chem_TE_ch03_FPL.fm Page 80 Monday, August 2, 2004 1:09 PM
3.3
3.3
1
Conversion Problems
FOCUS
Objectives
Guide for Reading
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.
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
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
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.
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.
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 measurements 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 m 100 cm 1
1m
1m
L2
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.)
or
100 cm
1m
100 cm 100 cm 1
conversion factors
Animation 3 Learn how to
select the proper conversion
factor and how to use it.
with ChemASAP
80 Chapter 3
Conversion Factors
1 dollar 4 quarters 10 dimes 20 nickels 100 pennies
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
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.
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.
80 Chapter 3
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
chem_TE_ch03_FPL.fm Page 81 Monday, August 2, 2004 1:09 PM
1 meter
100 centimeters
1m
1
Smaller number
=
10
m
100 cm
Larger number
20
30
40
50
60
Conversion Factors
70
80
90
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
Figure 3.11 The two parts of a
conversion factor, the numerator
and the denominator, are equal.
and
and
1 kg
1000 g
1m
109 nm
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?
Dimensional Analysis
Activity
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 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.
Expanding a Recipe
Common volumetric units used in chemistry include the liter and
the microliter. The relationship 1 L 106 µL yields the following conversion
factors.
1L
106 mL
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.
CLASS
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
1m
L1
Use Visuals
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.
Answers to...
Figure 3.12 1 m/109 nm
Checkpoint
unlimited
Scientific Measurement
81
chem_TE_ch03.fm Page 82 Wednesday, April 13, 2005 1:58 PM
Section 3.3 (continued)
SAMPLE PROBLEM 3.5
L2
Discuss
Using Dimensional Analysis
How many seconds are in a workday that lasts exactly eight hours?
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.
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.
Sample Problem 3.5
Answers
28. 1.0080 × 104 min
29. 1.44000 × 105 s
Practice Problems Plus
Math
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 help with dimensional
analysis, go to page R66.
8 h 60 min 60 s 28,800 s
1 min
1h
2.8800 104 s
Practice Problems
Evaluate 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.
Handbook
For a math refresher and practice,
direct students to dimensional
analysis, page R66.
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.
Practice Problems
Problem-Solving 3.29 Solve
Problem 29 with the help of an
interactive guided tutorial.
28. How many minutes are there
in exactly one week?
29. How many seconds are in
exactly a 40-hour work week?
with ChemASAP
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
chem_TE_ch03_FPL.fm Page 83 Wednesday, August 4, 2004 6:35 PM
Sample Problem 3.6
SAMPLE PROBLEM 3.6
Using Dimensional Analysis
Answers
Conversion Problems
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.
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.
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 ⫽ 27.174 students ⫽ 27 students
1.84 g Cu
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?
Problem-Solving 3.30 Solve
Problem 30 with the help of an
interactive guided tutorial.
with ChemASAP
Section 3.3 Conversion Problems
83
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.
Math
Handbook
For a math refresher and practice, direct students to conversion problems, page R66.
Differentiated Instruction
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
30. 67 students
31. 86.4°F
Less Proficient Readers
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.
Scientific Measurement
83
chem_TE_ch03_FPL.fm Page 84 Wednesday, August 4, 2004 6:37 PM
Converting Between
Units
CLASS
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.
Activity
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 available 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 students document their approach, including 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.
Sports Stats
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
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.
1g
10 dg
Note that the known unit is in the denominator and the unknown unit
is in the numerator.
750 dg ⫻
Sample Problem 3.7
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.
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 µg
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 µm)
d. 0.035 cm to millimeters
(3.5 × 10–1 mm)
84
Chapter 3
1g
⫽ 75 g
10 dg
Evaluate Does the result make sense?
Answers
Practice Problems Plus
Unknown
• mass ⫽ ? g
L2
Practice Problems
32. Using tables from this chapProblem-Solving 3.33 Solve
Problem 33 with the help of an
interactive guided tutorial.
with ChemASAP
84 Chapter 3
ter, 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
chem_TE_ch03.fm Page 85 Wednesday, April 13, 2005 1:59 PM
handled by breaking them down into 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 Problem 3.8 illustrates the use of multiple
conversion factors.
Checkpoint What problem-solving methods can help you solve complex
word problems?
SAMPLE PROBLEM 3.8
Converting Between Metric Units
What is 0.073 cm in micrometers?
Analyze List the knowns and the unknown.
Knowns
• length 0.073 cm 7.3 102 cm
• 102 cm 1 m
• 1 m 106 µm
Unknown
• length ? µm
Scientific Notation
It is often convenient to
express very large or very
small numbers in scientific
notation. The distance
between the sun and Earth is
150,000,000 km, which can
be written as 1.5 108 km.
The diameter of a gold atom
is 0.000 000 000 274 m, or
2.74 1010 m.
When multiplying numbers
written in scientific notation,
add the exponents. When
dividing numbers written in
scientific notation, subtract
the exponent in the denominator from the exponent in
the numerator.
Math
Handbook
For help with scientific
notation, go to page R56.
The desired conversion is from centimeters to micrometers. The problem can be solved in a two-step conversion.
Calculate Solve for the unknown.
First change centimeters to meters; then change meters to micrometers: centimeters ¡ meters ¡ micrometers. Each conversion
factor is written so that the unit in the denominator cancels the unit in
the numerator of the previous factor.
7.3 10 -2 cm 6
1 m 10 mm 7.3 102 mm
1m
102 cm
L2
Discuss
Multistep Problems Many complex tasks in your everyday life are best
Explain 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.
Sample Problem 3.8
Answers
Problem-Solving 3.35 Solve
Problem 35 with the help of an
interactive guided tutorial.
with ChemASAP
34. 2.27 × 10–8 cm
35. 1.3 × 108 dm
Practice Problems Plus
L2
Chapter 3 Assessment problem
70 is similar to Sample Problem 3.8.
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.
Remind students that writing a
number in scientific notation does
not change the actual size of the
number. In Sample Problem 3.8,
the given measurement 0.073 cm
can be written as 7.3 × 10–2 cm.
Practice Problems
34. The radius of a potassium
atom is 0.227 nm. Express this
radius in the unit centimeters.
35. The diameter of Earth is
1.3 104 km. What is the
diameter expressed in
decimeters?
Math
Section 3.3 Conversion Problems 85
Handbook
For a math refresher and practice, direct students to scientific
notation, page R56.
Differentiated Instruction
L1
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
• Find
Less Proficient Readers
Answers to...
Checkpoint Break the solution down into steps. Use more than
one conversion factor if necessary.
Scientific Measurement
85
chem_TE_ch03_FPL.fm Page 86 Wednesday, August 4, 2004 6:38 PM
Converting Complex Units Many common measurements are expressed
Sample Problem 3.9
as a ratio of two units. For example, the results of international car races
often give average lap speeds in kilometers per hour. You measure the densities of solids and liquids in grams per cubic centimeter. You measure the
gas mileage in a car in miles per gallon of gasoline. If you use dimensional
analysis, converting these complex units is just as easy as converting single
units. It will just take multiple steps to arrive at an answer.
Answers
36. 1.93 × 104 kg /m3
37. 7.0 × 1012 RBC/L
Practice Problems Plus
L2
1.00 L of neon gas contains 2.69 × 1022
neon atoms. How many neon atoms
are in 1.00 mm3 of neon gas under
the same conditions? (2.69 × 1016
atoms)
Math
SAMPLE PROBLEM 3.9
Converting Ratios of Units
The mass per unit volume of a substance is a property called density.
The density of manganese, a metallic element, is 7.21 g/cm3. What is the
density of manganese expressed in units kg/m3?
Analyze List the knowns and the unknown.
Handbook
Knowns
• density of manganese ⫽ 7.21 g/cm3
• 103 g ⫽ 1 kg
• 106 cm3 ⫽ 1 m3
For a math refresher and practice,
direct students to dimensional analysis, page R66.
Unknown
• density manganese ⫽ ? kg/m3
The desired conversion is g/cm3 ¡ kg/m3. 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: cm3 ¡ m3. Note that the relationship
between cm3 and m3 was determined from the relationship
between cm and m. Cubing the relationship 102 cm ⫽ 1 m yields
(102 cm)3 ⫽ (1 m)3, or 106 cm3 ⫽ 1 m3.
Quick LAB
L2
Objective After completing this activity, students will be able to
• develop conversion factors using
dimensional analysis.
Dimensional Analysis
Math
Handbook
Calculate Solve for the unknown.
1 kg
7.21 g
106 cm3
⫻ 3 ⫻
⫽ 7.21 ⫻ 103 kg/m3
1 cm3
10 g
1 m3
For help with dimensional
analysis, go to page R66.
Evaluate Does the result make sense?
Students may think that because a
conversion factor equals unity, it
doesn’t 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.
Because the physical size of the volume unit m3 is so much larger than
cm3 (106 times), the calculated value of the density should be larger
than the given value even though the mass unit is also larger (103
times). The units cancel, the conversion factors are correct, and the
answer has the correct ratio of units.
Practice Problems
36. Gold has a density of
Problem-Solving 3.37 Solve
Problem 37 with the help of an
interactive guided tutorial.
19.3 g/cm3. What is the
density in kilograms per
cubic meter?
37. There are 7.0 ⫻ 106 red blood
cells (RBC) in 1.0 mm3 of
blood. How many red blood
cells are in 1.0 L of blood?
with ChemASAP
Skills Focus Calculating
Prep Time 5 minutes
Materials 3 inch × 5 inch index cards
(or paper cut to approximately the
same size)
Class Time 15 minutes
Expected Outcome Students should
derive appropriate conversion factors
among metric units using standard
metric quantities and prefixes.
86
Chapter 3
86 Chapter 3
L3
Analyze and Conclude
For Enrichment
1. If a measurement of a quantity is being converted, the unit changes. If the equivalence of
a quantity is being found, the conversion factor changes the quantity itself.
2. a. 0.785 m b. 56 cm3 c. 7.7 × 107 mg
d. 9.8 × 10–10 dm e. 9.6 × 103 µm
f. 6.7 × 103 nm
Have students express the ratio in scientific notation between the largest and smallest units
listed in each table. (Table 3.3: 1 × 1012,
Table 3.4: 1 × 106; Table 3.5: 1 × 109 )
chem_TE_ch03.fm Page 87 Thursday, April 14, 2005 12:49 PM
Quick LAB
3
Dimensional Analysis
Evaluate Understanding
Purpose
Procedure
To apply the problemsolving technique of
dimensional analysis to
conversion problems.
A conversion factor is a ratio of equivalent
measurements. For any relationship, you
can write two ratios. On a conversion factor card you can write one ratio on each
side of the card.
Materials
• 3 inch 5 inch index
•
cards or paper cut to
approximately the same
size
pen
ASSESS
1. Make a conversion factor card for each
metric relationship shown in Tables 3.3,
3.4, and 3.5. Show the inverse of the
conversion factor on the back of
each card.
2. Use the appropriate conversion factor
cards to set up solutions to Sample
Problems 3.7 and 3.8. Notice that in
each solution, the unit in the denominator of the conversion factor cancels
the unit in the numerator of the previous conversion factor.
To determine students’ grasp of conversion factors, ask, What is 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.)
Analyze and Conclude
1. What is the effect of multiplying a
given measurement by one or more
conversion factors?
2. Use your conversion factor cards to set
up solutions to these problems.
a. 78.5 cm ? m
b. 0.056 L ? cm3
c. 77 kg ? mg
d. 0.098 nm ? dm
e. 0.96 cm ? µm
f. 0.0067 mm ? nm
39.
40.
Key Concept 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?
Key Concept Why is dimensional analysis
useful?
44. Convert the following. Express your answers in
scientific notation.
a. 7.5 104 J to kilojoules
b. 3.9 105 mg to decigrams
c. 2.21 104 dL to microliters
45. Light travels at a speed of 3.00 1010 cm/s.
Key Concept What types of problems can be
solved using dimensional analysis?
What is the speed of light in kilometers/hour?
41. What conversion factor would you use to convert
between these pairs of units?
L1
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 (2000) than the measurement expressed with the larger unit (L).
Connecting
3.3 Section Assessment
38.
L2
Concepts
Acceptable answers include
(a) analyze the known and
unknowns, devise a set of conversions steps that yields the desired
final units, and write a conversion
factor for each step; (b) calculate by
multiplying (making sure that the
appropriate units cancel); and (c)
evaluate the magnitude and units of
the calculated answer.
Problem-Solving Skills Reread the passage on
solving numeric problems in Section 1.4. Explain
how the three-step process might apply to conversion problems that involve dimensional analysis.
a. minutes to hours
b. grams to milligrams
c. cubic decimeters to milliliters
42. Make the following conversions. Express your
answers in standard exponential form.
a. 14.8 g to micrograms
b. 3.72 103 kg to grams
c. 66.3 L to cubic centimeters
43. An atom of gold has a mass of 3.271 1022 g. How
many atoms of gold are in 5.00 g of gold?
Assessment 3.3 Test yourself
on the concepts in Section 3.3.
with ChemASAP
If your class subscribes to the
Interactive Textbook, use it to
review key concepts in Section 3.3.
with ChemASAP
Section 3.3 Conversion Problems 87
Section 3.3 Assessment
38. The numerical value (and the unit)
changes; the actual size does not change.
39. Dimensional analysis provides an alternative approach to problem solving.
40. conversion problems
41. a. 1 hour / 60 min b. 103 mg / 1 g
c. 103 mL / 1 dm3
42. a. 1.48 × 107 µg b. 3.72 g
c. 6.63 × 104 cm3
43. 1.53 × 1022 atoms of gold
44. a. 7.5 × 101 kJ b. 3.9 × 103 dg
c. 2.21 × 101 µL
45. 1.08 × 109 km/h
Scientific Measurement
87