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Standards for Measurement
Chapter 2
Hein and Arena
Version 2.0
12th Edition
Eugene Passer
Chemistry Department
Bronx Community
1 College
© John Wiley and Sons, Inc
Chapter Outline
2.1 Scientific Notation
2.5 The Metric System
2.2 Measurement and
Uncertainty
2.6 Problem Solving
2.7 Measuring Mass and Volume
2.3 Significant Figures
2.4 Significant Figures in
Calculations
2.8 Measurement of Temperature
2.9 Density
2
2.1
Scientific Notation
3
• Very large and very small numbers are
often encountered in science.
602200000000000000000000
0.00000000000000000000625
• Very large and very small numbers like
these are awkward and difficult to
work with.
4
A method for representing these numbers
in a simpler form is called scientific
notation.
23
602200000000000000000000
6.022 x 10
-21
0.00000000000000000000625
6.25 x 10
5
Scientific Notation
• Move the decimal point in the original
number so that it is located after the first
nonzero digit.
• Follow the new number by a multiplication
sign and 10 with an exponent (power).
• The exponent is equal to the number of
places that the decimal point was shifted.
6
Write 6419 in scientific notation.
decimal after
first nonzero
digit
power of 10
1
2
3
6.419
641.9x10
64.19x10
6419.
6419
x 10
7
Write 0.000654 in scientific notation.
decimal after
first nonzero
digit
6.54 x
0.000654
0.00654
0.0654
0.654
power of 10
-4
-2
-1
-3
10
8
2.2
Measurement
and Uncertainty
9
Measurements
• Experiments are performed.
• Measurements are made.
10
Form of a Measurement
numerical value
70.0 kilograms = 154 pounds
unit
11
Significant Figures
• The number of digits that are known
plus one estimated digit are considered
significant in a measured quantity
known
5.16143estimated
12
Significant Figures
• The number of digits that are known
plus one estimated digit are considered
significant in a measured quantity
certain
6.06320 uncertain
13
Reading a Thermometer
14
The
temperature
Temperature
is
oC is expressed
21.2
estimated
to be
oC. The last 2 is
to
3 significant
21.2
figures.
uncertain.
15
The
temperature
Temperature
is
oC is expressed
22.0
estimated
to be
to
3 osignificant
22.0
C. The last 0 is
figures.
uncertain.
16
The
temperature
Temperature
is
oC isto
22.11
expressed
estimated
be
oC. The last 1
to
4 significant
22.11
figures.
is uncertain.
17
Exact Numbers
• Exact numbers have an infinite number
of significant figures.
• Exact numbers occur in simple
counting operations
12345
• Defined numbers are exact.
12 inches
100
centimeters
= 1 foot
= 1 meter
18
2.3
Significant Figures
19
Significant Figures
All nonzero numbers are significant.
461
20
Significant Figures
All nonzero numbers are significant.
461
21
Significant Figures
All nonzero numbers are significant.
461
22
Significant Figures
All nonzero numbers are significant.
3 Significant
Figures
461
23
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
401
24
Significant Figures
A zero is significant when it is between
nonzero digits.
5 Significant
Figures
93 . 006
25
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
9 . 03
26
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
55 . 000
27
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
2 . 1930
28
Significant Figures
A zero is not significant when it is before
the first nonzero digit.
1 Significant
Figure
0 . 006
29
Significant Figures
A zero is not significant when it is before
the first nonzero digit.
3 Significant
Figures
0 . 709
30
Significant Figures
A zero is not significant when it is at the
end of a number without a decimal point.
1 Significant
Figure
50000
31
Significant Figures
A zero is not significant when it is at the
end of a number without a decimal point.
4 Significant
Figures
68710
32
Rounding Off Numbers
33
• Often when calculations are performed on a
calculator extra digits are present in the
results.
• It is necessary to drop these extra digits so
as to express the answer to the correct
number of significant figures.
• When digits are dropped, the value of the
last digit retained is determined by a process
known as rounding off numbers.
34
Rules for Rounding Off
Rule 1. When the first digit after those you
want to retain is 4 or less, that digit and all
others to its right are dropped. The last digit
retained is not changed.
4 or less
80.873
35
Rounding Off Numbers
Rule 1. When the first digit after those you
want to retain is 4 or less, that digit and all
others to its right are dropped. The last digit
retained is not changed.
4 or less
1.875377
36
Rounding Off Numbers
Rule 2. When the first digit after those you
want to retain is 5 or greater, that digit and all
others to its right are dropped. The last digit
retained is increased by 1.
drop
5 or
these
greater
figures
increase by 1
6
5.459672
37
2.4
Significant Figures in
Calculations
38
The results of a calculation based on
measurements cannot be more precise
than the least precise measurement.
39
Multiplication or Division
40
In multiplication or division, the answer
must contain the same number of
significant figures as in the measurement
that has the least number of significant
figures.
41
2.3 has two significant
figures.
(190.6)(2.3) = 438.38
190.6 has four
significant figures.
Answer given
by calculator.
The answer should have two significant
figures because 2.3 is the number with
the fewest significant figures.
Round off this
digit to four.
Drop these three
digits.
438.38
The correct answer is 440 or 4.4 x 102
42
Addition or Subtraction
43
The results of an addition or a
subtraction must be expressed to the
same precision as the least precise
measurement.
44
The result must be rounded to the same
number of decimal places as the value
with the fewest decimal places.
45
Add 125.17, 129 and 52.2
Least precise number.
Answer given
by calculator.
125.17
129.
52.2
306.37
Round off to the
Correct answer.
nearest unit.
306.37
46
1.039 - 1.020
Calculate
1.039
Answer given
by calculator.
1.039 - 1.020
= 0.018286814
1.039
Two
1.039 - 1.020 = 0.019
0.019
= 0.018286814
1.039
significant
figures.
0.018286814
0.018
286814
The answer should have two significant
Drop these
Correct
answer.
figures because 0.019 is the number
6 digits.
with the fewest significant figures.
47
2.5
The Metric System
48
• The metric or International System (SI,
Systeme International) is a decimal system of
units.
• It is built around standard units.
• It uses prefixes representing powers of 10 to
express quantities that are larger or smaller
than the standard units.
49
International System’s
Standard Units of Measurement
Quantity
Length
Mass
Name of Unit
meter
kilogram
Temperature
Kelvin
Abbreviation
m
kg
K
Time
second
Amount of substance m
s
mole
Electric Current
ampere
A
Luminous Intensity
candela
cd50
Common Prefixes and Numerical Values for SI Units
Power of 10
Prefix
Symbol
Numerical Value
Equivalent
giga
G
1,000,000,000
109
mega
M
1,000,000
106
kilo
k
1,000
103
hecto
h
100
102
deca
da
10
101
—
—
1
100
51
Prefixes and Numerical Values for SI Units
Numerical Value
Power of 10
Equivalent
Prefix
Symbol
deci
d
0.1
10-1
centi
c
0.01
10-2
milli
m
0.001
10-3
micro

0.000001
10-6
nano
n
0.000000001
10-9
pico
p
0.000000000001
10-12
femto
f
0.00000000000001
10-15
52
Measurement of Length
53
The standard unit of length in the SI
system is the meter. 1 meter is the distance
1
that light travels in a vacuum during
299,792,458
of a second.
54
• 1 meter = 39.37 inches
• 1 meter is a little longer than a yard
55
Metric Units of Length
Unit
Abbreviation Metric Equivalent
Exponential
Equivalent
kilometer
meter
km
m
1,000 m
1m
103 m
100 m
decimeter
dm
0.1 m
10-1 m
centimeter
cm
0.01 m
10-2 m
millimeter
mm
0.001 m
10-3 m
micrometer
m
0.000001 m
10-6 m
nanometer
nm
0.000000001 m
10-9 m
angstrom
Å
0.0000000001 m
10-10 m
56
2.6
Problem Solving
57
Dimensional Analysis
Dimensional analysis converts one
unit to another by using conversion
factors.
unit1 x conversion factor = unit2
58
Basic Steps
1. Read the problem carefully. Determine
what is known and what is to be solved
for and write it down.
It is important to label all factors and
units with the proper labels.
59
Basic Steps
2. Determine which principles are
involved and which unit relationships
are needed to solve the problem.
– You may need to refer to tables for
needed data.
3. Set up the problem in a neat,
organized and logical fashion.
– Make sure unwanted units cancel.
– Use sample problems in the text as
guides for setting up the problem.
60
Basic Steps
4. Proceed with the necessary mathematical
operations.
– Make certain that your answer contains the
proper number of significant figures.
5. Check the answer to make sure it is
reasonable.
61
Length Conversion
62
How many millimeters are there in 2.5 meters?
The
conversion
factor
must
unit1 x conversion factor = unit2
accomplish two things:
m x conversion factor = mm
• It must cancel
meters.
• It must introduce
millimeters
63
The conversion factor takes a fractional
form.
mm
mx
= mm
m
64
The conversion factor
is derived from the
equality.
1 m = 1000 mm
Divide both sides by 1000 mm
1m
1000 mm
conversion
=
 1
factor
1000m 1000 mm
Divide both sides by 1 m
1 m 1000 mm
conversion
=
 1
factor
1m
1m
65
How many millimeters are there in 2.5 meters?
Use the conversion factor with millimeters in the
numerator and meters in the denominator.
1000 mm
1m
1000 mm
= 2500 mm
2.5 m x
3
1m
2.5 x 10 mm
66
Convert 16.0 inches to centimeters.
Use this
conversio
n factor
2.54 cm
1 in
2.54 cm
16.0 in x
= 40.6 cm
1 in
67
Convert 16.0 inches to centimeters.
68
Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by
writing down conversion factors in succession.
cm  m  meters
1m
10 μm
7
3.7 x 10 cm x
x
= 3.7 x 10 μm
1m
100 cm
6
3
69
Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by
a series of two conversion factors.
cm  m  meters
1m
1
3.7 x 10 cm x
= 3.7 x 10 m
100 cm
3
10 μm
7
3.7 x 10 m x
= 3.7 x 10 μm
1m
6
1
70
2.7
Measuring
Mass and Volume
71
Mass
72
The standard unit of mass in the SI
system is the kilogram. 1 kilogram is
equal to the mass of a platinum-iridium
cylinder kept in a vault at Sevres, France.
1 kg = 2.205 pounds
73
Metric Units of mass
Unit
Abbreviation Gram Equivalent
Exponential
Equivalent
kilogram
gram
kg
g
1,000 g
1g
103 g
100 g
decigram
dg
0.1 g
10-1 g
centigram
cg
0.01 g
10-2 g
milligram
mg
0.001 g
10-3 g
microgram
g
0.000001 g
10-6 g
74
Convert 45 decigrams to grams.
1 g = 10 dg
1g
45 dg x
= 4.5 g
10 dg
75
An atom of hydrogen weighs 1.674 x 10-24 g.
How many ounces does the atom weigh?
Grams can be converted to ounces using a series
of two conversion factors.
1 lb = 454 g
1.674 x 10
-24
1 lb
-27
gx
 3.69 x 10 lb
454 g
16 oz = 1 lb
3.69 x 10
-27
16 oz
-26

5.90
x
10
oz76
x
lb
1 lb
An atom of hydrogen weighs 1.674 x 10-24 g.
How many ounces does the atom weigh?
Grams can be converted to ounces using a single
linear expression by writing down conversion
factors in succession.
1 lb
16 oz
-26
x

5.90
x
10
oz
x
1.674 x 10 g
454 g
1 lb
-24
77
Volume
78
• Volume is the amount of space
occupied by matter.
• In the SI system the standard unit of
volume is the cubic meter (m3).
• The liter (L) and milliliter (mL) are the
standard units of volume used in most
chemical laboratories.
79
80
Convert 4.61 x 102 microliters to milliliters.
Microliters can be converted to milliliters using
a series of two conversion factors.
L  L  mL
1L
-4
x

4.61x10
L
4.61x10 μL
6
10 μL
2
1000 mL
-1
4.61x10 L x
= 4.61 x 10 mL
1L
81
-4
Convert 4.61 x 102 microliters to milliliters.
Microliters can be converted to milliliters using
a linear expression by writing down conversion
factors in succession.
L  L  mL
1L
1000 mL
-1
4.61x10 μL x 6
x
= 4.61 x 10 mL
10 μL
1L
2
82
2.8
Measurement of
Temperature
83
Heat
• A form of energy that is associated
with the motion of small particles of
matter.
• Heat refers to the quantity of this
energy associated with the system.
• The system is the entity that is being
heated or cooled.
84
Temperature
• A measure of the intensity of heat.
• It does not depend on the size of the
system.
• Heat always flows from a region of
higher temperature to a region of lower
temperature.
85
Temperature Measurement
• The SI unit of temperature is the Kelvin.
• There are three temperature scales: Kelvin,
Celsius and Fahrenheit.
• In the laboratory, temperature is commonly
measured with a thermometer.
86
Degree Symbols
degrees Celsius =
oC
Kelvin (absolute) = K
degrees Fahrenheit = oF
87
To convert between the scales, use the
following relationships:
o
K = C + 273.15
o
o
o
F = 1.8 x C + 32
o
F
32
o
o
o
o
= = 1.8 x C
F C- 32
1.8
88
180 Farenheit Degrees
= 100 Celcius degrees
180
=1.8
100
89
It is not uncommon for temperatures in the Canadian
plains to reach –60oF and below during the winter.
What is this temperature in oC and K?
o
o
o
F - 32
C=
1.8
60. - 32
o
C=
= -51 C
1.8
90
It is not uncommon for temperatures in the Canadian
planes to reach –60oF and below during the winter.
What is this temperature in oC and K?
o
K = C + 273.15
o
K = -51 C + 273.15 = 222 K
91
2.9
Density
92
Density is the ratio
of the mass of a
substance to the
volume occupied by
that substance.
mass
d=
volume
93
Mass
is usually
The density
of gases is
expressed
expressed in
in grams
grams per
and
liter.volume in mL
or cm3.
g
ddd=== 3
mL
cm
L
94
Density varies with temperature
d
d
4oC
H 2O
o
80 C
H 2O
1.0000 g
g
=
= 1.0000
1.0000 mL
mL
1.0000 g
g
=
= 0.97182
1.0290 mL
mL
95
96
97
Examples
98
A 13.5 mL sample of an unknown liquid has a
mass of 12.4 g. What is the density of the liquid?
M 12.4g
 0.919 g/mL
D 
V 13.5mL
99
A graduated cylinder is filled to the 35.0 mL mark with water.
A copper nugget weighing 98.1 grams is immersed into the
cylinder and the water level rises to the 46.0 mL. What is the
volume of the copper nugget? What is the density of copper?
Vcopper nugget = Vfinal - Vinitial = 46.0mL - 35.0mL = 11.0mL
M
98.1g
D

 8.92 g/mL
V 11.0 mL
46.0 mL
35.0 mL
98.1 g
100
The density of ether is 0.714 g/mL. What is the
mass of 25.0 milliliters of ether?
Method 1
(a) Solve the density equation for mass.
mass
d=
volume
mass = density x volume
(b) Substitute the data and calculate.
0.714 g
25.0 mL x
= 17.9 g
mL
101
The density of ether is 0.714 g/mL. What is the
mass of 25.0 milliliters of ether?
Method 2 Dimensional Analysis. Use density
as a conversion factor. Convert: mL → g
g
The conversion of units is mL x
=g
mL
0.714 g
25.0 ml x
= 17.9 g
mL
102
The density of oxygen at 0oC is 1.429 g/L. What is
the volume of 32.00 grams of oxygen at this
temperature?
Method 1
(a) Solve the density equation for volume.
mass
d=
volume
mass
volume =
density
(b) Substitute the data and calculate.
32.00 g O2
volume =
= 22.40 L
1.429 g O2 /L
103
The density of oxygen at 0oC is 1.429 g/L. What is
the volume of 32.00 grams of oxygen at this
temperature?
Method 2 Dimensional Analysis. Use density
as a conversion factor. Convert: g → L
The conversion of units is
L
gx =L
g
1L
32.00 g O2 x
= 22.40 L O2
1.429 g O2
104
105