.4, and 3.5 in Lesson 3.2. Have students express the ratio in scientific notation between the largest and smallest units listed in each table. (Table 3.3: ; Table 3.4: ; Table 3.5: ) Answers 41. a. 44 m b g c cg 42. a L b kg c ms d μg 43. a cm 3 b cm See answers for Problem g Scientific Measurement 89

7 LESSON 3.3 How many deciliters are in 6.5 hectoliters? (6,500 deciliters) Sample Problem 3.13 Multistep Problems Many complex tasks in your life are best handled by breaking them down into smaller, manageable parts. For example, if you were cleaning a car, you might first vacuum the inside, then wash the exterior, then dry the exterior, and finally put on a fresh coat of wax. Similarly, many complex word problems are more easily solved by breaking the solution down into steps. When converting between units, it is often necessary to use more than one conversion factor. Sample Problems 3.13 and 3.14 illustrate the use of multiple conversion factors. Converting Between Metric Units The diameter of a sewing needle is cm. What is the diameter in micrometers? The desired conversion is centimeters micrometers. The problem can be solved in a two-step conversion. First change centimeters to meters; then change meters to micrometers: centimeters meters micrometers. length cm cm 10 2 cm 1 m 1 m 10 6 m length? m Use the relationship 10 2 cm 1 m to write the first conversion factor. Use the relationship 1 m 10 6 m to write the second conversion factor. 1 m 10 2 cm 10 6 m 1 m Each conversion factor is written so that the unit in the denominator cancels the unit in the numerator of the previous factor. Multiply the known length by the conversion factors m 10 cm 6 m m 10 2 cm 1 m Evaluate Does the result make sense? Because a micrometer is a much smaller unit than a centimeter, the answer should be numerically larger than the given measurement. The units have canceled correctly, and the answer has the correct number of significant figures. 46. The radius of a potassium atom is nm. Express this radius in the unit centimeters. 47. The diameter of Earth is km. What is the diameter expressed in decimeters? Chapter 3 Lesson 3 Focus on ELL 4 LANGUAGE PRODUCTION Have students work in pairs to complete the Small-Scale Lab on page 92. Make sure each pair has ELLs of varied language proficiencies, so that more proficient students can help less proficient ones. Have students work according to their proficiency level. BEGINNING: LOW/HIGH Provide students a detailed step-by-step procedure to follow using pictures, words, and symbols. Be sure to show a balance with a question mark on top of it to indicate that students are to find the mass. Convey that the same procedure is to be followed three times. INTERMEDIATE: LOW/HIGH Have accelerated students perform the procedure for finding the mass of a drop of water using verbal questioning of what to do next. Have students mimic this procedure for finding the mass of the pre-1982 penny and the post-1892 penny. ADVANCED: LOW/HIGH Have students read out loud the questions in the Analyze and You re the Chemist sections. Then assist and/or edit the work of classmates with lower language proficiency.

8 M CHEM C U T T O R E ONLINE O P R O B L E M S R O M Sample Problem 3.14 Converting Ratios of Units The density of manganese, a metal, is 7.21 g/cm 3. What is the density of manganese expressed in units of kg/m 3? The desired conversion is g/cm 3 kg/m 3. The mass unit in the numerator must be changed from grams to kilograms: g kg. In the denominator, the volume unit must be changed from cubic centimeters to cubic meters: cm 3 m 3. Note that the relationship 10 6 cm 3 1 m 3 was derived by cubing the relationship 10 2 cm 1 m. That is, (10 2 cm) 3 (1 m) 3, or 10 6 cm 3 1 m 3. Multiply the known density by the correct conversion factors. 48. Gold has a density of 19.3 g/cm 3. What is the density in kilograms per cubic meter? 3.3 LessonCheck 50. Review What happens to the numerical value of a measurement that is multiplied by a conversion factor? What happens to the actual size of the quantity? 51. Review What types of problems can be solved using dimensional analysis? 52. Identify What conversion factor would you use to convert between these pairs of units? a. minutes to hours b. grams to milligrams c. cubic decimeters to milliliters density of manganese 7.21 g/cm g 1 kg 10 6 cm 3 1 m 3 density of manganese? kg/m g 1 kg 10 1 cm 3 6 cm g 1 m kg/m 3 Evaluate Does the result make sense? Because the physical size of the volume unit m 3 is so much larger than cm 3 (10 6 times), the calculated value of the density should be larger than the given value even though the mass unit is also larger (10 3 times). The units cancel, the conversion factors are correct, and the answer has the correct ratio of units. 49. There are red blood cells (RBCs) in 1.0 mm 3 of blood. How many red blood cells are in 1.0 L of blood? 53. Calculate Make the following conversions. Express your answers in scientific notation. a g? g d J? kj b g? kg e mg? dg c L? cm 3 f dl? L 54. Calculate What is the mass, in kilograms, of 14.0 L of gasoline? (Assume that the density of gasoline is g/cm 3.) 55. Apply Concepts Light travels at a speed of cm/s. What is the speed of light in kilometers/hour? In physics, acceleration is usually given in units of m/s 2. The acceleration of a falling object due to gravity is 9.8 m/s 2. What is the acceleration due to gravity in km/min 2? (35.28 km/min 2 ) Evaluate Informal Assessment To determine students grasp of conversion factors, ask students to orally explain the relationship between the numerator and the denominator of any measurement conversion factor. (They are equivalent so that the ratio of numerator to denominator equals 1.) Then have students complete the 3.3 Lesson Check. Reteach Model the conversion of 2 L to 2000 ml. Suggest that students check the answer by explaining that when using a conversion factor, such as 1 L = 1000 ml, the measurement expressed with the smaller unit (ml) should have a larger number associated with it than the measurement expressed with the larger unit (L). LESSON 3.3 Scientific Measurement 91 Lesson Check Answers 50. The numerical value (and the unit) changes; the actual size does not change. 51. conversion problems 52. a. 1 hour / 60 min b mg / 1 g c ml / 1 dm a μg b kg c cm 3 d kj e dg f μl kg km/h Answers cm dm kg/m RBC/L Scientific Measurement 91

9 SMALL-SCALE LAB Small-Scale Lab OBJECTIVE After completing this activity, students will be able to solve problems in divergent ways. SKILLS FOCUS measuring, calculating, evaluating, designing experiments PREP TIME 1 hour MATERIALS Calculators, small-scale pipets, meter sticks, water, mass balances, pre- and post-1982 pennies, dice, 8-well strips, aluminum cans, plastic cups ADVANCE PREP A day before doing the lab, obtain soda cans from the cafeteria. Wash and let air-dry overnight. CLASS TIME 40 minutes EXPECTED OUTCOME Students should find that the mass of the pre-1982 penny is 3.11 g and the mass of the post-1982 penny is 2.50 g. ANALYZE mg cm 3 ; ml; 19 μl mg/cm 3 ; 1000 mg/ml g Cu; 0.16 g Zn g Cu; 2.44 g Zn 6. The new penny is mostly zinc, which has a lower density than copper. YOU RE THE CHEMIST 1. at 90 o, mass of 1 drop: g at 45 o, mass of 1 drop: g at 0 o, mass of 1 drop: g Pipets give different results. 2. The pipet is easiest to control at 90 o. Expel the air bubble so that the first drop will be the same size as the rest. 3. Find the mass of the can and divide by the density of aluminum. Sample answer: mass of one can: g; density of Al: 2.70 g/cm 3 ; V = 5.47 cm 3 4. (1) Measure the mass before and after you fill the can with water. Use the mass and density of water to find the volume. (2) Measure the height and radius and calculate volume. V = πr 2 h (Can is not a perfect cylinder.) (3) Read label: 12 oz = 355 ml 5. Sample answer: V = 16.5 m 3.0 m 12.8 m = 630 m L/m 3 = 630,000 L. Assume 30 people with an average weight of 130 lb (1 kg = 2.2 lb) and a density of about 1.0 kg/l. Volume of 30 people = lb 1 kg/2.2 lb 1 L/1.0 kg = 1800 L. The volume of 30 chairs, 15 tables, and 2 desks is about that of 30 people or 1800 L. The volume of people and furniture is 3600 L. % error = (3600 L/630,000 L)(100%) = 0.57%. Small-Scale Lab Now What Do I Do? Purpose To solve problems by making accurate measurements and applying mathematics Materials pencil paper balance 92 water Procedure 1. Determine the mass, in grams, of one drop of water. To do this, measure the mass of an empty cup. Add 50 drops of water from a small-scale pipet to the cup and measure its mass again. Subtract the mass of the empty cup from the mass of the cup with water in it. To determine the average mass in grams of a single drop, divide the mass of the water by the number of drops (50). Repeat this experiment until your results are consistent. 2. Determine the mass of a pre-1982 penny and a post-1982 penny. Analyze Using your experimental data, record the answers to the following questions. 1. Calculate What is the average mass of a single drop of water in milligrams? (1 g 1000 mg) 2. Calculate The density of water is 1.00 g/cm 3. Calculate the volume of a single drop in cm 3 and ml. (1 ml 1 cm 3 ) What is the volume of a drop in microliters ( L)? (1000 L 1 ml) 3. Calculate What is the density of water in units of mg/cm 3 and mg/ml? (1 g 1000 mg) 4. Calculate Pennies made before 1982 consist of 95.0% copper and 5.0% zinc. Calculate the mass of copper and the mass of zinc in the pre-1982 penny. 6. If die measures 1.55 cm on a side: V = (1.55 cm) 3 = 3.72 cm 3 5. Calculate Pennies made after 1982 are made of zinc with a thin copper coating. They are 97.6% zinc and 2.4% copper. Calculate the mass of copper and the mass of zinc in the newer penny. 6. Why does one penny have less mass than the other? You re the Chemist The following small-scale activities allow you to develop your own procedures and analyze the results. 1. Design an Experiment Design an experiment to determine if the size of drops varies with the angle at which they are delivered from the pipet. Try vertical (90 ), horizontal (0 ), and halfway between (45 ). Repeat until your results are consistent. 2. Analyze Data What is the best angle to hold a pipet for ease of use and consistency of measurement?. Why is it important to expel the air bubbles before you begin the experiment? 3. Design an Experiment Make the necessary measurements to determine the volume of aluminum used to make an aluminum soda can. Hint: Look up the density of aluminum in your textbook. 4. Design an Experiment Design and carry out some experiments to determine the volume of liquid that an aluminum soda can will hold. 5. Design an Experiment Measure a room and calculate the volume of air it contains. Estimate the percent error associated with not taking into account the furniture in the room. 6. Design an Experiment Make the necessary measurements and do the necessary calculations to determine the volume of a pair of dice. First, ignore the volume of the dots on each face, and then account for the volume of the dots. What is your error and percent error when you ignore the holes? A die has 21 holes that are hemispheres with a radius of 0.20 cm. V of hemisphere = (2/3)πr 3 = cm 3 V of 21 hemispheres = 0.36 cm 3 V of die = 3.72 cm cm 3 = 3.36 cm 3 Error = 0.36 cm 3 % error = (0.36 cm 3 /3.36 cm 3 )(100%) = 11% 92 Chapter 3 Small-Scale Lab