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Mark S. CracoliceEdward I. Peters
http://academic.cengage.com/chemistry/cracolice
Mark S. Cracolice • The University of Montana
Chapter 3Measurement and
Chemical Calculations
Introduction to MeasurementMeasure
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 thedistance 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 MeasurementHow 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 MeasurementMeasurements 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 MeasurementThe SI system is defined by seven base units.
Examples of base units include:
Quantity Base Unit
Mass (weight) Kilogram
Length Meter
Temperature Kelvin
Time Second
Other measurement units are derived from the base units; accordingly, they are called derived units.
Exponential NotationGoal 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 NotationExponentials
BP
B is the base
p is the power or exponent
104 = 10 10 10 10 = 10,000
10–4 = = = = 0.0001
1104
110
110
110
110
110,000
Exponential NotationExponential Notation
a.bcd 10e
Coefficient: a.bcd
Usually 1 ≤ coefficient < 10
Exponent: e
A whole number
Exponential NotationConversion Between Decimal Numbers and
Standard Exponential Notation
Example:Convert 724,000 to standard exponential notation
7.24 10e
7 2 4 0 0 0 .
Five places
7.24 105
Exponential NotationConversion Between Decimal Numbers and
Standard Exponential Notation
Example:Convert 0.000427 to standard exponential notation
4.27 10e
0 . 0 0 0 4 2 7
Four places
4.27 10–4
Dimensional AnalysisGoal 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 AnalysisDimensional 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 AnalysisHow 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 GIVEN: 23 weeks
units.
Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 2:
Identify and write down the GIVEN: 23 weeks
units of the WANTED WANTED: days
quantity.
Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 3: GIVEN: 23 weeks
Write down the WANTED: days
PER/PATH. PER: 7 days/week
PATH: wk days
Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 4: GIVEN: 23 weeks
Write the calculation WANTED: days
setup. Include units. PER: 7 days/week
PATH: wk days
23 weeks =
7 daysweek
Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 5: GIVEN: 23 weeks
Calculate the answer. WANTED: days PER: 7 days/week
PATH: wk days
23 weeks = 161 days
7 daysweek
Dimensional AnalysisHow to Solve a Problem by Dimensional Analysis
Sample Problem:How many days are in23 weeks?
Step 6: GIVEN: 23 weeksCheck the answer to be sure WANTED: days both the
number and the PER: 7 days/weekunits make sense. PATH: wk days
23 weeks = 161 days
More days (smaller unit)than weeks (larger unit).OK.
7 daysweek
Metric UnitsGoal 4
Distinguish between mass and weight.
Goal 5
Identify the metric units of mass, length, and volume.
Metric UnitsMass 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 UnitsThe 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 UnitsIn 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 UnitsMetric Prefixes
Large Units Small Units
Metric Metric Metric Metric
Prefix Symbol Multiple Prefix Symbol Multiple
tera- T 1012 Unit 1
giga- G 109 deci- d 0.1
mega- M 106 centi- c 0.01
kilo- k 1000 milli- m 0.001
hecto- h 100 micro- µ 10–6
deca- da 10 nano- n 10–9
Unit 1 pico- p 10–12
Metric UnitsLength
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:
1 m 100 cm
m
1 in.2.54 cm
= 39.37 in.
Metric UnitsVolume
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 UnitsGoal 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 UnitsMetric 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 UnitsExample:
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
0.87 g = 87 cg
More centigrams (smaller unit) than grams (larger unit). OK.
100 cgg
Metric UnitsExample:
How many kilometers are in 2,335 meters?
Solution:
Use dimensional analysis.
GIVEN: 2335 m
WANTED: km
PER: 1000 m/km
PATH: m km
2335 m = 2.335 km
More meters (smaller unit) than kilometers (larger unit). OK.
1 km1000 m
Metric UnitsExample:
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
0.00339 L = 3.39 mL
More milliliters (smaller unit) than liters (larger unit). OK.
1000 mLL
Significant FiguresUncertainty 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
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 FiguresSignificant 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 FiguresSignificant 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 FiguresSignificant 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 FiguresSignificant 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 FiguresSignificant 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 FiguresSignificant 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 FiguresRounding 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 FiguresRounding 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 FiguresGoal 11
Add or subtract given quantities and express the result in the proper number of significant figures.
Significant FiguresSignificant Figure Rule
for Addition and Subtraction
Round off the answer to the first column
that has an uncertain digit.
Significant FiguresExample:
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:
2 2 6 g
3 3 . 5 g
5 8 9 g
1 1 . 8 8 g
8 6 0 . 3 8 g = 860 g = 8.60 102 g
Significant FiguresGoal 12
Multiply or divide given measurements and express the result in the proper number of significant figures.
Significant FiguresSignificant 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 FiguresExample:
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 ConversionsGoal 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 ConversionsConversions 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 ConversionsCitizens of the U.S. should know USCS–USCS conversions
Length1 ft 12 in.1 yd 3 ft
1 mi 5280 ft
Mass (Weight)1 lb = 16 oz
Volume1 qt = 32 fl oz1 gal = 4 qt
Metric–USCS ConversionsExample:
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
1.0 qt = 9.4 102 mL
CHECK: More milliliters (smaller unit) than quarts (larger unit). OK.
1 gal4 qt
3.785 Lgal
1000 mLL
TemperatureGoal 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.
TemperatureFahrenheit 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.
T°F – 32 = T°C
T°F – 32 = 1.8 T°C
180100
TemperatureKelvin (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
TemperatureExample:
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°C = = = 18°C
TK = T°C + 273 = 18 + 273 = 291 K
TF
Š 321.8
65 Š 321.8
Proportionality and DensityGoal 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 DensityGoal 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 DensityA 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 DensityWe 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 DensityThe 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 DensitySolving for the proportionality constant yields the defining equation for a physical property of a pure substance called
density:
D
In words, density is the mass per unit volume of a substance:
Density
mV
massvolume
Proportionality and DensityThe definition of density establishes its common units:
Density
The common laboratory unit for mass is grams.
The common laboratory unit for volume ismilliliters or cubic centimeters.
Density is therefore typically expressed in g/mL or g/cm3.
Since volume varies with temperature,density is temperature dependent.
massvolume
Proportionality and DensityDensities of Some Common Substances
(g/cm3 at 20°C and 1 atm)
Substance Density Substance Density
Helium 0.00017 Aluminum 2.7
Air 0.0012 Iron 7.8
Pine lumber 0.5 Copper 9.0
Maple lumber 0.6 Silver 10.5
Oak lumber 0.8 Lead 11.4
Water 1.0 Mercury 13.6
Glass 2.5 Gold 19.3
Proportionality and DensityWater 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 DensityExample:
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
15 g = 1.4 cm3
1 cm3
10.5 g
Strategy for Solving ProblemsThe 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 ProblemsA 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.
Reflective PracticeTo 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.