Standards for Measurement
Chapter 2
Hein and Arena
Version 1.1
Eugene Passer
Chemistry Department
1 College
Bronx Community
© John Wiley and Sons, Inc
Mass and Weight
2
• Matter: Anything that has mass and
occupies space.
• Mass : The quantity or amount of matter
that an object possesses.
– Fixed
– Independent of the object’s location
• Weight: A measure of the earth’s
gravitational attraction for an object.
– Not fixed
– Depends on the object’s location.
3
Data:
Measurements
and
Confidence
4
Components of a Measurement
numerical value
70.0 kilograms = 154 pounds
unit
A measurement has two components.
5
Confidence in a Numeric Value
The degree of confidence one has in the
accuracy of a numeric value is represented
by the number of significant figures the
numeric value has.
6
Significant Figures
Significant figures contain digits that are
known plus one estimated digit; all are
considered significant in a measured quantity
Here, 5 are known with certainty
5.16143 1 is estimated (i.e. uncertain)
7
Significant Figures
All nonzero numbers are significant.
3 Significant
Figures
461
8
Significant Figures
A zero is significant when it is between
nonzero digits.
3 Significant
Figures
401
9
Significant Figures
A zero is significant at the end of a number
that includes a decimal point.
5 Significant
Figures
55 . 000
10
Significant Figures
A zero is not significant when it is before
the first nonzero digit.
1 Significant
Figure
0 . 006
11
Significant Figures
A zero is not significant when it is at the
end of a number without a decimal point.
1 Significant
Figure
50000
12
Rounding
off Numbers
13
Rules for Rounding Off Numbers
• When calculations are performed, extra
digits may be present in the results.
• It is necessary to drop these extra digits
so the answer has 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.
14
Rounding Off Numbers
(to 4-Sig Figs)
Rule 1. When the first digit after those you
want to retain is 4 or less, that digit and all
digits after it are dropped. The last digit
retained is not changed.
4 or less
1.875377
15
Rounding Off Numbers
(to 3-Sig Figs)
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
16
Scientific Notation
of Numbers
17
A method for representing very large or
very small numbers in a simpler form is
called scientific notation (or exponential
notation).
23
602200000000000000000000
6.022 x 10
-21
0.00000000000000000000625
6.25 x 10
18
Write 6419 in scientific notation.
decimal after
first nonzero
digit
1
2
3
6.419
641.9x10
64.19x10
6419.
6419
x 10
When moving the decimal from right to left, the exponent is positive.
19
Write 0.000654 in scientific notation.
decimal after
first nonzero
digit
6.54 x
0.000654
0.00654
0.0654
0.654
-4
-2
-1
-3
10
When moving the decimal from left to right, the exponent is negative.
20
Significant Figures
in Calculations
21
The results of a calculation cannot be
more precise than the least precise data.
22
Multiplication or Division
23
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.
24
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
25
Addition or Subtraction
26
The results of addition or subtraction
must be expressed to the same precision
as the least precise data.
27
Add 125.17, 129 and 52.2
Least precise number.
Answer given
by calculator.
125.17
129.
+ 52.2
306.37
Hundredths
Unity
Tenths
Round off to the
Correct answer.
nearest unit.
306.37
28
Add 125.17, 120 and 52.2
Least precise number.
Answer given
by calculator.
125.17
120.
+ 52.2
297.37
Hundredths
Tens
Tenths
Correct answer.
3.0 x 102
29
Multiplication and/or
Division with Addition
and/or Subtraction
First perform all addition and/or subtraction and then
perform any multiplication and/or division.
30
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.
31
Question
Answer
What is the numeric value of the mathematical
operation given below to the proper number of
sig figs?
(0.0129 + 14.67)/(2.19 + 0.193)
14.68/2.38 = 6.17
32
The
Metric System
33
Measurement of Length
34
Metric Units of length
Unit
Abbreviation
Length Equivalent
kilometer
meter
km
m
1,000 m
1m
decimeter
10 dm
1m
centimeter
100 cm
1m
millimeter
1000 mm
1m
micrometer
1x106m
1m
nanometer
1x109nm
1m
35
Problem Solving
36
Dimensional Analysis
Dimensional analysis converts one
unit to another by using one or more
conversion factors.
unit2  unit1 x conversion factor = unit2
A conversion factor is an exact equivalent.
37
A Step by Step Strategy
Steps:
1. Write down the desired quantity (i.e. Unit 2).
2. Write down given quantities (i.e. Unit 1).
3. Write down needed conversion factors.
4. Unit conversion: L  mL  g  Mol.  Molecules.
5. Use “linear” setup to obtain desired unit(s) and
calculate answer.
Unit2  Unit1 x Conversion Factor(s) = Unit2
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
39
Measurement of Mass
40
Metric Units of mass
Unit
Abbreviation
Gram Equivalent
kilogram
gram
kg
g
1,000 g
1g
decigram
10 dg
1g
centigram
100 cg
1g
milligram
1000 mg
1g
microgram
1x106g
1g
41
Convert 45 decigrams to grams.
1 g = 10 dg
1g
45 dg x
= 4.5 g
10 dg
42
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 linear
expression by writing down conversion factors
in succession.
g lb  oz
1 lb
16 oz
-24
-26 oz
=
5.900x10
x
x
1.674 x 10 g
454 g
1 lb
43
Measurement of Volume
44
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
45
Measurement of
Temperature
46
Temperature
• A measure of heat.
• Heat always flows from the hot object to
the cold object.
47
There are three temperature scales:
Kelvin (absolute) = K
degrees Fahrenheit = oF
degrees Celsius =
oC
48
Conversion between temperature scales.
o
K = C + 273.15
o
o
o
F = 1.8 x C + 32
o
F - 32
o
C=
1.8
49
It is not uncommon for temperatures in the Canadian
planes to reach –51oC during the winter. What is this
temperature in K?
o
K = C + 273.15
o
K = -51 C + 273.15 = 222 K
50
Density
51
Density is the ratio
of the mass of a
substance to the
volume occupied by
that substance.
mass
d=
volume
52
Density varies with temperature
d
d
4 oC
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
53
54
Examples
55
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

D
 0.919 g/mL
V 13.5mL
56
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
57
The density of ether is 0.714 g/mL. What is the
mass of 25.0 milliliters of ether?
Dimensional Analysis. Use density as a
mL → g
conversion factor. Convert:
g
=g
The conversion of units is mL x
mL
0.714 g
25.0 ml x
= 17.9 g
mL
58
The density of oxygen at 0oC is 1.429 g/L. What is
the volume of 32.00 grams of oxygen at this
temperature?
Dimensional Analysis. Use density as a
g→L
conversion factor. Convert:
The conversion of units is
32.00 g O2 x
L
gx =L
g
1L
1.429 g O2
= 22.39 L
59
60