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1
Measurement
and
Significant Figures
2
Measurements
• Experiments are performed.
• Numerical values or data are obtained
from these measurements.
3
Significant Figures
• The number of digits that are known
plus one estimated digit are considered
significant in a measured quantity
known
5.16143estimated
4
Significant Figures
• The number of digits that are known
plus one estimated digit are considered
significant in a measured quantity
known
6.06320 estimated
5
Exact Numbers
• Exact numbers have an infinite
numbers of significant figures.
• Exact numbers occur in simple
counting operations
12345
• Defined numbers are exact.
12 inches
100
centimeters
= 1 foot
= 1 meter
6
Significant Figures
All nonzero numbers are significant.
461
7
Significant Figures
All nonzero numbers are significant.
461
8
Significant Figures
All nonzero numbers are significant.
461
9
Significant Figures
All nonzero numbers are significant.
3 Significant
Figures
461
10
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
401
11
Significant Figures
A zero is significant when it is between
nonzero digits.
5 Significant
Figures
93 . 006
12
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
9 . 03
13
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
55 . 000
14
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
2 . 1930
15
Significant Figures
A zero is not significant when it is before
the first nonzero digit.
1 Significant
Figure
0 . 006
16
Significant Figures
A zero is not significant when it is before
the first nonzero digit.
3 Significant
Figures
0 . 709
17
Significant Figures
A zero is not significant when it is at the
end of a number without a decimal point.
1 Significant
Figure
50000
18
Significant Figures
A zero is not significant when it is at the
end of a number without a decimal point.
4 Significant
Figures
68710
19
Rounding
off Numbers
20
Rounding Off Numbers
• Often when calculations are performed
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.
21
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
80.873
22
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
23
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
24
Scientific Notation
of Numbers
25
Scientific notation
If a number is larger than 1
Move decimal point X places left to get a
number between 1 and 10.
1 2 3 , 0 0 0 , 0 0 0. = 1.23 x 108
The resulting number is multiplied by 10X.
Scientific notation
If a number is smaller than 1
Move decimal point X places right to get a
number between 1 and 10.
0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7
The resulting number is multiplied by 10-X.
Examples
Write in Scientific Notation:
25 = 2.5 x 101
8931.5 = 8.9315 x 103
-4
5.93
x
10
0.000593 =
0.0000004 = 4 x 10-7
3
3.210
x
10
3,210. =
Scientific notation
1.44939 × 10-2 =
1.44939E -2
0.0144939
On Calculator
×10
1.44939 EE (-) 2
Means
×10
Change
Sign
29
Significant Figures
in Calculations
30
The results of a calculation cannot be
more precise than the least precise
measurement.
31
Multiplication or Division
32
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.
33
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
34
Addition or Subtraction
35
The results of an addition or a
subtraction must be expressed to the
same precision as the least precise
measurement.
36
The result must be rounded to the same
number of decimal places as the value
with the fewest decimal places.
37
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
38
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.
39
The
Metric System
40
• 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.
41
Standard Units of Measurement
Quantity
Length
Mass
Volume
Temperature
Energy
Pressure
Metric Unit (abbr.)
meter (m)
gram (g)
liter (L)
Celsius (ºC)
calorie (cal)
atmosphere (atm)
SI Unit (abbr.)
meter (m)
kilogram (kg)
cubic meter (m3)
Kelvin (K)
Joule (J)
pascal (Pa)
42
Prefixes and Numerical Values for SI Units
Numerical Value
Power of 10
Equivalent
Prefix
Symbol
exa
peta
E
P
1,000,000,000,000,000,000 1018
1,000,000,000,000,000
1015
tera
T
1,000,000,000,000
1012
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 43
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
atto
a
0.000000000000000001
10-18
44
Measurement of Length
45
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
46
The standard unit of length in the SI
system is the meter. 1 meter is the distance
that light travels in a vacuum during
1
of
a
second.
299,792,458
47
• 1 meter = 39.37 inches
• 1 meter is a little longer than a yard
48
Problem Solving
49
Problem Solving Using
Conversion Factors
Many problems require a change of
one unit to another unit by using
conversion factors (fractions).
unit1 × conversion factor = unit2
How many feet are there in 22.5 inches?
The conversion factor must
unit
x
conversion
factor
=
unit
1
2
accomplish two things:
in x conversion factor = ft
• It must cancel
inches.
• It must introduce
feet
51
The conversion factor takes a fractional
form.
ft
in 
= ft
in
53
Putting in the measured value and the
ratio of feet to inches produces:
1ft
22.5in 
= ft
12in
= 1.875 ft
= 1.88 ft
54
Convert
15
3.7×10
in to miles.
Inches can be converted to miles by writing
down conversion factors in succession.
in  ft  miles
3.7  10 in
15
1 mile
1 ft
10
x
x
= 5.8 10 miles
5280 ft
12 in
55
Convert
30
4.510
cm to kilometers.
Centimeters can be converted to kilometers by
writing down conversion factors in succession.
cm  m  km
1 km
1m
25
x
4.5  10 cm x
= 4.5  10 km
100 cm 1000 m
30
56
Conversion of units
Examples:
10.7 T = ? fl oz
62.04 mi = ? in
5.5 kg = ? mg
9.3 ft = ? cm
5.7 g/ml = ? lbs/qt
3.18 in2 = ? cm2
Conversion of units
Examples:
10.7 T = ? fl oz
1 fl oz
10.7 T x
= 5.35 fl oz
2T
62.04 mi x
62.04 mi = ? in
5.5 kg = ? mg
9.3 ft = ? cm
or 3931000 in
5.5 kg x
9.3 ft
5.7 g/ml = ? lbs/qt
3.18
in2
=?
cm2
5280 ft 12 in
x
= 3.931106 in
1 mi
1 ft
x
1000 g 1000 mg
x
= 5.5 106 mg
1 kg
1g
12 in 2.54 cm
x
= 280 in (not 280.)
1 ft
1 in
12 lb
1 lb
1L
5.7 g
1000 ml

x
x
x
1.06 qt 1 qt
1 ml 454 g
1L
2
2
2.54
cm
2
3.18 in 2 x
=
20.5
cm
12 in 2
Measurement of Mass
59
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
60
Measurement of Volume
61
• 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. 1 mL = 1 cm3 = 1cc
62
63
Measurement of
Temperature
64
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.
• System is the entity that is being
heated or cooled.
65
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 colder
temperature.
66
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.
67
Gabriel Daniel Fahrenheit
Anders Celsius
William Thomson
68
Degree Symbols
degrees Celsius =
oC
Kelvin (absolute) = K
degrees Fahrenheit = oF
69
To convert between the scales use the
following relationships.
o
K = C + 273.15
o
F = 1.8  C + 32
o
o
o
(
F
32)
FC-=32 = 1.8  C
1.8
o
o
70
It is not uncommon for temperatures in the Canadian
planes to reach –60.oF 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
–
71
It is not uncommon for temperatures in the Canadian
planes to reach –60.oF 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
72
Density
73
Density is the ratio
of the mass of a
substance to the
volume occupied by
that substance.
mass
d=
volume
74
Mass
is usually
The density
of gases is
expressed in grams per
and volume in ml or
liter.
cm3.
g
ddd=== 3
L
mL
cm
75
76
Examples
77
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
78
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
46.0 mL
35.0 mL
98.1 g
79
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
80
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
mass = density  volume
d=
volume
(b) Substitute the data and calculate.
0.714 g
25.0 mL 
= 17.9 g
mL
81
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 
=g
mL
0.714 g
25.0 ml 
= 17.9 g
mL
82
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 =
= 22.39 L
1.429 g O2 /L
83
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
L
=L
The conversion of units is g 
g
1L
32.00 g O2 
= 22.39 L O2
1.429 g O2
84
85