Transcript Unit 1 - Chapter 2

Chapter 2 - Measurements
Section 2.1
Units of Measurement
1
Stating a Measurement
In a measurement, a number is followed by a unit.
Observe the following examples of measurements:
Number
35
0.25
225
3.4
2
Unit
m
L
lb
hr
The Metric System (SI)
The metric system or SI (international system) is
• A decimal system based on 10.
• Used in most of the world.
• Used everywhere by scientists.
3
Units in the Metric System
In the metric (SI) system, one unit is used for each
type of measurement:
Measurement
length
volume
mass
time
temperature
4
Metric
meter (m)
liter (L)
gram (g)
second (s)
Celsius (C)
SI
meter (m)
cubic meter (m3)
kilogram (kg)
second (s)
Kelvin (K)
Length Measurement
Length
• Is measured using a
meter stick.
• Has the unit of meter
(m) in the metric (SI)
system.
5
Volume Measurement
Volume
• Is the space occupied by a substance.
• Has the unit liter (L) in metric system.
1 L = 1.057 qt
• Uses the unit m3(cubic meter) in the SI system.
• Is measured using a graduated cylinder.
6
Mass Measurement
The mass of an object
• Is the quantity of material it contains.
• Is measured on a balance.
• Has the unit gram(g) in the metric system.
• Has the unit kilogram(kg) in the SI system.
7
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.
• In the SI system uses the Kelvin(K) scale.
8
Time Measurement
Time measurement
• Has the unit second(s) in both the metric and
SI systems.
• Is based on an atomic clock that uses a
frequency emitted by cesium atoms.
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Chapter 2 - Measurements
Section 2.2
Scientific Notation
10
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 as
8 x 10-6 m
• For a large number such as 2 500 000 s is
written as
2.5 x 106 s
11
Scientific Notation
• A number written in scientific notation contains a
coefficient and a power of 10.
coefficient
1.5
power of ten
x
102
coefficient
7.35
power of ten
x 10-4
• To write a number in scientific notation, the
decimal point is moved after the first digit.
• The spaces moved are shown as a power of ten.
52 000. = 5.2 x 104
0.00378 = 3.78 x 10-3
4 spaces left
12
3 spaces right
Comparing Numbers in
Standard and Scientific
Notation
Number in
Standard Format
Scientific Notation
Diameter of the Earth
12 800 000 m
1.28 x 107 m
Mass of a human
68 kg
6.8 x 101 kg
Mass of a hummingbird
0.002 kg
2 x 10-3 kg
Length of a pox virus
0.000 000 3 cm
3 x 10-7 cm
13
Learning Check
Write each number using the correct scientific
notation for each.
A. 0.000 008
B. 72 000
14
Learning Check
Write each as a standard number.
A. 2.0 x 10-2
B. 1.8 x 105
15
Chapter 2 - Measurements
Section 2.3
Measured Numbers
and Significant Figures
16
Measured Numbers
A measuring tool
• Is used to determine
a quantity such as
height or the mass of
an object.
• Provides numbers for
a measurement
called measured
numbers.
17
Reading a Meter Stick
. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm
• The markings on the meter stick at the end of
the 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 might 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.
18
Known + Estimated Digits
In the length reported as 2.76 cm,
• The digits 2 and 7 are certain (known)
• The final digit 6 was estimated (uncertain)
• All three digits (2.76) are significant including
the estimated digit
19
Significant Figures
in Measured Numbers
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.
20
Significant Figures
21
Counting Significant Figures
All non-zero numbers in a measured number are
significant.
Measurement
38.15 cm
5.6 ft
65.6 lb
122.55 m
22
Number of
Significant Figures
4
2
3
5
Sandwiched Zeros
Sandwiched zeros
• Occur between nonzero numbers.
• Are significant.
Measurement
50.8 mm
2001 min
0.0702 lb
0.40505 m
23
Number of
Significant Figures
3
4
3
5
Trailing Zeros
Trailing zeros
• Follow non-zero numbers in numbers without
decimal points.
• Are usually place holders.
• Are not significant.
Measurement
Number of Significant
Figures
25 000 cm
2
200 kg
1
48 600 mL
3
25 005 000 g
5
24
Leading Zeros
Leading zeros
• Precede non-zero digits in a decimal number.
• Are not significant.
Measurement
0.008 mm
0.0156 oz
0.0042 lb
0.000262 mL
25
Number of
Significant Figures
1
3
2
3
Significant Figures in
Scientific Notation
In scientific notation
• All digits including zeros in the coefficient are
significant.
Scientific Notation
8 x 104 m
8.0 x 104 m
8.00 x 104 m
26
Number of
Significant Figures
1
2
3
Learning Check
State the number of significant figures in each
of the following measurements:
A.
0.030 m
B.
4.050 L
C.
0.0008 g
D.
2.80 m
27
Learning Check
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
1) 0.00307
2) 25.300
3) 2.050 x 103
C. The number of significant figures in 5.80 x 102 is
1) one
3) two
3) three
28
Learning Check
In which set(s) do both numbers contain the
same number of significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150 000
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Examples of Exact Numbers
An exact number is obtained
• When objects are counted.
Counting objects
2 soccer balls
4 pizzas
• From numbers in a defined relationship.
Defined relationships
1 foot = 12 inches
1 meter = 100 cm
30
Exact Numbers
31
Learning Check
Classify each of the following as (1) exact or
(2) measured numbers.
A.__Gold melts at 1064°C.
B.__1 yard = 3 feet
C.__The diameter of a red blood cell is 6 x 10-4 cm.
D.__There are 6 hats on the shelf.
E.__A can of soda contains 355 mL of soda.
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Chapter 2 - Measurements
Section 2.4
Significant Figures in
Calculations
33
Rounding Off Answers
Often, the correct answer
• Requires that a calculated
answer be rounded off.
• Uses rounding rules to
obtain the correct answer
with the proper number of
significant figures.
34
Rounding Off Calculated
Answers
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
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)
35
Adding Significant Zeros
• Occasionally, a calculated answer requires
more significant digits. Then one or more
zeros are added.
Calculated
answer
4
1.5
0.2
12
36
Zeros added to
give 3 significant figures
4.00
1.50
0.200
12.0
Learning Check
Adjust the following calculated answers to give
answers with three significant figures:
A. 824.75 cm
B. 0.112486 g
C. 8.2 L
37
Calculations with Measured
Numbers
In calculations with measured numbers:
• Significant figures are counted or place
values are determined to give the correct
number of figures in the final answer.
38
Multiplication and Division
When multiplying or dividing
• The answer has the same number of significant
figures as in the measurement with the fewest
significant figures.
• Rounding rules are used to obtain the correct
number of significant figures.
Example:
110.5
4 SF
39
x
0.048 = 5.304
2 SF
=
calculator
5.3 (rounded)
2 SF
Learning Check
Give an answer for each with the correct
number of significant figures:
A. 2.19 x 4.2
=
B. 4.311 ÷ 0.07
=
C. 2.54 x 0.0028 =
0.0105 x 0.060
40
Addition and Subtraction
When adding or subtracting use
• The same number of digits as the measurement
whose last digit has the highest place value.
• Use rounding rules to adjust the number of digits
in the answer.
25.2
+ 1.34
26.54
26.5
41
one decimal place
two decimal places
calculated answer
answer with one decimal place
Learning Check
For each calculation, round the answer to give
the correct number of digits.
A. 235.05 + 19.6 + 2 =
B.
42
58.925 - 18.2 =
Chapter 2 - Measurements
Section 2.5
Prefixes and Equalities
43
Prefixes
A prefix
• In front of a unit increases or decreases the size of
that unit.
• Make units larger or smaller than the initial unit by
one or more factors of 10.
• Indicates a numerical value.
prefix
1 kilometer
1 kilogram
44
=
=
=
value
1000 meters
1000 grams
Metric and SI Prefixes
45
Learning Check
Indicate the unit that matches the description:
1. A mass that is 1000 times greater than 1 gram.
2. A length that is 1/100 of 1 meter.
3. A unit of time that is 1/1000 of a second.
46
Metric Equalities
An equality
• States the same measurement in two different units.
• 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
100 cm =
1000 mm
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Metric Equalities for Volume
48
Metric Equalities for Mass
• Several equalities can be written for mass in the
metric (SI) system
1 kg
1g
1 mg
1 mg
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=
=
=
=
1000 g
1000 mg
0.001 g
1000 µg
Learning Check
Indicate the unit that completes each of the following
equalities:
A. 1000 m =
B. 0.001 g =
C. 0.1 s
=
D. 0.01 m =
50
Chapter 2 - Measurements
Section 2.6
Writing Conversion Factors
51
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
2.205 lb =
52
1 kg
Exact and Measured
Numbers in Equalities
Equalities between units of
• The same system are definitions and use exact
numbers.
• Different systems (metric and U.S.) use
measured numbers and count as significant
figures.
53
Some Common Equalities
54
Conversion Factors
A conversion factor
• Is a fraction obtained from an equality.
Equality: 1 in. = 2.54 cm
• Is written as a ratio with a numerator and
denominator.
• Can be inverted to give two conversion factors
for every equality.
1 in.
and 2.54 cm
2.54 cm
1 in.
55
Learning Check
Write conversion factors for each pair of units:
A. liters and mL
B. hours and minutes
C. meters and kilometers
56
Factors with Powers
A conversion factor
• Can be squared or cubed on both sides of the
equality.
Equality
1 in. = 2.54 cm
1 in.
and 2.54 cm
2.54 cm
1 in.
Squared
(1 in.)2 = (2.54 cm)2
(1 in.)2
and (2.54 cm)2
(2.54 cm)2
(1 in.)2
Cubed
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(1 in.)3 = (2.54 cm)3
(1 in.)3
and (2.54 cm)3
(2.54 cm)3
(1 in.)3
Conversion Factors in a
Problem
A conversion factor
• May be obtained from information in a word
problem.
• Is written for that problem only.
Example:
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
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Chapter 2 - Measurements
Section 2.7
Problem Solving
59
Given and Needed Units
To solve a problem
• Identify the given unit.
• Identify the needed unit.
Problem:
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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Setting up a Problem
How many minutes are 2.5 hours?
given unit
=
needed unit =
plan
61
=
Learning Check
A rattlesnake is 2.44 m long. How long is the snake
in centimeters?
62
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 can be placed in the
setup 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
63
Example: Problem Solving
How many minutes are in 1.4 days?
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Check the Unit Cancellation
• Be sure to check your 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 x 1 hr
24 hr
60 min
Units =
day2/min is Not the needed unit
Units don’t cancel properly.
65
Learning Check
What is 165 lb in kg?
66
Learning Check
A bucket contains 4.65 L water. How many
gallons of water is that?
67
Learning Check
If a ski pole is 3.0 feet in length, how long is the
ski pole in mm?
68
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?
69
Chapter 2 - Measurements
Section 2.8
Density
70
Density
Density
• Compares the mass of an object to its
volume.
• Is the mass of a substance divided by its
volume.
• Note: 1 mL = 1 cm3
• Is expressed as
D = mass = g or g = g/cm3
volume
mL
cm3
71
Densities of Common
Substances
72
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?
73
Volume by Displacement
• A solid completely
submerged in water
displaces its own
volume of water.
• The volume of the
solid is calculated
from the volume
difference.
45.0 mL - 35.5 mL
= 9.5 mL
= 9.5 cm3
74
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?
25.0 mL
75
33.0 mL
object
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.
76
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)
77
1
2
3
V
W
K
W
K
V
K
V
W
Density as a Conversion
Factor
Density can be written as an equality.
• For a density of 3.8 g/mL, the equality is:
3.8 g = 1 mL
• From this equality, two conversion factors can be written:
Conversion 3.8 g
factors
1 mL
78
and
1 mL
3.8 g
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?
79
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?
80
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?
81
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
82
2
45 g of gold
19.3 g/mL
3
75 g of lead
11.3 g/mL