Transcript Problem Solving Using Dimensional Analysis

Chapter 1
Measurements
1.6
Writing Conversion Factors
Copyright © 2009 by Pearson Education, Inc.
1
Equalities
Equalities
• 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 =
1 kg
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Exact and Measured Numbers in
Equalities
Equalities between units in
• the same system of measurement are definitions that use
exact numbers.
• 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.
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Some Common Equalities
39.4 in.
1.06 qt
946 mL = 1 qt
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Equalities on Food Labels
The contents of packaged foods
• in the U.S. are listed in both metric and U.S. units.
• indicate the same amount of a substance in two
different units.
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Conversion Factors
A conversion factor is
• obtained from an equality.
Equality:
1 in. = 2.54 cm
• written as a fraction (ratio) with a numerator and
denominator.
• inverted to give two conversion factors for every equality.
1 in.
and 2.54 cm
2.54 cm
1 in.
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Learning Check
Write conversion factors from the equality for each of the
following.
A. liters and mL
B. hours and minutes
C. meters and kilometers
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Conversion Factors in a Problem
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
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Percent as a Conversion Factor
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.
30 g fat
100 g food
and
100 g food
30 g fat
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Percent Factor in a Problem
The thickness of the skin fold at
the waist indicates 11% body
fat. What factors can be
written for percent body fat (in
kg)?
Percent factors using kg:
11 kg fat
100 kg mass
and 100 kg mass
11 kg fat
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Smaller Percents: ppm and ppb
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
=  g part/kg whole
Example: The EPA allows10 ppb arsenic in public water
10  g arsenic = 1 kg water
• Parts per billion (ppb)
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Arsenic in Water
Write the conversion factors for 10 ppb arsenic
in public water from the equality
10  g arsenic = 1 kg water.
Conversion factors:
10  g arsenic
1 kg water
and
1 kg water
10  g arsenic
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Study Tip: Conversion
Factors
An equality
• is written as a fraction (ratio).
• provides two conversion factors that are the inverse of
each other.
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Learning Check
Write the equality and conversion factors for each of the
following.
A. meters and centimeters
B. jewelry that contains 18% gold
C. One gallon of gas is $2.89
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Risk-Benefit Assessment
A measurement of toxicity is
• LD50 or “lethal dose.”
• the concentration of the substance that causes death
in 50% of the test animals.
• in milligrams per kilogram (mg/kg or ppm) of body mass.
• in micrograms per kilogram ( g/kg or ppb) of body mass.
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Learning Check
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
C. 94 000 g
Copyright © 2009 by Pearson Education, Inc.
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Chapter 1
Measurements
1.7
Problem Solving
Copyright © 2009 by Pearson Education, Inc.
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Given and Needed Units
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.)
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Learning Check
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.
Given unit
= _______
Needed unit = _______
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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.
Use conversion factors to cancel the given unit and provide the
needed unit.
Unit 1
Given
unit
x
x
Unit 2
Unit 1
Conversion
factor
= Unit 2
= Needed
unit
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Setting Up a Problem
How many minutes are in 2.5 hours?
Given unit
=
2.5 h
Needed unit =
min
Plan =
h
min
Set Up Problem
Given Conversion
unit
factor
Needed unit
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Learning Check
A rattlesnake is 2.44 m long. How many cm long is the
snake?
1)
2)
3)
2440 cm
244 cm
24.4 cm
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Using Two or More Factors
• Often, two or more conversion factors are required to
obtain the unit needed for the answer.
Unit 1
Unit 2
Unit 3
• Additional conversion factors are placed in the setup
problem to cancel each preceding unit.
Given unit x factor 1 x factor 2
= needed unit
Unit 1
x Unit 2 x Unit 3
= Unit 3
Unit 1
Unit 2
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Example: Problem Solving
How many minutes are in 1.4 days?
Given unit: 1.4 days
Factor 1
Plan:
days
Factor 2
h
min
Set Up Problem:
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Study Tip: Check Unit
Cancellation
• Be sure to check the unit cancellation in the setup.
• The units in the conversion factors must cancel to give the
correct unit for the answer.
What is wrong with the following setup?
1.4 day
x 1 day
24 h
x
1h
60 min
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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:
Equalities:
Set Up Problem:
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Learning Check
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?
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Percent Factor in a Problem
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?
percent factor
86 kg mass x
11 kg fat
100 kg mass
Copyright © 2009 by Pearson Education, Inc.
= 9.5 kg of fat
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Learning Check
How many lb of sugar are in 120 g of candy if the candy
is 25% (by mass) sugar?
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Chapter 1
Measurements
1.8
Density
Copyright © 2009 by Pearson Education, Inc.
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Density
Density
• compares the mass of an object to its volume.
• is the mass of a substance divided by its volume.
Density Expression
Density = mass = g or g
volume
mL
cm3
= g/cm3
Note: 1 mL = 1 cm3
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Densities of Common Substances
(at 4 °C)
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Learning Check
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?
1) 2.25 g/cm3
2) 22.5 g/cm3
3) 111 g/cm3
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Volume by Displacement
• A solid completely
submerged in water
displaces its own volume
of water.
• The volume of the
object is calculated from
the difference in volume.
45.0 mL - 35.5 mL
= 9.5 mL
= 9.5 cm3
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Density Using Volume Displacement
The density of the zinc object is
then calculated from its mass
and volume.
Density =
mass = 68.60 g = 7.2 g/cm3
volume 9.5 cm3
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Learning Check
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
25.0 mL
2) 6.0 g/cm3 3) 380 g/cm3
33.0 mL
object
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Sink or Float
• Ice floats in water
because the density
of ice is less than
the density of
water.
• Aluminum sinks
because its density
is greater than the
density of water.
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Learning Check
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)
1
2
3
V
W
K
W
K
V
K
V
W
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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) 0.614 kg
2) 614 kg
3) 1.25 kg
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Study Tip: Density as a
Conversion Factor
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
• From this equality, two conversion factors can be written for
density.
Conversion 3.8 g
factors
and
1 mL
1 mL
3.8 g
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Learning Check
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)
2)
3)
0.26 L
0.31 L
310 L
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Learning Check
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
2) 2.0 L
3) 4.0 L
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Learning Check
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
3
75 g of lead
11.3 g/mL
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