Chapter 2
Measurements
1
CHAPTER OUTLINE








Scientific Notation
Error in Measurements
Significant Figures
Rounding Off Numbers
SI Units
Conversion of Factors
Conversion of Units
Volume & Density
2
What is a Measurement?
 quantitative
observation
 comparison to an
agreed upon
standard
 every measurement
has a number and a
unit
A Measurement
 the unit tells you what standard you
are comparing your object to
 the number tells you
1. what multiple of the standard the
object measures
2. the uncertainty in the measurement
Scientists have measured the
average global temperature rise
over the past century to be 0.6°C
 °C tells you that the
temperature is being
compared to the Celsius
temperature scale
 0.6 tells you that
1. the average temperature rise is
0.6 times the standard unit
2. the uncertainty in the
measurement is such that we
know the measurement is
between 0.5 and 0.7°C
SCIENTIFIC
NOTATION
 Scientific Notation is a convenient way to express
very large or very small quantities.
 Its general form is
A x 10n
coefficient
n = exponent
1  A < 10
6
SCIENTIFIC
NOTATION
To convert from decimal to scientific notation:
Move the
point by
in the
original number
 Follow
thedecimal
new number
a multiplication
signso
that 10
it iswith
located
after the(power).
first nonzero digit.
and
an exponent
 The exponent is equal to the number of places that
the decimal point was shifted.
75000000
7.5 x 10 7
7
Scientific Notation:
Writing Large and Small Numbers
 A positive exponent means 1 multiplied by 10 n times.
 A negative exponent (–n) means 1 divided by 10 n
times.
SCIENTIFIC
NOTATION
 For numbers smaller than 1, the decimal moves
to the left and the power becomes negative.
0 00642
6.42 x 10
3
9
Examples:
1. Write 6419 in scientific notation.
decimal after
first nonzero
digit
power of 10
64.19x10
641.9x10
6419.
6419
6.419
x 10213
10
Examples:
2. Write 0.000654 in scientific notation.
decimal after
first nonzero
digit
power of 10
-1
-2
-3
-4
0.000654
0.00654
x
10
0.0654
0.654
10
6.54 xx10
11
CALCULATIONS WITH
SCIENTIFIC NOTATION
 To perform multiplication or division with
scientific notation:
1. Change numbers to exponential form.
2. Multiply or divide coefficients.
3. Add exponents if multiplying, or subtract
exponents if dividing.
4. If needed, reconstruct answer in standard
exponential form.
12
Example 1:
Multiply 30,000 by 600,000
Convert
Multiply
Reconstruct
Add
to exponential
exponents
coefficients
answerform
(3 x 104) (6 x 105) = 18 x 10 9
1.8 x 1010
13
Example 2:
Divided 30,000 by 0.006
Convert
Subtract
Reconstruct
Divide
to exponential
coefficients
exponents
answerform
4 – (-3)
(3 x 104)
(6 x 10-3)
= 0.5 x 10
7
5 x 106
14
Follow-up Problems:
(5.5x103)(3.1x105) = 17.05x108 = 1.7x109
(9.7x1014)(4.3x1020) = 41.71x106
= 4.2x105
6
2.6x10
4
0.4483x10
=
5.8x102
1.7x10
8.2x10
= 4.5x103
5
8
= 0.2073x103
= 2.1x102
(3.7x106)(4.0x108) = 14.8x102 = 1.5x103
15
Follow-up Problems:
(8.75x1014)(3.6x108) = 31.5x1022 = 3.2x1023
28
1.48x10
41
=
0.2041x10
7.25x1013
= 2.04x1042
16
ACCURACY & PRECISION
 Precision is the reproducibility of a measurement
Is this
compared to other similar measurements.
measurement
 Precision describes how close measurements
are
precise?
to one another.
 Precision is affected by random errors.
Avg mass = 3.12± 0.01 g
This measurement has high precision because
the deviation of multiple trials is small.
17
ACCURACY & PRECISION
 Accuracy is the closeness of a measurement
to an
Is this
measurement
accepted value (external standard).
accurate?
 Accuracy describes how true a measurement is.
 Accuracy is affected by systematic errors.
Avg mass = 3.12± 0.01 g
True mass = 3.03 g
cannothas
below
determined
ThisAccuracy
measurement
accuracywithout
because
of the
accepted
the knowledge
deviation from
true
value isvalue.
large.
18
ACCURACY & PRECISION
Poor precision
Good precision
Good accuracy
Poor accuracy
Good precision
Poor precision
Good accuracy
Poor accuracy
19
ACCURACY & PRECISION
 Two types of error can affect measurements:
 Systematic errors: those errors that are controllable, and cause
measurements to be either higher or lower than the actual value.
 Random errors: those errors that are uncontrollable, and cause
measurements to be both higher and lower than the average
value.
20
ERROR IN
MEASUREMENTS
 Two kinds of numbers are used in science:
Counted or defined:
exact numbers; have no uncertainty
Measured:
are subject to error; have uncertainty
 Every measurement has uncertainty because of
instrument limitations and human error.
21
ERROR IN
MEASUREMENTS
certain
certain
8.65
8.6
uncertain
uncertain

What
Theislast
thisdigit
measurement?
in any
measurement
is the
What
is this measurement?
estimated one.
22
RECORDING MEASUREMENTS
TO THE PROPER NO OF DIGITS
What is the correct value
for each measurement?
a) 28ml (1 certain, 1 uncertain)
b) 28.2ml (2 certain, 1 uncertain)
c) 28.31ml (3 certain, 1 uncertain)
23
SIGNIFICANT
FIGURES RULES
1. Significant
figures
figures
are
arecertain
used toand
determine
uncertain
All non-zero
digitsrules
arethe
significant.
which
digits in
digits
a measurement.
are significant and which are not.
2. All sandwiched zeros are significant.
3. Leading zeros (before or after a decimal) are
NOT significant.
4. Trailing zeros (after a decimal) are significant.
0
.
0
0
4
0
0
4
5
0
0
24
Examples:
Determine the number of significant figures in each
of the following measurements.
461 cm
3 sig figs
1025 g
4 sig figs
0.705 mL
3 sig figs
93.500 g
5 sig figs
0.006 m
1 sig fig
5500 km
2 sig figs
25
ROUNDING OFF
NUMBERS
 If rounded digit is less than 5, the digit is dropped.
51.234
Round to 3 sig figs
1.875377
Less than 5Round
to 4 sig figs
Less than 5
26
ROUNDING OFF
NUMBERS
 If rounded digit is equal to or more than 5, the
digit is increased by 1.
4
51.369
Round to 3 sig figs
1
5.4505
More than Round
5
to 4 sig figs
Equal to 5
27
SIGNIFICANT
FIGURES & CALCULATIONS
 The results of a calculation cannot be more
precise than the least precise measurement.
 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.
 For addition and subtraction, the answer must
have the same number of decimal places as
there are in the measurement with the fewest
decimal places.
28
MULTIPLICATION
& DIVISION
3 sig figs
4 sig figs
Calculator
answer
(9.2)(6.80)(0.3744) = 23.4225
2 sig figs
The answer should have two significant
figures because 9.2 is the number with
the fewest significant figures.
The correct answer is 23
29
ADDITION &
SUBTRACTION
Add 83.5 and 23.28
Least precise number
Calculator
answer
Correct answer
83.5
23.28
106.78
106.8
30
Example 1:
5.008 + 16.2 + 13.48 = 34.688
34.7
Least precise
number
Round to
31
Example 2:
3 sig figs
3.15 x 1.53
= 6.1788
0.78
6.2
2 sig figs
Round to
32
SI UNITS
 Measurements are made by scientists to
determine size, length and other properties of
matter.
 For measurements to be useful, a measurement
standard must be used.
 A standard is an exact quantity that people agree
to use for comparison.
 SI is the standard system of measurement used
worldwide by scientists.
33
SI (METRIC)
BASE UNITS
Quantity Measured
Length
Metric
Units
Meter
m
English
Units
yd
Symbol
Mass
Kilogram
kg
lb
Time
Seconds
s
s
Temperature
Kelvin
K
F
Mole
mol
mol
Amount of
substance
34
Basic Units of Measurement
 The kilogram is a measure of mass, which
is different from weight.
 The mass of an object is a measure of
the quantity of matter within it.
 The weight of an object is a measure of
the gravitational pull on that matter.
 Consequently, weight depends on gravity
while mass does not.
Derived Units
 A derived unit is formed from other units.
 Many units of volume, a measure of
space, are derived units.
 Any unit of length, when cubed (raised to
the third power), becomes a unit of
volume.
 Cubic meters (m3), cubic centimeters
(cm3), and cubic millimeters (mm3) are
all units of volume.
DERIVED UNITS
 In addition to the base units, several derived
units are commonly used in SI system.
Quantity Measured
Units
Symbol
Volume
Liter
L
Density
grams/cc
g/cm3
37
SI PREFIXES
 The
Common
SI system
prefixes
of units
are
used
is easy
with
to the
use base
because
units
it is
to
SI Prefixes
indicate
multiple
ten that the unit represents.
based onthe
multiples
ofof
ten.
Prefixes
mega-
Symbol
M
kilocentimilli-
k
c
m
micro-

Multiplying factor
1,000,000
106
1000
0.01
0.001
103
10-2
10-3
0.000,001 10-6
38
SI UNITS &
PREFIXES
 SI system used a common set of prefixes for use
with the base units.
106
103
Base Unit
103
106
10 10 10 10 10 10 10 10 10 10 10 10
micro
Smaller units
milli
deci
centi
kilo
mega
Larger units
39
SI CONVERSION
FACTORS
106
10 10
micro
Base Unit
103
10
10
milli
10
10
deci
centi
10
103
10
10
106
10
kilo
10
10
mega
103 mm
or
1 mm =
103 m
1 mm = 103 m
or
1 m =
103 mm
1m=
40
SI PREFIXES
How many cm
mmare
areininaakm?
cm?
100000
10x10x10x10x10
10
or 105
41
Prefix Multipliers
 Choose the prefix multiplier that is most
convenient for a particular
measurement.
 Pick a unit similar in size to (or smaller
than) the quantity you are measuring.
 A short chemical bond is about 1.2 ×
10–10 m. Which prefix multiplier should
you use?
 The most convenient one is probably the
picometer. Chemical bonds measure
about 120 pm.
CONVERSION
FACTORS
 Many problems in chemistry and related fields
require a change of units.
 Any unit can be converted into another by use of
the appropriate conversion factor.
 Any equality in units can be written inMetric-Metric
the form of a
Factor
fraction called a conversion factor. For example:
Equality
Conversion Factors
1 m = 100 cm
1m
100 cm
or
100 cm
1m
43
CONVERSION
FACTORS Metric-English
Factor
Equality
1 kg = 2.20 lb
Conversion Factors
1 kg
2.20 lb
or
2.20 lb
1 kg
 Sometimes a conversion factor is given Percentage
as a percentage.
For example:
Factor
Percent quantity:
Conversion
Factors
18% body fat by mass
18 kg body fat
100 kg body mass
or
100 kg body mass
18 kg body fat
44
CONVERSION
OF UNITS
 Problems involving conversion of units and other
chemistry problems can be solved using the
following step-wise method:
4.
2.
3.
Write
Set
Planupathe
the
sequence
conversion
problem
of steps
by
factor
arranging
to convert
forand
each
cancelling
the
units
initial
change
units
unit
in
in
tothe
the
1. Determine
the
intial
unit
given
the
final
unit
needed.
final unit.
your
numerator
plan. and denominator of the steps involved.
beginning unit x
final unit
= final unit
beginning unit
Conversion factor
45
Example 1:
Convert 164 lb to kg (1 kg = 2.20 lb)
Step 1:
Step 2:
Step 3:
Step 4:
Given: 164 lb
lb
1 kg
2.20 lb
Need: kg
Metric-English
factor
or
kg
2.20 lb
1 kg
1 kg
164 lb x
= 74.5 kg
2.20 lb
46
Example 2:
The thickness of a book is 2.5 cm. What is this measurement
in mm?
Step 1:
Given: 2.5 cm
Step 2:
cm
Step 3:
1 cm
10 mm
Step 4:
Need: mm
Metric-Metric
factor
or
mm
10 mm
1 cm
10 mm
2.5 cm x
= 25 mm
1 cm
47
Example 3:
How many centimeters are in 2.0 ft? (1 in=2.54 cm)
Step 1:
Given: 2.0 ft
Step 2:
ft
Step 3:
Step 4:
English-English
factor
1 ft
12 in
and
Need: cm
in
Metric-English
factor
cm
1 in
2.54 cm
12 in 2.54 cm
61 cm
cm
= 60.96
2.0 ft x
x
1 in
1 ft
48
Example 4:
Bronze is 80.0% by mass copper and 20.0% by mass tin.
A sculptor is preparing to case a figure that requires
1.75 lb of bronze. How many grams of copper are
needed for the brass figure (1lb = 454g)?
Step 1:
Step 2:
Given: 1.75 lb bronze
lb
brz
English-Metric
factor
g
brz
Need: g of copper
Percentage
factor
g
Cu
49
Example 4:
Step 3:
Step 4:
1 lb
454 g
1.75 lb brz x
and
454 g
1 lb
80.0 g Cu
100 g brz
80.0 g Cu
x
==
635.6
636 g
100 g brz
50
VOLUME
 Volume is the amount of space an object
occupies.
 Common units are cm3 or liter (L) and
milliliter (mL).
1 L = 1000 mL
1 mL = 1 cm3
51
VOLUME
 Volume of various regular shapes can be
calculated as follows:
Cube V = s x s x s
Cylinder V = π x r2 x h
Rect. V = l x w x h
Sphere V =4/3 πr3
52
DENSITY
 Density is mass per unit volume of a material.
 Common units are g/cm3 (solids) or g/mL (liquids).
Density is directly
Density is indirectly
related to the
related to the
mass of an object.
volume of an object.
Which has greatest density?
53
Example 1:
A copper sample has a mass of 44.65 g and a volume
of 5.0 mL. What is the density of copper?
m = 44.65 g
m
44.65 g
d=
=
= 8.9
8.93g/mL
g/mL
v
5.0 mL
v = 5.0 mL
d = ???
Round to
2 sig figs
54
Example 2:
A silver bar with a volume of 28.0 cm3 has a mass of
294 g. What is the density of this bar?
m = 294 g
m
294 g
d=
=
= 10.5 g/mL
v
28.0 mL
v = 28.0 mL
d = ???
3 sig figs
55
Example 3:
If the density of gold is 19.3 g/cm3, how many grams
does a 5.00 cm3 nugget weigh?
Step 1:
Step 2:
Step 3:
Given: 5.00 cm3
cm3
g
density
5.00
cm3
Need: g
x
19.3 g
1 cm 3
= 96.5 g
56
Example 4:
If the density of milk is 1.04 g/mL, what is the mass of
0.50 qt of milk? (1L = 1.06 qt)
Step 1:
Step 2:
Step 3:
Given: 0.5 qt
qt
English-Metric
factor
1L
1.06 qt
Need: g
L
and
Metric-Metric
factor
103 mL
1L
mL
and
density
1.04 g
1 mL
1 L 103 mL 1.04 g
x
x
Step 4: 0.50 qt x
== 490
490.57
g g
1.06 qt
1 mL
1L
57
g
Example 5:
What volume of mercury has a mass of 60.0 g if its
density is 13.6 g/mL?
1
mL
4.41 mL
=
60.0 g x
13.6 g
inverse of
density
58
IS UNIT CONVERSION
IMPORTANT?
 Further
In 1999 investigation
Mars Climateshowed

orbiter
was lost
space
that
engineers
at in
Lockheed
becausewhich
engineers
Martin,
builtfailed
the to
make a simple
conversion
aircraft,
calculated
from Englishmeasurements
units to metricin
navigational
units, an
embarrassing
lapse
English
units.
When NASA’s
thatengineers
sent the $125
million
JPL
received
the
craft they
fatally
close tothe
the
data,
assumed
Martian surface.
information
was in metric
units, causing the confusion.
59
THE END
60