3.3 Conversion Problems - Henry County Schools

[Pages:8]3.3

1 FOCUS

Objectives

3.3.1 Construct conversion factors from equivalent measurements.

3.3.2 Apply the technique of dimensional analysis to a variety of conversion problems.

3.3.3 Solve problems by breaking the solution into steps.

3.3.4 Convert complex units, using dimensional analysis.

Guide for Reading

Build Vocabulary

L2

Paraphrase Have students write definitions of the vocabulary terms in their own words. (Acceptable answers include conversion factor: a ratio of equivalent measurements used to convert a quantity from one unit to another, and dimensional analysis: a technique of problem-solving that uses the units that are part of a measurement to solve the problem.)

Reading Strategy

L2

Sequence As the students read the Analyze and Calculate sections of Sample Problems 3.5?3.9, have them write word sequences using the appropriate conversion factors for each problem.

2 INSTRUCT

Have students examine the photograph that opens the section. Ask if any of them has ever noticed a chart or table in a bank or in the newspaper relating the values of foreign currency to the U.S. dollar. Explain that these are conversion tables that allow people to relate one currency to another. Ask, How would you decide which amount of money would be worth more--75 euros or 75 British pounds? (Convert these values to a familiar currency--U.S. dollars.)

3.3 Conversion Problems

Guide for Reading

Key Concepts

? What happens when a measurement is multiplied by a conversion factor?

? Why is dimensional analysis useful?

? What types of problems are easily solved by using dimensional analysis?

Vocabulary

conversion factor dimensional analysis

Reading Strategy

Monitoring Your Understanding Preview the Key Concepts, the section heads, and boldfaced terms. List three things you expect to learn. After reading, state what you learned about each item listed.

Animation 3 Learn how to select the proper conversion factor and how to use it.

with ChemASAP

80 Chapter 3

Perhaps you have traveled abroad 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 essential to the enjoyment of your trip. After all, you must pay for your meals, hotel, transportation, gift purchases, and tickets to exhibits and events. Because each country's currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problem-solving approach called dimensional analysis.

Conversion Factors

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 mea-

surements 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.

1 1

m m

100 cm 1 m

1

or

1 m 100 cm

100 100

cm cm

1

conversion factors

A conversion factor is a ratio of equivalent measurements. The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors. In a conversion factor, the measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom). The conversion factors above are read "one hundred centimeters per meter" and "one meter per hundred centimeters." Figure 3.11 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.

Section Resources

Print ? Guided Reading and Study Workbook, Section 3.3 ? Core Teaching Resources, Section 3.3 Review ? Transparencies, T31?T37

Technology ? Interactive Textbook with ChemASAP, Animation 3, Problem-Solving 3.28, 3.30, 3.33, 3.35, 3.37, Assessment 3.3

80 Chapter 3

1 meter

Smaller number Larger number

1m =

100 centimeters 10 20 30 40 50 60 70 80 90

1m 100 cm

Larger unit Smaller unit

A Conversion Factor

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

The scale of the micrograph in Figure 3.12 is in nanometers. Using the relationship 109 nm 1 m, you can write the following conversion factors.

109 nm and 1 m

1 m

109 nm

Common volumetric units used in chemistry include the liter and the microliter. The relationship 1 L 106 ?L yields the following conversion factors.

1L 106 mL

and

106 mL 1L

Based on what you know about metric prefixes, you should be able to easily write conversion factors that relate equivalent metric quantities.

Checkpoint How many significant figures does a conversion factor within a system of measurement have?

Figure 3.11 The two parts of a conversion factor, the numerator and the denominator, are equal.

Figure 3.12 In this computer image of atoms, distance is marked off in nanometers (nm). Inferring What conversion factor would you use to convert nanometers to meters?

Dimensional Analysis

No single method is best for solving every type of problem. Several good approaches are available, and generally one of the best is 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 problem-solving technique is to use it to solve an everyday situation.

Section 3.3 Conversion Problems 81

Facts and Figures

Monetary Exchange Rates 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.

Conversion Factors

Use Visuals

L1

Figure 3.11 Have students inspect the 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, every conversion factor must contain two numbers and two units so that one number and its unit equal another number and its unit.

Dimensional Analysis

CLASS Activity

Expanding a Recipe

L2

Purpose To 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.

Explain that the students 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 stu-

dents 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 tea-

spoons/1 tablespoon, and then rewrite

the recipe using larger measurements.

Answers to... Figure 3.12 1 m/109 nm

Checkpoint unlimited

Scientific Measurement 81

Section 3.3 (continued)

Discuss

L2

Explain 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. Explain 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 cm3.

Sample Problem 3.5

Answers

28. 1.0080 ? 104 min 29. 1.44000 ? 105 s

Practice Problems Plus

L2

At Earth's farthest point from the sun,

sunlight takes 8.5 minutes to reach

Earth. How many weeks is this? (8.4 ? 10?4 weeks)

Math Handbook

For a math refresher and practice, direct students to dimensional analysis, page R66.

Math Handbook For help with dimensional analysis, go to page R66.

Problem-Solving 3.29 Solve Problem 29 with the help of an interactive guided tutorial.

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SAMPLE PROBLEM 3.5

Using Dimensional Analysis

How many seconds are in a workday that lasts exactly eight hours?

Analyze List the knowns and the unknown.

Knowns ? time worked 8 h ? 1 hour 60 min ? 1 minute 60 s Unknown ? seconds worked ? s

The first conversion factor must be written with the unit hours in the denominator. The second conversion factor must be written with the unit minutes in the denominator. This will provide the desired unit (seconds) in the answer.

Calculate Solve for the unknown. Start with the known, 8 hours. Use the first relationship (1 hour 60 minutes) to write a conversion factor that expresses 8 hours as minutes. The unit hours must be in the denominator so that the known unit will cancel. Then use the second conversion factor to change the unit minutes into the unit seconds. This conversion factor must have the unit minutes in the denominator. The two conversion factors can be used together in a simple overall calculation.

8

h

60 min 1 h

60 s 1 min

28,800

s

2.8800 104 s

Practice Evaluate

Problems

Does the result

make

sense?

The answer has the desired unit (seconds). Since the second is a small

unit of time, you should expect a large number of seconds in 8 hours.

Before you do the actual arithmetic, it is 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. The answer is exact since the

given measurement and each of the conversion factors is exact.

Practice Problems

28. How many minutes are there 29. How many seconds are in

in exactly one week?

exactly a 40-hour work week?

There is usually more than one way to solve a problem. When you first read Sample Problem 3.5, you may have thought about different and equally correct ways to approach and solve the problem. Some problems are easily worked with simple algebra. Dimensional analysis provides you with an alternative approach to problem solving. In either case, you should choose the problem-solving method that works best.

82 Chapter 3

82 Chapter 3

SAMPLE PROBLEM 3.6

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?

Analyze List the knowns and the unknown.

Knowns ? mass of copper available 50.0 g Cu ? each student needs 1.84 grams of copper, or 1.84 g Cu .

student

Unknown ? number of students ?

From the known mass of copper, calculate the number of students that can do the experiment by using the appropriate conversion factor. The desired conversion is mass of copper ? number of students.

Calculate Solve for the unknown. Because students is the desired unit for the answer, the conversion factor should be written with students in the numerator. Multiply the mass of copper by the conversion factor.

50.0

g

Cu

1 student 1.84 g Cu

27.174

students

27

students

Note that because students cannot be fractional, the result is shown rounded down to a whole number.

Evaluate Does the result make sense? The unit of the answer (students) is the one desired. The number of students (27) seems to be a reasonable answer. 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.

Practice Problems

30. 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?

31. 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 on the Fahrenheit scale?

Conversion Problems

A conversion factor is a ratio of two quantities that are equal to one another. When doing conversions, write the conversion factors so that the unit of a given measurement cancels, leaving the correct unit for your answer. Note that the equalities needed to write a particular conversion may be given in the problem. In other cases, you will need to know or look up the necessary equalities.

Math Handbook For help with conversion problems, go to page R66.

Problem-Solving 3.30 Solve Problem 30 with the help of an interactive guided tutorial.

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Sample Problem 3.6

Answers

30. 67 students 31. 86.4?F

Practice Problems Plus

L2

Chapter 3 Assessment problem 84 is related to Sample Problem 3.6.

Conversion Problems Remind students that they often convert from one unit to another, both inside and outside of chemistry class.They convert money from cents to dollars and 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, you may wish to have them list several conversion factors on the chalkboard. 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.

Section 3.3 Conversion Problems 83

Differentiated Instruction

Less Proficient Readers

L1

Provide as much class time as possible for

students to work on problem assignments in

cooperative learning groups. Have students

explore their own problem-solving styles.

Encourage students to draw a diagram or

picture of the problem to be solved whenever possible. Some students may want to read the problem aloud or have a partner read it to them. Some may want to work with symbols and equations.

Math Handbook

For a math refresher and practice, direct students to conversion problems, page R66.

Scientific Measurement 83

Converting Between Units

CLASS Activity

Sports Stats

L2

Purpose To use dimensional analysis

to convert between English and metric

units

Materials copies of media guides

containing vital statistics, such as

heights and weights, of players on a

sports team (These guides are avail-

able from local sports franchises.)

Procedure Group students. Distribute

the media guides and assign each

group a set of players. Ask the group to

convert heights and weights into

heights and masses expressed in meters

and kilograms, respectively. Have stu-

dents document their approach, includ-

ing dimensional analysis expressions,

conversion factors, and calculations.

Expected Outcome Students

should use conversion factors, such

as 2.54 cm/1 inch and 454 g/1 lb to

convert their measurements.

Sample Problem 3.7

Answers

32. a. 44 m b. 4.6 x 10?3 g

c. 10.7 cg 33. a. 1.5 ? 10?2 L b. 7.38 ? 10?3 kg

c. 6.7 ? 103 ms d. 9.45 ? 107 ?g

Practice Problems Plus

L2

Make the following conversions.

a. 0.045 L to cubic centimeters (4.5 ? 101 cm3) b. 14.3 mg to grams (1.43 ? 10?2 g)

c. 0.0056 m to micrometers (5.6 ? 103 ?m)

d. 0.035 cm to millimeters (3.5 ? 10?1 mm)

Converting Between Units

In chemistry, as in many other subjects, you often need to express a measurement in a unit different from the one given or measured initially.

Problems in which a measurement with one unit is converted to an equivalent measurement with another unit are easily solved using dimensional analysis.

Suppose that a laboratory experiment requires 7.5 dg of magnesium metal, and 100 students will do the experiment. How many grams of magnesium should your teacher have on hand? Multiplying 100 students by 7.5 dg/student gives you 750 dg. But then you must convert dg to grams. Sample Problem 3.7 shows you how to do the conversion.

Problem-Solving 3.33 Solve Problem 33 with the help of an interactive guided tutorial.

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84 Chapter 3

SAMPLE PROBLEM 3.7

Converting Between Metric Units

Express 750 dg in grams.

Analyze List the knowns and the unknown.

Knowns ? mass 750 dg ? 1 g 10 dg

Unknown ? mass ? g

The desired conversion is decigrams ? grams. Using the expression relating the units, 10 dg 1 g, multiply the given mass by the proper conversion factor.

Calculate Solve for the unknown. The correct conversion factor is shown below.

1 g 10 dg

Note that the known unit is in the denominator and the unknown unit is in the numerator.

1 g 750 dg 10 dg 75 g

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 unit of the known (dg) cancels, and the answer has the correct unit (g). The answer also has the correct number of significant figures.

Practice Problems

32. Using tables from this chapter, convert the following. a. 0.044 km to meters b. 4.6 mg to grams c. 0.107 g to centigrams

33. Convert the following. a. 15 cm3 to liters b. 7.38 g to kilograms c. 6.7 s to milliseconds d. 94.5 g to micrograms

84 Chapter 3

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