Fundamentals of General, Organic, and
Biological Chemistry
5th Edition
Chapter Two
Measurements in
Chemistry
James E. Mayhugh
Oklahoma City University
2007 Prentice Hall, Inc.
Outline
► 2.1 Physical Quantities
► 2.2 Measuring Mass
► 2.3 Measuring Length and Volume
► 2.4 Measurement and Significant Figures
► 2.5 Scientific Notation
► 2.6 Rounding Off Numbers
► 2.7 Converting a Quantity from One Unit to Another
► 2.8 Problem Solving: Estimating Answers
► 2.9 Measuring Temperature
► 2.10 Energy and Heat
► 2.11 Density
► 2.12 Specific Gravity
Chapter Two
2
2.1 Physical Quantities
Physical properties such as height, volume, and
temperature that can be measured are called
physical quantities. Both a number and a unit of
defined size is required to describe physical quantity.
Chapter Two
3
► A number without a unit is meaningless.
► To avoid confusion, scientists have agreed on a standard set
of units.
► Scientists use SI or the closely related metric units.
Prentice Hall © 2007
Chapter Two
4
► Scientists work with both very large and very
small numbers.
► Prefixes are applied to units to make saying and
writing measurements much easier.
► The prefix pico (p) means “a trillionth of.”
► The radius of a lithium atom is 0.000000000152
meter (m). Try to say it.
► The radius of a lithium atom is 152 picometers
(pm). Try to say it.
Chapter Two
5
Frequently used prefixes are shown below.
Chapter Two
6
2.2 Measuring Mass
► Mass is a measure of the amount of matter in an
object. Mass does not depend on location.
► Weight is a measure of the gravitational force
acting on an object. Weight depends on location.
► A scale responds to weight.
► At the same location, two objects with identical
masses have identical weights.
► The mass of an object can be determined by
comparing the weight of the object to the weight
of a reference standard of known mass.
Chapter Two
7
a) The single-pan balance with sliding
counterweights. (b) A modern electronic balance.
Chapter Two
8
Relationships between metric units of mass and the
mass units commonly used in the United States are
shown below.
Chapter Two
9
2.3 Measuring Length and Volume
► The meter (m) is the standard measure of length
or distance in both the SI and the metric system.
► Volume is the amount of space occupied by an
object. A volume can be described as a length3.
► The SI unit for volume is the cubic meter (m3).
Chapter Two
10
Relationships between metric units of length and
volume and the length and volume units commonly
used in the United States are shown below and on
the next slide.
Chapter Two
11
A m3 is the volume of a cube 1 m or 10 dm on edge.
Each m3 contains (10 dm)3 = 1000 dm3 or liters. Each
liter or dm3 = (10cm)3 =1000 cm3 or milliliters. Thus,
there are 1000 mL in a liter and 1000 L in a m3.
Chapter Two
12
The metric system is based on factors of 10 and is
much easier to use than common U.S. units. Does
anyone know how many teaspoons are in a
gallon?
Chapter Two
13
Exact Numbers
An exact number is obtained when you count objects
or use a defined relationship.
Counting objects are always exact
2 soccer balls
4 pizzas
Exact relationships, predefined values, not measured
1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No
1 ft is EXACTLY 12 inches.
14
Learning Check
A. Exact numbers are obtained by
1. using a measuring tool
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
2. counting
3. definition
15
Solution
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
16
Learning Check
Classify each of the following as an exact or a
measured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
17
Solution
Classify each of the following as an exact (1) or a
measured(2) number.
This is a defined relationship.
A measuring tool is used to determine length.
The number of hats is obtained by counting.
A measuring tool is required.
18
2.4 Measurement and Significant
Figures
► Every experimental
measurement has a
degree of uncertainty.
► The volume, V, at right
is certain in the 10’s
place, 10mL<V<20mL
► The 1’s digit is also
certain, 17mL<V<18mL
► A best guess is needed
for the tenths place.
Chapter Two
19
What is the Length?
1
2
3
►We can see the markings between 1.6-1.7cm
►We can’t see the markings between the .6-.7
►We must guess between .6 & .7
►We record 1.67 cm as our measurement
►The last digit an 7 was our guess...stop there
20
4 cm
Learning Check
What is the length of the wooden stick?
1) 4.5 cm
2) 4.54 cm
3) 4.547 cm
? 8.00 cm or 3 (2.2/8)
22
► To indicate the precision of a measurement, the
value recorded should use all the digits known
with certainty, plus one additional estimated
digit that usually is considered uncertain by plus
or minus 1.
► No further insignificant digits should be
recorded.
► The total number of digits used to express such
a measurement is called the number of
significant figures.
► All but one of the significant figures are known
with certainty. The last significant figure is only
the best possible estimate.
Chapter Two
23
Below are two measurements of the mass of the
same object. The same quantity is being described
at two different levels of precision or certainty.
Chapter Two
24
► When reading a measured value, all nonzero digits
should be counted as significant. There is a set of
rules for determining if a zero in a measurement is
significant or not.
► RULE 1. Zeros in the middle of a number are like
any other digit; they are always significant. Thus,
94.072 g has five significant figures.
► RULE 2. Zeros at the beginning of a number are
not significant; they act only to locate the decimal
point. Thus, 0.0834 cm has three significant
figures, and 0.029 07 mL has four.
Chapter Two
25
► RULE 3. Zeros at the end of a number and after
the decimal point are significant. It is assumed
that these zeros would not be shown unless they
were significant. 138.200 m has six significant
figures. If the value were known to only four
significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before
an implied decimal point may or may not be
significant. We cannot tell whether they are part
of the measurement or whether they act only to
locate the unwritten but implied decimal point.
Chapter Two
26
Practice Rule #1 Zeros
45.8736
6
•All digits count
.000239
3
•Leading 0’s don’t
.00023900 5
•Trailing 0’s do
48000.
5
•0’s count in decimal form
48000
2
•0’s don’t count w/o decimal
3.982106 4
1.00040
6
•All digits count
•0’s between digits count as well
as trailing in decimal form
2.5 Scientific Notation
► Scientific notation is a convenient way to
write a very small or a very large number.
► Numbers are written as a product of a number
between 1 and 10, times the number 10
raised to power.
► 215 is written in scientific notation as:
215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102
Chapter Two
28
Two examples of converting standard notation to
scientific notation are shown below.
Chapter Two
29
Two examples of converting scientific notation back to
standard notation are shown below.
Chapter Two
30
► Scientific notation is helpful for indicating how
many significant figures are present in a number
that has zeros at the end but to the left of a decimal
point.
► The distance from the Earth to the Sun is
150,000,000 km. Written in standard notation this
number could have anywhere from 2 to 9 significant
figures.
► Scientific notation can indicate how many digits are
significant. Writing 150,000,000 as 1.5 x 108
indicates 2 and writing it as 1.500 x 108 indicates 4.
► Scientific notation can make doing arithmetic easier.
Rules for doing arithmetic with numbers written in
scientific notation are reviewed in Appendix A.
Chapter Two
31
2.6 Rounding Off Numbers
► Often when doing arithmetic on a pocket
calculator, the answer is displayed with more
significant figures than are really justified.
► How do you decide how many digits to keep?
► Simple rules exist to tell you how.
Chapter Two
32
► Once you decide how many digits to retain, the
rules for rounding off numbers are straightforward:
► RULE 1. If the first digit you remove is 4 or less, drop
it and all following digits. 2.4271 becomes 2.4 when
rounded off to two significant figures because the
first dropped digit (a 2) is 4 or less.
► RULE 2. If the first digit removed is 5 or greater,
round up by adding 1 to the last digit kept. 4.5832 is
4.6 when rounded off to 2 significant figures since
the first dropped digit (an 8) is 5 or greater.
► If a calculation has several steps, it is best to round
off at the end.
Chapter Two
33
Practice Rule #2 Rounding
Make the following into a 3 Sig Fig number
1.5587
1.56
.0037421
.00374
1367
1370
128,522
129,000
1.6683 106
1.67 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
Examples of Rounding
For example you want a 4 Sig Fig number
0 is dropped, it is <5
4965.03
4965
780,582
780,600 8 is dropped, it is >5; Note you
must include the 0’s
1999.5
2000.
5 is dropped it is = 5; note you
need a 4 Sig Fig
RULE 1. In carrying out a multiplication or division,
the answer cannot have more significant figures than
either of the original numbers.
Chapter Two
36
►RULE 2. In carrying out an addition or
subtraction, the answer cannot have more digits
after the decimal point than either of the
original numbers.
Chapter Two
37
Multiplication and division
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
3.2650106  4.858 = 1.586137  107
1.586 107
6.0221023  1.66110-24 = 1.000000
1.000
Addition/Subtraction
25.5
+34.270
59.770
59.8
32.72
- 0.0049
32.7151
32.72
320
+ 12.5
332.5
330
Addition and Subtraction
.56
__ + .153
___ = .713
__ .71
82000 + 5.32 = 82005.32
82000
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for the
last important
digit
Mixed Order of Operation
8.52 + 4.1586  18.73 + 153.2 =
= 8.52 + 77.89 + 153.2 = 239.61 =
239.6
(8.52 + 4.1586)  (18.73 + 153.2) =
= 12.68  171.9 = 2179.692 =
2180.
Try
Find the standard deviation for the following
numbers: 7.691 g, 7.23 g, 7.892 g
xi
x
i n


 x x 

i
 i

s
n

1



2







1
2
Try
xi
x
i n
7.691 g
7.23 g
7.892 g
22.813 g
7.691 g, 7.23 g, 7.892 g
22.81 g = 7.603 g
3
Try


 x x 

i
 i

s
n

1



2







1
2
= 7.603 g
7.691 g – 7.603 g = .088 g
7.23 g – 7.603 g = -.37 g
7.892 g – 7.603 g = .289 g
.01 g
Try
.25


 x x 

i
 i

s
n

1



2







1
2
.01 = .01 g
 .01 
s

 2 
2
1/ 2
 .007
2.7 Problem Solving: Converting a
Quantity from One Unit to Another
► Factor-Label Method: A quantity in one unit is
converted to an equivalent quantity in a different
unit by using a conversion factor that expresses the
relationship between units.
(Starting quantity) x (Conversion factor) = Equivalent quantity
Chapter Two
46
Writing 1 km = 0.6214 mi as a fraction restates it in
the form of a conversion factor. This and all other
conversion factors are numerically equal to 1.
The numerator is equal to the denominator.
Multiplying by a conversion factor is equivalent to
multiplying by 1 and so causes no change in value.
Chapter Two
47
When solving a problem, the idea is to set up an
equation so that all unwanted units cancel, leaving
only the desired units.
Chapter Two
49
2.8 Problem Solving: Estimating
Answers
► STEP 1: Identify the information given.
► STEP 2: Identify the information needed to answer.
► STEP 3: Find the relationship(s) between the known
information and unknown answer, and plan a series
of steps, including conversion factors, for getting
from one to the other.
► STEP 4: Solve the problem.
► BALLPARK CHECK: Make a rough estimate to be
sure the value and the units of your calculated
answer are reasonable.
Chapter Two
51
Example: Problem Solving
How many minutes are in 1.4 days?
Initial unit: 1.4 days
Unit plan: days
hr
min
Set up problem:
1.4 days x 24 hr x 60 min = 2.0 x 103 min
1 day
1 hr
54
Learning Check
An adult human has 4650 mL of blood.
How many gallons of blood is that?
Unit plan: mL
qt
gallon
Equalities: 1 quart = 946 mL
1 gallon = 4 quarts
55
Solution
Unit plan:
mL
qt
Setup:
4650 mL x 1 qt
x 1 gal
946 mL
4 qt
3 SF
3 SF
exact
gallon
= 1.23 gal
3 SF
56
►150 pounds (American) is how many stones (British)?
1 pound = 265 dram
1 Gram = 1.71 dram
1.13 pennyweights = 1 Gram
1.21 scruples = 1 pennyweight
17.2 scruples = 1 stone
?
57
►The following relationships are British liquid units.
1 hogshead = 7 firkin
18 pottle = 1 firkin
140 pottle = 1 puncheon
504 pottle = 1 tun
How many hogsheads in 12.5 tuns?
1 firkin

18 pottle
50.0hogsheads
58
►A 60 watt light bulb is the how many horsepower?
3.41 Btu/hr = 1 watt
4.20 calorie/minute = 1 Btu/hr
.0514 Foot-pound-force/second = 1 calorie/minute
1.825×10-3 horsepower = 1 Foot-pound-force/second
59
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?
►The dosage ordered is 485 mg of Erythromycin four
times a day (q.i.d). If the oral suspension contains
200 mg Erythromycin/5 mL, how many mL will be
given in a day?
►The dimensions of a box are 12 inch by 11 inch by
5.5 inch. Calculate the volume of the box in cm3.
There is 2.54 cm per inch.
60
2.9 Measuring Temperature
► Temperature is commonly reported either in
degrees Fahrenheit (oF) or degrees Celsius (oC).
► The SI unit of temperature is the Kelvin (K).
► 1 Kelvin, no degree, is the same size as 1 oC.
► 0 K is the lowest possible temperature, 0 oC =
273.15 K is the normal freezing point of water.
To convert, adjust for the zero offset.
► Temperature in K = Temperature in oC + 273.15
► Temperature in oC = Temperature in K - 273.15
Chapter Two
61
Freezing point of H2O
32oF
0oC
Boiling point of H2O
212oF
100oC
212oF - 32oF = 180oF covers the same range of
temperature as 100oC - 0oC = 100oC covers.
Therefore, a Celsius degree is exactly 180/100 = 1.8
times as large as a Fahrenheit degree. The zeros on
the two scales are separated by 32oF.
Chapter Two
62
Fahrenheit, Celsius, and Kelvin temperature scales.
Chapter Two
63
► Converting between Fahrenheit and Celsius
scales is similar to converting between different
units of length or volume, but is a little more
complex. The different size of the degree and the
zero offset must both be accounted for.
►
►
oF
= (1.8 x oC) + 32
oC = (oF – 32)/1.8
Chapter Two
64
2.10 Energy and Heat
► Energy: The capacity to do work or supply heat.
► Energy is measured in SI units by the Joule (J); the
calorie is another unit often used to measure
energy.
► One calorie (cal) is the amount of heat necessary to
raise the temperature of 1 g of water by 1°C.
► A kilocalorie (kcal) = 1000 cal. A Calorie, with a
capital C, used by nutritionists, equals 1000 cal.
► An important energy conversion factor is:
1 cal = 4.184 J
Chapter Two
65
► Not all substances have their temperatures raised to
the same extent when equal amounts of heat
energy are added.
► One calorie raises the temperature of 1 g of water
by 1°C but raises the temperature of 1 g of iron by
10°C.
► The amount of heat needed to raise the
temperature of 1 g of a substance by 1°C is called
the specific heat of the substance.
► Specific heat is measured in units of cal/gC
Chapter Two
66
► Knowing the mass and specific heat of a substance
makes it possible to calculate how much heat must
be added or removed to accomplish a given
temperature change.
► (Heat Change) = (Mass) x (Specific Heat) x
(Temperature Change)
► Using the symbols Δ for change, H for heat, m for
mass, C for specific heat, and T for temperature, a
more compact form is:
ΔH = m×C×Δ T
Chapter Two
67
Learning Check
1. How much energy is required to change the
temperature of 15.0 g Fe from 18.5 C to 56.8 C?
The specific heat of iron is 0.451 J/g·K.
2. (2.67) copper has specific heat of .092 cal/(g∙°C).
When 52.7 cal of heat is added to a piece of copper,
the temperature increases from 22.4 °C to 38.6 °C.
What is the mass of the piece of copper?
3. If 34.8 J is required to change the temperature of
10.0 g of mercury by 25 K, what is the specific heat
of mercury?
68
Learning Check
1. How much energy is required to change the
temperature of 15.0 g Fe from 18.5 C to 56.8 C?
The specific heat of iron is 0.451 J/g·K.
ΔH = m×C×Δ T
ΔH = 15.0 g×0.451 J×(56.8-18.5)C
g·K
ΔH = 259 J
69
Learning Check
2. (2.67) copper has specific heat of .092 cal/(g∙°C).
When 52.7 cal of heat is added to a piece of copper,
the temperature increases from 22.4 °C to 38.6 °C.
What is the mass of the piece of copper?
ΔH = m×C×Δ T
52.7 cal = m×.092 cal× (38.6-22.4)C
gC
70
Learning Check
3. If 34.8 J is required to change the temperature of
10.0 g of mercury by 25 K, what is the specific heat
of mercury?
ΔH = m×C×Δ T
34.8 J = 10.0 g×C×25 K
71
2.11 Density
Density relates the mass of an object to its volume.
Density is usually expressed in units of grams per cubic
centimeter (g/cm3) for solids, and grams per milliliter
(g/mL) for liquids.
Density =
Mass (g)
Volume (mL or cm3)
Chapter Two
72
►Which is heavier, a ton
of feathers or a ton of
bricks?
►Which is larger?
►If two objects have the
same mass, the one
with the higher density
will be smaller.
Chapter Two
73
Mercury has a density of 13.6 g/mL. How many
milliliters of mercury weigh 475 grams?
1.
2.
3.
4.
0.000155 mL
0.0286 mL
34.9 mL
6460 mL
Learning Check
►(2.75) What is the density of lithium metal ( in
g/cm3) if a cylindrical wire with a diameter of 2.40
mm and a length of 15.0 cm has a mass of .3624 g;
vcyln=r2l.
►The density of acetic acid is 1.05 g/mL. What is the
volume of 275 g of acetic acid?
►A cube of iron has a mass of 15.37 g. If each side of
the cube has dimensions of 1.25 cm, what is the
density of iron?
75
Learning Check
►(2.75) What is the density of lithium metal ( in g/cm3) if
a cylindrical wire with a diameter of 2.40 mm and a
length of 15.0 cm has a mass of .3624 g; vcyln=r2l.
Density =
Mass (g)
Volume (mL or cm3)
76
Learning Check
►The density of acetic acid is 1.05 g/mL. What is the
volume of 275 g of acetic acid?
Density =
Mass (g)
Volume (mL or cm3)
77
Learning Check
►A cube of iron has a mass of 15.37 g. If each side of
the cube has dimensions of 1.25 cm, what is the
density of iron?
Density =
Mass (g)
Volume (mL or cm3)
78
2. 12 Specific Gravity
Specific gravity (sp gr): density of a substance
divided by the density of water at the same
temperature. Specific gravity is unitless. The
density of water is so close to 1 g/mL that the
specific gravity of a substance at normal
temperature is numerically equal to the density.
Density of substance (g/ml)
Specific gravity =
Density of water at the same temperature (g/ml)
Chapter Two
79
The specific gravity of a liquid
can be measured using an
instrument called a
hydrometer, which consists of
a weighted bulb on the end of
a calibrated glass tube. The
depth to which the
hydrometer sinks when
placed in a fluid indicates the
fluid’s specific gravity.
Chapter Two
80
►Galileo’s
Thermometer
As temperature
changes so do
the density's of
the solutions in
the floating
bulbs.
81
Chapter Summary
► Physical quantities require a number and a unit.
► Preferred units are either SI units or metric units.
► Mass, the amount of matter an object contains, is
measured in kilograms (kg) or grams (g).
► Length is measured in meters (m). Volume is
measured in cubic meters in the SI system and in
liters (L) or milliliters (mL) in the metric system.
► Temperature is measured in Kelvin (K) in the SI
system and in degrees Celsius (°C) in the metric
system.
Chapter Two
82
Chapter Summary Cont.
► The exactness of a measurement is indicated by
using the correct number of significant figures.
► Significant figures in a number are all known with
certainty except for the final estimated digit.
► Small and large quantities are usually written in
scientific notation as the product of a number
between 1 and 10, times a power of 10.
► A measurement in one unit can be converted to
another unit by multiplying by a conversion factor
that expresses the exact relationship between the
units.
Chapter Two
83
Chapter Summary Cont.
► Problems are solved by the factor-label method.
► Units can be multiplied and divided like numbers.
► Temperature measures how hot or cold an object is.
► Specific heat is the amount of heat necessary to
raise the temperature of 1 g of a substance by 1°C.
► Density relates mass to volume in units of g/mL for
a liquid or g/cm3 for a solid.
► Specific gravity is density of a substance divided by
the density of water at the same temperature.
Chapter Two
84
End of Chapter 2
Chapter Two
85