Mark S. Cracolice
Edward I. Peters
http://academic.cengage.com/chemistry/cracolice
Chapter 3
Measurement and
Chemical Calculations
Mark S. Cracolice • The University of Montana
Introduction to Measurement
Measure
Comparison of the dimensions, quantity, or capacity of
something with a standard.
Thus, to measure something, members of a society have to first
agree on a standard for comparison.
For example, in the United States, people agree on the
distance represented by the unit called the inch.
The dimensions of objects can then be expressed in inches,
and all members of society understand
the meaning of the measurement.
Introduction to Measurement
How tall are you?
To answer this question, you use an agreed-upon standard to
express the value of the measured quantity:
Feet and inches are typically used in the
United States to express a person’s height.
Centimeters are typically used in the
rest of the world to express a person’s height.
Introduction to Measurement
Measurements everywhere in the world, with the
exception of the U.S., are made in the metric system.
U.S. scientists, as well as all scientists in every country, also
make and express measurements in the metric system.
SI units are a subset of all metric units.
SI is an abbreviation for the French name for the International
System of Units (the metric system was invented in France).
Introduction to Measurement
The SI system is defined by seven base units.
Examples of base units include:
Quantity
Mass (weight)
Length
Temperature
Time
Base Unit
Kilogram
Meter
Kelvin
Second
Other measurement units are derived from the base units;
accordingly, they are called derived units.
Exponential Notation
Goal 1
Write in exponential notation a number given in ordinary
decimal form; write in ordinary decimal form a
number given in exponential notation.
Goal 2
Using a calculator, add, subtract, multiply, and divide
numbers expressed in exponential notation.
Exponential Notation
Exponentials
BP
B is the base
p is the power or exponent
104 = 10  10  10  10 = 10,000
10–4 =

1 = 1 1  1  1 =
1
= 0.0001
4
10
10 10 10 10 10,000





Exponential Notation
Exponential Notation
a.bcd  10e
Coefficient: a.bcd
Usually 1 ≤ coefficient < 10
Exponent: e
A whole number
Exponential Notation
Conversion Between Decimal Numbers and
Standard Exponential Notation
Example:
Convert 724,000 to standard exponential notation
7.24  10e
724000.
Five places
7.24  105
Exponential Notation
Conversion Between Decimal Numbers and
Standard Exponential Notation
Example:
Convert 0.000427 to standard exponential notation
4.27  10e
0.000427
Four places
4.27  10–4
Dimensional Analysis
Goal 3
In a problem, identify given and wanted quantities that
are related by a PER expression. Set up and solve the
problem by dimensional analysis.
Dimensional Analysis
Dimensional Analysis
A quantitative problem-solving method featuring algebraic
cancellation of units and the use of PER expressions.
PER Expression
A mathematical statement expressing the relationship between
two quantities that are directly proportional to one another.
Examples:
24 hours PER day
10 cents PER dime
Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 1:
Identify and write down the
GIVEN quantity, including
units.
GIVEN: 23 weeks
Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 2:
Identify and write down the
units of the WANTED
quantity.
GIVEN: 23 weeks
WANTED: days
Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 3:
Write down the
PER/PATH.
GIVEN: 23 weeks
WANTED: days
PER:
7 days/week
PATH: wk
days
Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 4:
Write the calculation
setup. Include units.
GIVEN: 23 weeks
WANTED: days
PER:
7 days/week
PATH: wk
23 weeks  7 days =
week

days
Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 5:
Calculate the answer.
GIVEN: 23 weeks
WANTED: days
PER:
7 days/week
PATH: wk
days
23 weeks  7 days = 161 days
week

Dimensional Analysis
How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 6:
GIVEN: 23 weeks
Check the answer to be sure
WANTED: days
both the number and the PER:
7 days/week
units make sense.
PATH: wk
days
23 weeks  7 days = 161 days
week

More days
(smaller unit)
than weeks (larger unit).
OK.
Metric Units
Goal 4
Distinguish between mass and weight.
Goal 5
Identify the metric units of mass, length, and volume.
Metric Units
Mass and Weight
Mass is a measure of quantity of matter.
Weight is a measure of the force of gravitational attraction.
Mass and weight are directly proportional to each other.
Metric Units
The SI unit of mass is the
kilogram, kg.
It is defined as the mass of a
platinum-iridium cylinder stored in
a vault in France.
A kilogram weighs 2.2 pounds.
Metric Units
For most laboratory work, the basic metric mass unit is used:
the gram, g.
Metric Units
In the metric system, units that are larger than the
basic unit are larger by multiples of 10.
For example, the kilo- unit is 1000 times
larger than the basic unit.
Units that are smaller than the basic unit are smaller
by fractions that are also multiples of 10.
For example, the milli- unit is 1/1000 times
smaller than the basic unit.
Metric Units
Metric Prefixes
Large Units
Metric
Metric
Prefix
Symbol
teraT
gigaG
megaM
kilok
hectoh
decada
Unit
Multiple
1012
109
106
1000
100
10
1
Small Units
Metric
Metric
Prefix
Symbol
Unit
decid
centic
millim
microµ
nanon
picop
Multiple
1
0.1
0.01
0.001
10–6
10–9
10–12
Metric Units
Length
The SI unit of length is the meter, m.
It is defined as the distance light travels in a vacuum in
1/299,792,458 second
The meter is 39.37 inches:
100 cm
1 in.
1m 

= 39.37 in.
m
2.54 cm
Metric Units
One inch is defined as 2.54 centimeters.
Metric Units
Volume
The SI unit of volume is the cubic meter, m3.
This is a derived unit. V = l  w  h.
A more practical unit for laboratory work is the
cubic centimeter, cm3.
Metric Units
One liter (L) is defined as exactly 1000 cubic centimeters.
1 mL = 0.001 L = 1 cm3
Metric Units
Goal 6
State and write with appropriate metric prefixes the relationship
between any metric unit and its corresponding kilounit, centiunit,
and milliunit.
Goal 7
Using Table 3.1, state and write with appropriate metric prefixes the
relationship between any metric unit and other larger and smaller
metric units.
Goal 8
Given a mass, length, or volume expresed in metric units, kilounits,
centiunits, or milliunits, express that quantity in the other three
units.
Metric Units
Metric Relationship
Example
1000 units per kilounit
1000 meters per kilometer
1000 m/km
100 centiunits per unit
100 centigrams per gram
100 cg/g
1000 milliunits per unit
1000 milliliters per liter
1000 mL/L
Metric Units
Example:
How many centigrams are in 0.87 gram?
Solution:
Use dimensional analysis.
GIVEN: 0.87 g
WANTED: cg
PER:
100 cg/g
PATH: g
cg
100 cg
0.87 g  g
= 87 cg
More centigrams (smaller unit) than grams (larger unit). OK.

Metric Units
Example:
How many kilometers are in 2,335 meters?
Solution:
Use dimensional analysis.
GIVEN: 2335 m
WANTED: km
PER:
1000 m/km
PATH: m
km
1 km
2335 m  1000 m = 2.335 km
More meters (smaller unit) than kilometers (larger unit). OK.

Metric Units
Example:
How many milliliters are in 0.00339 liter?
Solution:
Use dimensional analysis.
GIVEN: 0.00339 L
WANTED: mL
PER:
1000 mL/L
PATH: L
mL
1000 mL
0.00339 L 
= 3.39 mL
L
More milliliters (smaller unit) than liters (larger unit). OK.

Significant Figures
Goal 9
State the number of significant figures in a given
quantity.
Significant Figures
Uncertainty in Measurement
No measurement is exact.
In scientific writing, the uncertainty associated with a measured
quantity is always included.
By convention, a measured quantity is expressed by stating all
digits known accurately plus one uncertain digit.
Significant Figures
Significant Figures
The bottom board is one meter long.
How long is the top board?
More than half as long as the meter stick,
but less than one meter—about 6/10 of a meter.
The uncertain digit is the last digit written: 0.6 m
Significant Figures
Now the meter stick has marks every 0.1 m,
numbered in centimeters. How long is the board?
Between 0.6 m and 0.7 m with certainty, and the uncertain digit
must be estimated—the board is about 4/10 of the way
between 0.6 m and 0.7 m: 0.64 m.
Significant Figures
The measuring instrument now has centimeter marks.
How long is the board?
Between 0.64 m and 0.65 m with certainty.
It is about 3/10 of the way between the two marks,
so we record 0.643 m as the length of the board.
Significant Figures
The measuring instrument now has millimeter marks.
We could estimate between the millimeter marks, but the
alignment of the board and the meter stick has an uncertainty
of a millimeter or so.
We have reached the limit of this measuring instrument: 0.643 m.
Significant Figures
Significant Figures
Significant figures are applied to measurements
and quantities calculated from measurements.
They do not apply to exact numbers.
An exact number has no uncertainty.
Types of exact numbers:
Counting numbers
Numbers fixed by definition
Significant Figures
Significant Figures
The number of significant figures in a quantity is the number of
digits that are known accurately plus the one that is
uncertain—the uncertain digit.
The uncertain digit is the last digit written
when expressing a scientific measurement.
Significant Figures
Significant Figures
The measurement process, not the unit in which
the result is expressed, determines the
number of significant figures in a quantity.
The length of the board in the previous illustrations
was 0.643 m. Expressed in centimeters, it is 64.3 cm.
They are the same measurement with the same uncertainty.
Both must have the same number of significant figures.
Significant Figures
Significant Figures
The location of the decimal point has nothing
to do with significant figures.
The same 0.643 m board is 0.000643 km.
The three zeros before the decimal point are not significant.
Begin counting significant figures at the first nonzero digit,
not at the decimal point.
Significant Figures
Significant Figures
The uncertain digit is the last digit written.
If the uncertain digit is a zero to the right of the decimal point,
that zero must be written.
If the mass of a sample on a triple beam balance is 15.10 g, and
the balance is accurate to ±0.01 g, the last digit recorded must
be zero to indicate the correct uncertainty.
Significant Figures
Significant Figures
Exponential notation must be used for very large numbers to
show if final zeros are significant.
If the length of the 0.643 m board is expressed
in micrometers, its length is 643,000 µm.
The uncertainty is ±1,000 µm.
The ordinary decimal number makes this ambiguous.
Writing 6.43  105 µm shows clearly the correct
location of the uncertain digit.
Significant Figures
Goal 10
Round off given numbers to a specified number of
significant figures.
Significant Figures
Rounding a Calculated Number
If the first digit to be dropped is less than 5,
leave the digit before it unchanged.
Examples:
Round to three significant figures.
Answers
1.743 m
1.74 m
0.041239 kg
0.0412 kg
Significant Figures
Rounding a Calculated Number
If the first digit to be dropped is 5 or more,
increase the digit before it by 1.
Examples:
Round to three significant figures.
Answers
32.88 mL
32.9 mL
0.0097761 km
0.00978 km
Significant Figures
Goal 11
Add or subtract given quantities and express the result
in the proper number of significant figures.
Significant Figures
Significant Figure Rule
for Addition and Subtraction
Round off the answer to the first column
that has an uncertain digit.
Significant Figures
Example:
The following is a list of masses of items to be shipped. What is
the total mass of the package?
Carton: 226 g; Item 1: 33.5 g; Item 2: 589 g; Packaging: 11.88 g
Answer:
226
g
33.5 g
589
g
11.88g
860.38g
= 860 g = 8.60  102 g
Significant Figures
Goal 12
Multiply or divide given measurements and express the
result in the proper number of significant figures.
Significant Figures
Significant Figure Rule
For Multiplication and Division
Round off the answer to the same number of significant figures
as the smallest number of significant figures in any factor.
Significant Figures
Example:
What is the volume of a cube that is 34.49 cm long, 23.0 cm wide
and 15 cm high?
Solution:
Use algebra because the GIVENS and WANTED are related by a
formula, volume = length  width  height.
V = l  w  h = 34.49 cm  23.0 cm  15 cm
4 sf
3 sf
2 sf
= 11,899.05 cm3 (unrounded)
The answer is rounded to 2 sf, 1.2  104 cm3
Metric–USCS Conversions
Goal 13
Given a metric–USCS conversion factor and a quantity
expressed in any unit in Table 3.2, express that
quantity in corresponding units in the other system.
Metric–USCS Conversions
Conversions between the United States Customary System
(USCS) and the metric system are made by applying
dimensional analysis.
Length
1 in.  2.54 cm (definition of an inch)
Mass
1 lb  453.59237 g (definition of a pound)
Volume
1 gal  3.785411784 L (exactly)
Metric–USCS Conversions
Citizens of the U.S. should know USCS–USCS conversions
Length
1 ft  12 in.
1 yd  3 ft
1 mi  5280 ft
Mass (Weight)
1 lb = 16 oz
Volume
1 qt = 32 fl oz
1 gal = 4 qt
Metric–USCS Conversions
Example:
How many milliliters are in 1.0 quart?
Solution:
PER:
1 gal/4 qt
3.785 L/gal 1000 mL/L
PATH: qt
gal
L
mL
1 gal 3.785 L 1000 mL
1.0 qt  4 qt 

= 9.4  102 mL
L
gal



CHECK: More milliliters (smaller unit) than quarts (larger unit). OK.
Temperature
Goal 14
Given a temperature in either Celsius or Fahrenheit
degrees, convert it to the other scale.
Goal 15
Given a temperature in Celsius degrees or kelvins,
convert it to the other scale.
Temperature
Temperature
Fahrenheit Temperature Scale
Water freezes at 32°F and boils at 212°F.
There are 180 Fahrenheit degrees between freezing and boiling.
Celsius Temperature Scale
Water freezes at 0°C and boils at 100°C.
There are 100 Celsius degrees between freezing and boiling.
180
T°F – 32 = 100 T°C

T°F – 32 = 1.8 T°C
Temperature
Kelvin (Absolute) Temperature Scale
The degree is the same size as a Celsius degree,
but 0 on the Kelvin scale is set at the lowest
temperature possible, which is –273°C.
TK = T°C + 273
Temperature
Example:
Convert 65°F to its equivalent in degrees Celsius and kelvins.
Solution:
GIVEN: 65°F
WANTED: °C and K
EQUATIONS: T°F – 32 = 1.8 T°C and TK = T°C + 273
T F Š 32
65 Š 32
T°C =
=
1.8 = 18°C
1.8


TK = T°C + 273 = 18 + 273 = 291 K
Proportionality and Density
Goal 16
Write a mathematical expression indicating that one quantity is
directly proportional to another quantity.
Goal 17
Use a proportionality constant to convert a proportionality to an
equation.
Goal 18
Given the values of two quantities that are directly proportional to
each other, calculate the proportionality constant, including its
units.
Proportionality and Density
Goal 19
Write the defining equation for a proportionality constant
and identify units in which it might be expressed.
Goal 20
Given two of the following for a sample of a pure
substance, calculate the third: mass, volume, and
density.
Proportionality and Density
A direct proportionality exists between two quantities when
they increase or decrease at the same rate.
If a graph of two related measurements is a straight line that
passes through the origin, the measured quantities are directly
proportional to each other.
ab
a is proportional to b
Proportionality and Density
We can describe direct proportionalities between measured
quantities with PER expressions.
Direct proportionalities between measured quantities yield two
conversion factors between the quantities.
Given either quantity in a direct proportionality and the
conversion factor between the quantities, we can calculate the
other quantity with dimensional analysis.
Proportionality and Density
The mass and volume of any pure substance at a given
temperature are directly proportional:
mass is proportional to volume
mass  volume
mV
A proportionality is changed into an equation by inserting a
multiplier called a proportionality constant.
Let D be the proportionality constant:
m=DV
Proportionality and Density
Solving for the proportionality constant yields the defining
equation for a physical property of a pure substance called
density:
m
D V

In words, density is the mass per unit volume of a substance:
mass
Density  volume

Proportionality and Density
The definition of density establishes its common units:
Density 
mass
volume

The common laboratory
unit for mass is grams.
The common laboratory unit for volume is
milliliters or cubic centimeters.
Density is therefore typically expressed in g/mL or g/cm3.
Since volume varies with temperature,
density is temperature dependent.
Proportionality and Density
Densities of Some Common Substances
(g/cm3 at 20°C and 1 atm)
Substance
Helium
Air
Pine lumber
Maple lumber
Oak lumber
Water
Glass
Density
0.00017
0.0012
0.5
0.6
0.8
1.0
2.5
Substance
Aluminum
Iron
Copper
Silver
Lead
Mercury
Gold
Density
2.7
7.8
9.0
10.5
11.4
13.6
19.3
Proportionality and Density
Proportionality and Density
Water is unusual in that its solid
phase, ice, will float on its liquid
phase.
Solid ethanol sinks to the bottom of
the liquid. The solid form of almost
all substances is more dense than
the liquid phase.
Proportionality and Density
Example:
What is the volume of 15 g of silver?
Solution:
GIVEN: 15 g silver
WANTED: volume (assume cm3)
PER:
10.5 g/cm3 (from table)
PATH: g
cm3
1 cm3
15 g  10.5 g = 1.4 cm3

Strategy for Solving Problems
The only way to learn how to solve problems is to
solve them for yourself.
However, we can provide you with some general
guidelines for solving chemistry problems.
Strategy for Solving Problems
A problem can be solved by dimensional analysis if the
GIVEN and WANTED can be linked by one or more
PER expressions and you know or can find the
conversion factor for each expression.
A problem can be solved by algebra if the GIVEN and
WANTED appear in an algebraic equation
in which the WANTED is the only unknown.
Strategy for Solving Problems
Reflective Practice
To become a better problem solver, you must practice solving
problems. When you solve a problem and check the answer
section of the textbook, reflect on both the chemistry concept
and the strategy you used in solving the problem.
The reason for solving problems is not to get the same answers
as the textbook authors, but to learn how to solve the problem.