Download Chapter 1 Measurements

Chapter 1
Measurements
Measurement
You make a measurement
every time you
1.1
Units of Measurement
•
•
•
•
measure your height.
read your watch.
take your temperature.
weigh a cantaloupe.
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
1
Measurement in Chemistry
2
Measurement
In chemistry we
In a measurement
•
•
•
•
• a measuring tool is used
•
measure quantities.
do experiments.
calculate results.
use numbers to report
measurements.
compare results to
standards.
to compare some
dimension of an object
to a standard.
• of the thickness of the
skin fold at the waist,
calipers are used.
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
3
Stating a Measurement
4
The Metric System (SI)
In every measurement, a number is followed by a unit.
The metric system or SI (international system) is
Observe the following examples of measurements:
• a decimal system based on 10.
• used in most of the world.
• used everywhere by scientists.
Number and Unit
35
m
0.25
L
225
lb
3.4
kg
5
6
Units in the Metric System
Length Measurement
In the metric and SI systems, one unit is used for each
type of measurement:
Measurement
Length
Volume
Mass
Time
Temperature
Metric
meter (m)
liter (L)
gram (g)
second (s)
Celsius (°C)
Length
• is measured using a
SI
meter (m)
cubic meter (m3)
kilogram (kg)
second (s)
Kelvin (K)
meterstick.
• uses the unit of meter
(m) in both the metric
and SI systems.
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7
Inches and Centimeters
8
Volume Measurement
Volume
• is the space occupied by a
The unit of an inch is
equal to exactly 2.54
centimeters in the metric
(SI) system.
substance.
• uses the unit liter (L) in the
metric system.
•
1 qt = 946 mL
• uses the unit m3 (cubic
1 in. = 2.54 cm
meter) in the SI system.
• is measured using a
graduated cylinder.
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
9
Mass Measurement
10
Temperature Measurement
The temperature of a substance
• indicates how hot or cold it is.
• is measured on the Celsius
(°C) scale in the metric
system.
• on this thermometer is 18 ºC
or 64 ºF.
• in the SI system uses the
Kelvin (K) scale.
The mass of an object
• is the quantity of material it
contains.
• is measured on a balance.
• uses the unit gram (g) in the
metric system.
• uses the unit kilogram (kg)
in the SI system.
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
11
12
Time Measurement
Summary of Units of Measurement
Time measurement
• uses the unit second (s)
in both the metric and SI
systems.
• is based on an atomic
clock that uses the
frequency of cesium
atoms.
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13
Learning Check
14
Solution
For each of the following, indicate whether the unit
describes 1) length, 2) mass, or 3) volume.
For each of the following, indicate whether the unit
describes 1) length, 2) mass, or 3) volume.
____ A.
A bag of tomatoes is 4.6 kg.
2
A. A bag of tomatoes is 4.6 kg.
____ B.
A person is 2.0 m tall.
1
B. A person is 2.0 m tall.
____ C.
A medication contains 0.50 g aspirin.
2
C. A medication contains 0.50 g aspirin.
____ D.
A bottle contains 1.5 L of water.
3
D. A bottle contains 1.5 L of water.
15
Learning Check
Solution
Identify the measurement that has an SI unit.
A. John’s height is
1) 1.5 yd.
2) 6 ft .
3) 2.1 m.
B. The race was won in
1) 19.6 s.
2) 14.2 min.
C. The mass of a lemon is
1) 12 oz.
2) 0.145 kg.
D. The temperature is
1) 85 °C.
2) 255 K.
16
A. John’s height is
3) 2.1 m.
B. The race was won in
1) 19.6 s.
3) 3.5 h.
C. The mass of a lemon is
2) 0.145 kg.
3) 0.6 lb.
D. The temperature is
2) 255 K.
3) 45 °F.
17
18
Scientific Notation
Measurements
1.2
Scientific Notation
Scientific Notation
• is used to write very large
or very small numbers.
• for the width of a human
hair of 0.000 008 m is
written 8 x 10-6 m.
• of a large number such
as 4 500 000 s is written
4.5 x 106 s.
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Copyright © 2009 by Pearson Education, Inc.
20
Writing Numbers in
Scientific Notation
Numbers in Scientific Notation
A number written in scientific notation contains a
• coefficient.
• power of 10.
To write a number in scientific notation,
Examples:
Examples:
coefficient
1.5
power of ten
x
coefficient
102
7.35
• move the decimal point to give a number 1-9.
• show the spaces moved as a power of 10.
power of ten
x
52 000. = 5.2 x 10 4
10-4
4 spaces left
0.00178 = 1.78 x 10-3
3 spaces right
21
22
Comparing Numbers in Standard
and Scientific Notation
Some Powers of 10
Here are some numbers written in standard format
and in scientific notation.
Number in
Number in
Standard Format
Scientific Notation
Diameter of the Earth
12 800 000 m
1.28 x 107 m
Mass of a typical human
68 kg
6.8 x 101 kg
Length of a pox virus
0.000 03 cm
3 x 10-5 cm
23
24
Study Tip: Scientific Notation
Learning Check
Select the correct scientific notation for each.
In a number 10 or larger, the decimal point
• is moved to the left to give a positive power of 10
A. 0.000 008 m
1) 8 x 106 m,
In a number less than 1, the decimal point
• is moved to the right to give a negative power of 10
2) 8 x 10-6 m, 3) 0.8 x 10-5 m
B. 72 000 g
1) 7.2 x 104 g, 2) 72 x 103 g, 3) 7.2 x 10-4 g
25
Solution
26
Learning Check
Select the correct scientific notation for each.
Write each as a standard number.
A. 0.000 008 m
2) 8 x 10-6 m
A. 2.0 x 10-2 L
1) 200 L,
2) 0.0020 L,
3) 0.020 L
B. 72 000 g
1) 7.2 x 104 g
B. 1.8 x 105 g
1) 180 000 g,
2) 0.000 018 g,
3) 18 000 g
27
Solution
28
Chapter 1
Measurements
1.3
Measured Numbers and
Significant Figures
Write each as a standard number.
A. 2.0 x 10-2 L
3) 0.020 L
B. 1.8 x 105 g
1) 180 000 g
Copyright © 2009 by Pearson Education, Inc.
29
30
Measured Numbers
Reading a Meterstick
A measuring tool
. l2. . . . l . . . . l3 . . . . l . . . . l4. .
• The markings on the meterstick at the end of the
• is used to determine
•
a quantity such as
height or the mass of
an object.
provides numbers for
a measurement
called measured
numbers.
cm
•
•
Copyright © 2009 by Pearson Education, Inc.
orange line are read as:
the first digit
2
plus the second digit
2.7
The last digit is obtained by estimating.
The end of the line may be estimated between 2.7–
2.8 as half way (0.5) or a little more (0.6), which gives
a reported length of 2.75 cm or 2.76 cm.
31
32
Learning Check
Known & Estimated Digits
If the length is reported as 2.76 cm,
. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
• the digits 2 and 7 are certain (known).
• the final digit, 6, is estimated (uncertain).
• all three digits (2, 7, and 6) are significant, including
What is the length of the orange line?
1) 9.0 cm
the estimated digit.
2) 9.04 cm
3) 9.05 cm
33
Solution
Zero as a Measured Number
. l3. . . . l . . . . l4. . . . l . . . . l5. . cm
. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
The length of the orange line could be reported as
• For this measurement, the first and second known
digits are 4 and 5.
2) 9.04 cm
or
34
3) 9.05 cm
• When a measurement ends on a mark, the estimated
digit in the hundredths place is 0.
The estimated digit may be slightly different. Both
readings are acceptable.
• This measurement is reported as 4.50 cm.
35
36
Significant Figures in
Measured Numbers
Significant Figures
Significant Figures
• obtained from a measurement include all
of the known digits plus the estimated
digit.
• reported in a measurement depend on the
measuring tool.
37
Counting Significant Figures
Sandwiched Zeros
All nonzero numbers in a measured number are
significant.
Measurement
38.15 cm
5.6 ft
65.6 lb
122.55 m
38
Sandwiched Zeros
• occur between nonzero numbers.
• are significant.
Number of
Significant Figures
4
2
3
5
Measurement
50.8 mm
2001 min
0.0702 lb
0.405 05 m
Number of
Significant Figures
3
4
3
5
39
Trailing Zeros
40
Leading Zeros
Leading Zeros
• precede nonzero digits in a decimal number.
• are not significant.
Trailing Zeros
• follow nonzero numbers in numbers without
decimal points.
• are usually placeholders.
• are not significant.
Number of
Measurement
Significant Figures
25 000 cm
2
200 kg
1
48 600 mL
3
25 005 000 g
5
Measurement
0.008 mm
0.0156 oz
0.0042 lb
0.000 262 mL
41
Number of
Significant Figures
1
3
2
3
42
Solution
Learning Check
State the number of significant figures in each of
the following measurements.
State the number of significant figures in each of
the following measurements.
A. 0.030 m
A. 0.030 m
2
B. 4.050 L
B. 4.050 L
4
C. 0.0008 g
C. 0.0008 g
1
D. 2.80 m
D. 2.80 m
3
43
Significant Figures in
Scientific Notation
Study Tip: Significant Figures
In scientific notation all digits in the coefficient
including zeros are significant.
The significant figures in a measured number are
• all the nonzero numbers.
12.56 m
4 significant figures
• zeros between nonzero numbers.
4.05 g
3 significant figures
• zeros that follow nonzero numbers in a decimal
number.
25.800 L
5 significant figures
Number of
Significant Figures
1
2
3
Measurement
8 x 104 m
8.0 x 104 m
8.00 x 104 m
44
45
Learning Check
46
Solution
A. Which answer(s) contain 3 significant figures?
2) 0.00476
3) 4.76 x 103
A. Which answer(s) contain 3 significant figures?
1) 0.4760
2) 0.00476
3) 4.76 x 103
B. All the zeros are significant in
B. All the zeros are significant in
1) 0.00307.
2) 25.300.
2) 25.300.
3) 2.050 x 103.
3) 2.050 x 103.
C. The number of significant figures in 5.80 x 102 is
3) three (3).
102 is
C. The number of significant figures in 5.80 x
1) one (1).
2) two (2).
3) three (3).
47
48
Learning Check
Solution
In which set(s) do both numbers contain the
same number of significant figures?
In which set(s) do both numbers contain the same
number of significant figures?
1) 22.0 and 22.00
3) 0.000 015 and 150 000
2) 400.0 and 40
3) 0.000 015 and 150 000
Both numbers contain 2 significant figures.
49
Examples of Exact Numbers
50
Exact Numbers
An exact number is obtained
• when objects are counted.
Counted objects
2 soccer balls
4 pizzas
• from numbers in a defined relationship.
Defined relationships
1 foot = 12 inches
1 meter = 100 cm
51
Learning Check
52
Solution
A. Exact numbers are obtained by
1. using a measuring tool.
2. counting.
3. definition.
A. Exact numbers are obtained by
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
53
54
Learning Check
Solution
Classify each of the following as (1) exact or (2) measured
numbers.
Classify each of the following as (1) exact or (2) measured
numbers.
A.__Gold melts at 1064 °C.
A. 2
A measuring tool is required.
B.__1 yard = 3 feet
B. 1
This is a defined relationship.
C.__The diameter of a red blood cell is 6 x 10-4 cm.
C. 2
A measuring tool is used to determine
length.
D.__There are 6 hats on the shelf.
E.__A can of soda contains 355 mL of soda.
D. 1
The number of hats is obtained by counting.
E. 2
The volume of soda is measured.
55
Chapter 1
Measurements
56
Rounding Off Calculated Answers
1.4
Significant Figures in
Calculations
In calculations,
•
answers must have the
same number of significant
figures as the measured
numbers.
•
a calculator answer often
must be rounded off.
•
rounding rules are used to
obtain the correct number of
significant figures.
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
57
Rounding Off Calculated
Answers
58
Adding Significant Zeros
When the first digit dropped is 4 or less,
•
the retained numbers remain the same.
•
45.832 rounded to 3 significant figures
drops the digits 32 = 45.8
Sometimes a calculated answer requires more
significant digits. Then, one or more zeros are
added.
Calculated
Answer
4
1.5
0.2
12
When the first digit dropped is 5 or greater,
the last retained digit is increased by 1.
2.4884 rounded to 2 significant figures
drops the digits 884 = 2.5 (increase by 0.1)
•
59
Zeros Added to
Give 3 Significant Figures
4.00
1.50
0.200
12.0
60
Learning Check
Solution
Round off or add zeros to the following calculated
answers to give three significant figures.
Adjust the following calculated answers to give answers
with 3 significant figures.
A. 824.75 cm
A. 825 cm
First digit dropped is greater than 5.
B. 0.112g
First digit dropped is 4.
C. 8.20 L
Significant zero is added.
B. 0.112486 g
C. 8.2 L
61
Calculations with Measured Numbers
62
Multiplication and Division
When multiplying or dividing
In calculations with
measured numbers,
significant figures or
decimal places are
counted to determine
the number of figures in
the final answer.
•
the final answer must have the same number of
significant figures as the measurement with the
fewest significant figures.
•
use rounding rules to obtain the correct number of
significant figures.
Example:
110.5
Copyright © 2009 by Pearson Education, Inc.
4 SF
x
0.048 = 5.304
2 SF
=
calculator
5.3 (rounded)
2 SF
63
Learning Check
64
Solution
Select the answer with the correct number of
significant figures.
A. 2.19 x 4.2
1) 9
=
2) 9.2
3) 9.198
A. 2.19 x 4.2 = 2) 9.2
B. 4.311 ÷ 0.07
= 3) 60
C. 2.54 x 0.0028 = 2) 11
0.0105 x 0.060
B. 4.311 ÷ 0.07
1) 61.59
=
2) 62
3) 60
On a calculator, enter each number, followed by
the operation key.
C. 2.54 x 0.0028 =
0.0105 x 0.060
1) 11.3
2) 11
2.54 x 0.0028 ÷ 0.0105 ÷ 0.060 = 11.28888889
= 11 (rounded)
3) 0.041
65
66
Addition and Subtraction
Learning Check
When adding or subtracting
•
the final answer must have the same number of
decimal places as the measurement with the fewest
decimal places.
•
use rounding rules to adjust the number of digits in
the answer.
25.2
+ 1.34
26.54
26.5
For each calculation, round off the calculated answer
to give a final answer with the correct number of
significant figures.
A. 235.05 + 19.6 + 2 =
1) 257
2) 256.7
B.
one decimal place
two decimal places
calculated answer
final answer with one decimal place
58.925 - 18.2 =
1) 40.725
2) 40.73
3) 256.65
3) 40.7
67
68
Solution
Chapter 1
A. 235.05
+19.6
+ 2
256.65 round to 257
B.
58.925
-18.2
40.725 round to 40.7
Measurements
1.5
Prefixes and Equalities
Answer (1)
Answer (3)
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69
70
Prefixes
Metric and SI Prefixes
A prefix
l in front of a unit increases or decreases the size of that unit.
l makes units larger or smaller than the initial unit by one or more
factors of 10.
l indicates a numerical value.
prefix
1 kilometer
=
=
value
1000 meters
1 kilogram
=
1000 grams
71
72
Learning Check
Solution
Indicate the unit that matches the description.
Indicate the unit that matches the description.
1. A mass that is 1000 times greater than 1 gram.
1) kilogram
2) milligram
3) megagram
1. A mass that is 1000 times greater than 1 gram.
1) kilogram
2. A length that is 1/100 of 1 meter.
1) decimeter
2) centimeter
2. A length that is 1/100 of 1 meter.
2) centimeter
3) millimeter
3. A unit of time that is 1/1000 of a second.
3) millisecond
3. A unit of time that is 1/1000 of a second.
1) nanosecond 2) microsecond
3) millisecond
73
Solution
Learning Check
Select the unit you would use to measure
A. your height.
1) millimeters
A. your height.
2) meters
2) meters
3) kilometers
2) grams
3) kilograms
B. your mass.
3) kilograms
C. the distance between two cities.
3) kilometers
D. the width of an artery.
1) millimeters
B. your mass.
1) milligrams
C. the distance between two cities.
1) millimeters
2) meters
3) kilometers
D. the width of an artery.
1) millimeters
74
2) meters
3) kilometers
75
Metric Equalities
76
Measuring Length
An equality
l states the same measurement in two different units.
l can be written using the relationships between two metric
units.
Example: 1 meter is the same as 100 cm and 1000 mm.
1 m = 100 cm
1 m = 1000 mm
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77
78
Measuring Volume
Measuring Mass
l Several equalities can be
written for mass in the
metric (SI) system
1 kg = 1000 g
1 g = 1000 mg
1 mg = 0.001 g
1 mg = 1000 µg
Copyright © 2009 by Pearson Education, Inc.
79
Learning Check
80
Solution
Indicate the unit that completes each of the following
equalities.
Indicate the unit that completes each of the following
equalities.
A.
1000 m = ___
1) 1 mm
2) 1 km
2) 1 dm
A. 2)
1000 m = 1 km
B.
0.001 g = ___
1) 1 mg
2) 1 kg
2) 1 dg
B. 1)
0.001 g = 1 mg
C.
0.1 s
1) 1 ms
2) 1 cs
2) 1 ds
C. 3)
0.1 s
D.
0.01 m = ___
1) 1 mm
2) 1 cm
2) 1 dm
D. 2)
0.01 m = 1 cm
= ___
= 1 ds
81
Learning Check
82
Solution
Complete each of the following equalities.
Complete each of the following equalities.
A. 1 kg =
___
1) 10 g
2) 100 g
3) 1000 g
B. 1 mm =
___
1) 0.001 m
2) 0.01 m
3) 0.1 m
A. 1 kg
= 1000 g
B. 1 mm = 0.001 m
83
(3)
(1)
84
Chapter 1
Measurements
Equalities
Equalities
1.6
Writing Conversion Factors
• use two different units to describe the same measured
amount.
• are written for relationships between units of the metric
system, U.S. units, or between metric and U.S. units.
For example,
1m
=
1000 mm
1 lb
=
16 oz
2.20 lb =
Copyright © 2009 by Pearson Education, Inc.
1 kg
85
Exact and Measured Numbers in
Equalities
86
Some Common Equalities
Equalities between units in
• the same system of measurement are definitions
39.4 in.
that use exact numbers.
1.06 qt
946 mL = 1 qt
• different systems of measurement (metric and U.S.)
use measured numbers that have significant figures.
Exception:
The equality 1 in. = 2.54 cm has been defined as an
exact relationship. Thus, 2.54 is an exact number.
87
Equalities on Food Labels
88
Conversion Factors
The contents of packaged foods
A conversion factor is
• obtained from an equality.
• in the U.S. are listed in both metric and U.S.
units.
Equality: 1 in. = 2.54 cm
written as a fraction (ratio) with a numerator and
denominator.
• indicate the same amount of a substance in
two different units.
• inverted to give two conversion factors for every
equality.
1 in.
and 2.54 cm
2.54 cm
1 in.
Copyright © 2009 by Pearson Education, Inc.
89
90
Learning Check
Solution
Write conversion factors from the equality for each
of the following.
Write conversion factors from the equality for each of the
following.
A. liters and mL
A. 1 L = 1000 mL
1L
and 1000 mL
1000 mL
1L
B. hours and minutes
B. 1 h = 60 min
and
1h
60 min
60 min
1h
C. 1 km = 1000 m
1 km
1000 m
1000 m
1 km
C. meters and kilometers
and
91
Conversion Factors in a Problem
92
Percent as a Conversion Factor
A conversion factor
• may be obtained from information in a word problem.
• is written for that problem only.
Example 1:
The price of one pound (1 lb) of red peppers is $2.39.
1 lb red peppers and $2.39
$2.39
1 lb red peppers
Example 2:
The cost of one gallon (1 gal) of gas is $2.89.
1 gallon of gas and $2.89
$2.89
1 gallon of gas
A percent factor
• gives the ratio of the parts to the whole.
parts x 100
whole
uses the same unit in the numerator and denominator.
uses the value 100.
can be written as two factors.
Example: A food contains 30% (by mass) fat.
and 100 g food
30 g fat
100 g food
30 g fat
%
•
•
•
=
93
Percent Factor in a Problem
Smaller Percents: ppm and ppb
Small percents are shown as ppm and ppb.
The thickness of the skin fold at
the waist indicates 11% body
fat. What factors can be
written for percent body fat (in
kg)?
• Parts per million (ppm) = mg part/kg whole
Example: The EPA allows 15 ppm cadmium in food
colors
15 mg cadmium = 1 kg food color
Parts per billion ppb = μ g part/kg whole
Example: The EPA allows10 ppb arsenic in public
water
10 μ g arsenic = 1 kg water
Percent factors using kg:
11 kg fat
and 100 kg mass
100 kg mass
11 kg fat
94
Copyright © 2009 by Pearson Education, Inc.
95
96
Smaller Percents: ppm and ppb
Arsenic in Water
Small percents are shown as ppm and ppb.
• Parts per million (ppm) = mg part/kg whole
Example: The EPA allows 15 ppm cadmium in food
colors
15 mg cadmium = 1 kg food color
• Parts per billion ppb = μ g part/kg whole
Example: The EPA allows10 ppb arsenic in public
water
10 μ g arsenic = 1 kg water
Write the conversion factors for 10 ppb arsenic
in public water from the equality
10 μ g arsenic = 1 kg water.
Conversion factors:
10 μ g arsenic
1 kg water
and
1 kg water
10 μ g arsenic
97
Study Tip: Conversion Factors
98
Learning Check
Write the equality and conversion factors for each of the
following.
An equality
• is written as a fraction (ratio).
• provides two conversion factors that are the
A. meters and centimeters
inverse of each other.
B. jewelry that contains 18% gold
C. One gallon of gas is $2.89
99
Risk-Benefit Assessment
Solution
A. 1 m = 100 cm
1m
and
100 cm
A measurement of toxicity is
100 cm
1m
• LD50 or “lethal dose.”
B. 100 g jewelry = 18 g gold
• the concentration of the substance that causes death
18 g gold and 100 g jewelry
100 g jewelry
18 g gold
in 50% of the test animals.
C. 1 gal gas = $2.89
1 gal
$2.89
100
and
• in milligrams per kilogram (mg/kg or ppm) of body mass.
$2.89
1 gal
• in micrograms per kilogram (μ g/kg or ppb) of body
mass.
101
102
Learning Check
Solution
The LD50 for aspirin is 1100 ppm. How many grams of
aspirin would be lethal in 50% of persons with a body
mass of 85 kg?
The LD50 for aspirin is 1100 ppm. How many grams of
aspirin would be lethal in 50% of persons with a body
mass of 85 kg?
A. 9.4 g
B. 94 g
B. 94 g
1100 ppm = 1100 mg/kg body mass
85 kg ×
C. 94 000 g
Copyright © 2009 by Pearson Education, Inc.
1100 mg
1g
×
= 94 g
kg
1000 mg
103
Chapter 1
Measurements
104
Given and Needed Units
1.7
Problem Solving
To solve a problem,
• identify the given unit.
• identify the needed unit.
Example:
A person has a height of 2.0 meters.
What is that height in inches?
The given unit is the initial unit of height.
given unit = meters (m)
The needed unit is the unit for the answer.
needed unit = inches (in.)
Copyright © 2009 by Pearson Education, Inc.
105
Learning Check
106
Solution
An injured person loses 0.30 pints of blood. How
many milliliters of blood would that be?
An injured person loses 0.30 pints of blood. How
many milliliters of blood would that be?
Identify the given and needed units given in this
problem.
Identify the given and needed units given in this
problem.
Given unit
= _______
Given unit
Needed unit = _______
=
Needed unit =
107
pints
milliliters
108
Problem Setup
Problem Setup
• Write the given and needed units.
• Write a plan to convert the given unit to the needed unit.
• Write equalities and conversion factors that connect the
units.
• Write the given and needed units.
• Write a plan to convert the given unit to the needed unit.
• Write equalities and conversion factors that connect the
units.
• Use conversion factors to cancel the given unit and
• Use conversion factors to cancel the given unit and
provide the needed unit.
Unit 1
x
Given
unit
x
provide the needed unit.
Unit 2
= Unit 2
Unit 1
Conversion = Needed
factor
unit
Unit 1
x
Given
unit
x
Unit 2
= Unit 2
Unit 1
Conversion = Needed
factor
unit
109
Setting Up a Problem
Learning Check
How many minutes are in 2.5
hours?
Given unit =
2.5 h
Needed unit =
min
Plan =
h
min
A rattlesnake is 2.44 m long. How many cm long is
the snake?
1) 2440 cm
2) 244 cm
3) 24.4 cm
Set Up Problem
Given Conversion
unit
factor
110
Needed unit
2.5 h x 60 min = 150 min (2 SF)
1h
Copyright © 2009 by Pearson Education, Inc.
111
Solution
112
Using Two or More Factors
• Often, two or more conversion factors are required
A rattlesnake is 2.44 m long. How many cm long
is the snake?
2)
to obtain the unit needed for the answer.
Unit 1
Unit 2
Unit 3
244 cm
• Additional conversion factors are placed in the
setup problem to cancel each preceding unit.
Given
Conversion
Needed
unit
factor
unit
2.44 m x 100 cm = 244 cm
1m
Given unit x factor 1
Unit 1
x Unit 2
Unit 1
113
x factor 2
x Unit 3
Unit 2
= needed unit
= Unit 3
114
Study Tip: Check Unit
Cancellation
Example: Problem Solving
• Be sure to check the unit cancellation in the setup.
• The units in the conversion factors must cancel to
How many minutes are in 1.4 days?
Given unit: 1.4 days
give the correct unit for the answer.
Factor 1
Plan:
Factor 2
days
h
Set Up Problem:
1.4 days x 24 h x 60 min
1 day
1h
2 SF
Exact
What is wrong with the following setup?
min
1.4 day
= 2.0 x 103 min
(rounded)
Exact
= 2 SF
x 1 day x 1 h
24 h
60 min
=
day2/min is not the unit needed
Units don’t cancel properly.
115
Solution
Learning Check
A bucket contains 4.65 L of water. Write the setup
for the problem and calculate the gallons of water in
the bucket.
Plan:
L
Equalities:
1.06 qt = 1 L
1 gal = 4 qt
116
qt
Set Up Problem:
4.65 L x x 1.06 qt
Given: 4.65 L
Needed: gallons
Plan:
qt
L
gallon
Equalities: 1.06 qt = 1 L; 1 gal = 4 qt
gallon
Set Up Problem:
x
1L
1 gal
4.65 L x x 1.06 qt
1L
3 SF
3 SF
= 1.23 gal
4 qt
x
1 gal
4 qt
exact
= 1.23 gal
3 SF
117
118
Solution
Learning Check
Equalities:
If a ski pole is 3.0 feet in length, how long is the
ski pole in mm?
1 ft = 12 in.
1 in. = 2.54 cm
Set Up Problem:
3.0 ft x 12 in.
1 ft
1 cm = 10 mm
x 2.54 cm x 10 mm =
1 in.
1 cm
Calculator answer = 914.4 mm
Final answer
= 910 mm
(2 SF rounded)
Check Factors in Setup:
Check Final Unit:
119
Units cancel properly
mm
120
Learning Check
Solution
Given: 7500 ft
If your pace on a treadmill is 65 meters per minute,
how many minutes will it take for you to walk a
distance of 7500 feet?
Plan: ft
65 m/min
in.
cm
cm
Equalities: 1 ft = 12 in.
Need: min
m
1 in. = 2.54 cm
min
1 m = 100 cm
1 min = 65 m (walking pace)
Set Up Problem:
7500 ft x
12 in. x
1 ft
2.54 cm
1 in.
x 1m
x 1 min
100 cm 65 m
= 35 min final answer (2 SF)
121
Percent Factor in a Problem
122
Learning Check
If the thickness of the skin fold
at the waist indicates an 11%
body fat, how much fat is in a
person with a mass of 86 kg?
How many lb of sugar are in 120 g of candy if the
candy is 25% (by mass) sugar?
percent factor
86 kg mass x
11 kg fat
100 kg mass
Copyright © 2009 by Pearson Education, Inc.
= 9.5 kg of fat
123
124
Chapter 1
Solution
Measurements
1.8
Density
How many lb of sugar are in 120 g of candy if the
candy is 25%(by mass) sugar?
percent factor
120 g candy x 1 lb candy
454 g candy
x 25 lb sugar
100 lb candy
= 0.066 lb of sugar
Copyright © 2009 by Pearson Education, Inc.
125
126
Density
Densities of Common Substances
Density
• compares the mass of an object to its volume.
• is the mass of a substance divided by its
(at 4 °C)
volume.
Density Expression
Density = mass = g or g
volume
mL
cm3
= g/cm3
Note: 1 mL = 1 cm3
127
Learning Check
Solution
Given: mass = 50.0 g
volume = 2.22 cm3
Plan: Place the mass and volume of the osmium metal
in the density expression.
Osmium is a very dense metal. What is its density
in g/cm3 if 50.0 g of osmium has a volume of 2.22
cm3?
2)
1) 2.25 g/cm3
2) 22.5
128
D = mass
g/cm3
3) 111 g/cm3
=
volume
calculator answer
final answer
50.0 g
2.22 cm3
= 22.522522 g/cm3
= 22.5 g/cm3
129
Volume by Displacement
Density Using Volume Displacement
• A solid completely
The density of the zinc object is
then calculated from its mass
and volume.
submerged in water
displaces its own
volume of water.
• The volume of the
Density =
object is calculated
from the difference in
volume.
45.0 mL - 35.5 mL
= 9.5 mL
= 9.5 cm3
130
mass = 68.60 g = 7.2 g/cm3
volume 9.5 cm3
Copyright © 2009 by Pearson Education, Inc.
Copyright © 2009 by Pearson Education, Inc.
131
132
Learning Check
Solution
Given: 48.0 g
Volume of water
= 25.0 mL
Volume of water + metal
= 33.0 mL
Need: Density (g/mL)
Plan: Calculate the volume difference in cm3 and place
in density expression.
What is the density (g/cm3) of 48.0 g of a metal if the
level of water in a graduated cylinder rises from 25.0
mL to 33.0 mL after the metal is added?
1) 0.17 g/cm3
2) 6.0 g/cm3
3) 380 g/cm3
33.0 mL - 25.0 mL
8.0 mL x
Set Up Problem:
Density = 48.0 g =
8.0 cm3
33.0 mL
25.0 mL
1 cm3
1 mL
object
=
8.0 mL
=
8.0 cm3
6.0 g = 6.0 g/cm3
1 cm3
133
Sink or Float
134
Learning Check
• Ice floats in
Which diagram correctly represents the liquid layers
in the cylinder? Karo (K) syrup (1.4 g/mL); vegetable
(V) oil (0.91 g/mL); water (W) (1.0 g/mL)
water because
the density of ice
is less than the
density of water.
1
2
3
V
W
K
W
K
V
K
V
W
• Aluminum sinks
because its
density is
greater than the
density of water.
Copyright © 2009 by Pearson Education, Inc.
135
Solution
136
Learning Check
The density of octane, a component of gasoline, is
0.702 g/mL. What is the mass, in kg, of 875 mL of
octane?
1)
1) 0.614 kg
V
W
K
vegetable oil 0.91 g/mL
2) 614 kg
water 1.0 g/mL
3) 1.25 kg
Karo syrup 1.4 g/mL
137
138
Study Tip: Density as a
Conversion Factor
Solution
1) 0.614 kg
Density can be written as an equality.
• For a substance with a density of 3.8 g/mL, the
equality is
3.8 g = 1 mL
Given: D = 0.702 g/mL
Plan:
Equalities: density
and
• From this equality, two conversion factors can be
V= 875 mL
mL → g → kg
0.702 g
= 1 mL
1 kg = 1000 g
written for density.
Conversion
factors
3.8 g
1 mL
and
Setup: 875 mL x 0.702 g x 1 kg = 0.614 kg
1 mL
1000 g
1 mL
3.8 g
density
factor
metric
factor
139
Learning Check
140
Solution
2) 0.31 L
Given: D = 0.92 g/mL
mass = 285 g
Need: volume in liters
Plan: g → mL → L
Equalities: 1 mL = 0.92 g and 1 L = 1000 mL
Set Up Problem:
If olive oil has a density of 0.92 g/mL, how many
liters of olive oil are in 285 g of olive oil?
1) 0.26 L
2) 0.31 L
3) 310 L
285 g x 1 mL x
0.92 g
density
factor
inverted
1L
=
1000 mL
metric
factor
141
Learning Check
2) 2.0 L
142
Solution
A group of students collected 125 empty aluminum
cans to take to the recycling center. If 21 cans
make 1.0 lb aluminum, how many liters of
aluminum (D=2.70 g/cm3) are obtained from the
cans?
1) 1.0 L
0.31 L
1) 1.0 L
1L
125 cans x 1.0 lb x 454 g x 1 cm3 x 1 mL x
21 cans 1 lb
2.70 g 1 cm3 1000 mL
density
factor
inverted
3) 4.0 L
= 1.0 L
143
144
Learning Check
Solution
Which of the following samples of metals will displace
the greatest volume of water?
1
25 g of aluminum
2.70 g/mL
2
45 g of gold
19.3 g/mL
1)
25 g of aluminum
2.70 g/mL
Plan: Calculate the volume for each metal and select
the metal sample with the greatest volume.
=
9.3 mL aluminum
1) 25 g x 1 mL
2.70 g
=
2.3 mL gold
2) 45 g x 1 mL
19.3 g
=
6.6 mL lead
3) 75 g x 1 mL
11.3 g
3
75 g of lead
11.3 g/mL
density
factors
145
146