Note: This completed assignment is due in Recitation during Week 2!
“Anything worth measuring is worth measuring
well.”
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Partner(s)______________________________
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Chemistry 221
ChemActivity 0
Mathematics & Measurements
To determine if a runner broke the world’s record for a marathon, you must carefully
measure the time that passed between the start and finish of the race and compare it to
the record time for that marathon. Since time can be measured and expressed as an
amount, it is called a quantity. Ten seconds, two minutes, and five hours are examples
of quantities of time. Other familiar quantities that are important in chemistry include
mass (similar to the more familiar weight), length, volume, and density.
The International System of Units
In 1960, a group of scientists from many fields and many countries agreed upon a set of
metric units that would serve as a standard for scientific communication. This standard
set of units is known as the International System of Units and is abbreviated SI (the
abbreviation is derived from the French spelling le Systeme International d’ Unites).
Seven quantities are the foundation for SI, and each has a base unit in which that
quantity is expressed. Table 1 lists the base units for length, mass, and time, along with
their abbreviations and their relationships to common United States units.
Table 1. Three SI Base Units Commonly Used in General Chemistry
Quantity
Base Unit
Abbreviation
U.S. Equivalent
Length
meter
m
39.37 inches
Mass
kilogram
kg
2.205 pounds
Time
second
s
1 second
The three SI base units listed in Table 1 were chosen because they correspond to
magnitudes which are convenient for everyday measurement. They are well suited for
measurements at the macroscopic scale. However, chemists often work with tiny
quantities such as those used to express the diameter of a hydrogen atom or huge
quantities such as the number of particles in a kilogram of carbon. These numbers are
beyond the range of our senses and cannot be conveniently expressed in standard
notation in SI units. Thus, the system of scientific notation is used to express very small
and very large quantities.
Scientific notation is a method of expressing numbers as a product of two factors. The
first factor is a number that is greater than or equal to 1 but less than 10. The second
factor is 10 raised to a power. The power of 10, or exponent, is positive for numbers
greater than 10 and negative for numbers less than 1. Table 2 gives examples showing
both the ordinary decimal form and the exponential form for some quantities. Notice
how scientific notation eliminates the need to write a long list of zeros in very small and
very large numbers.
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Table 2. Examples of Quantities Expressed in Scientific Notation
Quantity
Ordinary Decimal Form
Diameter of a
hydrogen atom
Mass of a
hydrogen atom
Number of
molecules in
2.0 g of hydrogen
Scientific
Notation
0.000 000 000 074 m
7.4 x 10–11 m
0.000 000 000 000 000 000 000 000 11 kg
1.1 x 10–25 kg
600 000 000 000 000 000 000 000
6.0 x 1023
Metric Prefixes
To further simplify the expression of measured quantities, scientists
use prefixes with metric units to represent powers of ten. Table 3 lists metric prefixes
frequently used in chemistry. Notice that each prefix has an abbreviation and an
equivalent power of ten. Your instructor may require you to memorize some or all of the
prefixes and their associated powers of ten.
Table 3. Common Metric Prefixes and Equivalent Powers of Ten
Prefix
Abbreviation
Power of Ten
picop
10–12
nanon
10–9
microµ
10–6
millim
10–3
centic
10–2
kilok
103
megaM
106
gigaG
109
Example
picogram, pg
nanometer, nm
microsecond, µs
milliliter, mL
centimeter, cm
kilometer, km
megahertz, MHz
gigabyte, GB
Mathematical Operations in Scientific Notation
Addition and Subtraction
To add or subtract numbers that are expressed in
scientific notation, use the “enter exponent” key on your calculator to express the power
of ten. It is usually labeled “EE” or “EXP.” Do not use the 10x or yx key for scientific
notation exponents. The calculator usually will express the result in scientific notation
automatically. For example:
3.25 x 103 + 4.66 x 104 =
Calculator key sequence:
3
.
2
5
EE
3
+
4
.
6
6
EE
4
Result: 4.985 x 104
Multiplication
To multiply numbers that are expressed in scientific notation, use
your calculator. For example:
(3.1 x 102) x (5.2 x 104) =
Calculator key sequence:
3
.
1
EE
2
Result: 1.612 x 107
2

5
.
2
EE
4
=
=
ChemActivity 0
Division
To divide numbers that are expressed in scientific notation, again, let your
calculator do the work. For example:
7.5x10 2
5.9x10 4
=
Calculator key sequence:
7
.
5
EE
2
÷
5
.
9
EE
4
+/
–
=
Result: 1.271... x 106
Powers
To raise a number expressed in scientific notation to a power, use the y x key
on your calculator. For example:
(2.5x 102)3 =
Calculator key sequence:
2
.
5
EE
2
yx
3
=
Result: 1.5625 x 107
Measured Quantities and Significant Figures
There is a degree of uncertainty in every measurement, and this uncertainty, by
convention, is reflected in the last recorded digit of any measured quantity. If several
people measure the same distance with a ruler, their measurements will probably differ
in the last digit. However, these measurements will cluster around the true value. Some
will be equal to the true value, some will be higher, and some will be lower. The average
of a series of measurements is generally considered to be the most accurate value.
The number of significant figures in a measurement is the total of the number of digits
known with certainty plus the one uncertain digit. The digits known with certainty are
those that can be read precisely from the measuring instrument. The last digit recorded
in any measured quantity—the uncertain digit—is estimated.
When a measurement is written in scientific notation, the first factor represents the
significant digits in the measured quantity. The second factor, the power of ten,
represents the magnitude of the measurement, or the number of decimal places, and
therefore has nothing to do with the number of significant figures in the quantity. If the
three-significant-figure measured quantity 134 pounds is written in scientific notation
as 1.34 x 102 pounds, the number of significant figures cannot change, so this must
also have three significant figures.
The following rules summarize the conventions used by chemists when working with
measured quantities:
1.
The last digit expressed in a measured quantity is the uncertain, or estimated,
digit.
2.
Zeros that serve as place holders in ordinary decimal notation are not
significant. For example, the measured quantity 0.001 m has one significant
figure. This becomes more apparent when we write the quantity in scientific
notation: 1 x 10–3 m.
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3.
When adding and subtracting numbers representing measured quantities, the
number of decimal places in the sum or difference are limited by the number of
decimal places in the measured quantity that has the least number of decimal
places. For example, 0.152 g + 0.26 g = 0.41 g.
4.
When multiplying and/or dividing numbers representing measured quantities,
the number of significant figures in the result is the number of significant
figures in the factor with the least number of significant figures. For example,
(1.5 x 102) + (1.350 x 103) = 2.0 x 105.
Self Test
1.
Express the following in ordinary decimal form or scientific notation. The first
line has been completed as an example.
Ordinary Decimal Form
Scientific Notation
0.683 kg
6.83 x 10–1 kg
1365 s
1.034 x 101 m
300 000 000 m · s–1
(1.75 x 105) (2.0 x 105)
1605 + 3.22 x 102
2.
How many significant figures would be appropriate for each of the following?
a) If you were to measure your height in inches. In cm.
b) If you were to measure your weight in pounds. In kg.
c) The speed of a car as read from a speedometer (mi/hr).
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ChemActivity 0
3.
Evaluate each of the following expressions. Write your answers in scientific
notation and with the correct number of significant figures.
a)
b)
6.42 x 104 + 3.5 x 103 =
2.00x10 5
4.0x103
=
c)
(5 x 102) x (8.0 x 103) =
d)
(4.00 x 10–2)3 =
Answers:
1. 1.365 x 103 s; 10.34 m; 3 x 108 ms-1 (or m/s); 3.5 x 1010; 1.927 x 103
3. a) 6.77 x 104
b) 5.0 x 101
c) 4 x 106
d) 6.40 x 10-5
Unit Conversion—Dimensional Analysis
It is often necessary to convert a measurement from one system of units to another,
particularly for citizens and residents of the United States. In spite of the fact that all
other countries of the world and all scientists use the metric system to express
measured quantities, the U.S. still clings to the archaic British system of measurement.
Even Great Britain herself has converted to the metric system.
For example, when your physician prescribes medication, he or she needs to convert
your body weight to kilograms because dosages are usually expressed as milligrams of
medication per kilogram of body weight. To convert a quantity from one system of units
to another, medical personnel, scientists, and engineers frequently use a procedure
called dimensional analysis.
Measured quantities are always represented by a number and its associated unit, such
as 1.9 pounds or 3.5 inches. If you think of the number as a factor that multiplies the
unit, you can apply standard algebraic conventions when you convert a measured
quantity from one system of units to another. For example, to convert 3.45 kilograms to
pounds, you multiply the given unit, kilograms, by a conversion factor that algebraically
cancels the kilogram unit and yields pounds. Here’s the conversion:
3.45 kg x
2.205 lb
= 7.61 lb
1 kg
Dimensional analysis works because the given unit is always multiplied by a conversion
factor that is equal to one. The conversion factor comes from an equation that relates
the given unit to the wanted, or desired, unit. For example, the equation
1 kg = 2.205 lb
defines the relationship between kilograms and pounds. If we divide both sides of this
equation by 1 kg, we get a fraction that is equivalent to one:
1 kg
2.205 lb
1 kg = 1 =
1 kg
The expression 2.205 lb/1 kg is a conversion factor that changes kilograms to pounds
or vice versa. The “1 kg” quantity in this conversion factor is exactly 1 kilogram.
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Therefore when you use this conversion factor, the number of significant figures is
determined by the number of significant figures in 2.205 lb.
Dimensional Analysis in Three Simple Steps
You can use the following three-step process to convert any given unit to the unit you
need:
1.
Write a conversion factor that relates the given unit to the wanted unit. If you
cannot relate the two units directly with a single conversion factor, write a
conversion factor that relates the given unit to an intermediate unit.
2.
Multiply the given unit by the conversion factor from Step 1. Follow the algebraic
rules from multiplication of fractions.
3.
If the result of Step 2 is the wanted unit, the conversion is finished. If not, you
must convert the intermediate unit. If another conversion is necessary, repeat
Steps 1 and 2 until you arrive at the wanted unit.
Here is an example of a conversion that requires two steps:
Convert 3.00 feet to centimeters. 1 foot = 12 inches; 1 inch  2.54 centimeters.
3.00 ft x
12 in.
2.54 cm
x
= 91.4 cm
1 ft
1 in.
1.
Find the number of centimeters in 1.00 x 102 yards.
2.
Determine the number of meters in 1.00 mile.
3.
The speed of light is 1.86 x 105 miles per second. How many meters will light
travel in 1.0 second?
4.
Calculate the number of seconds in a year.
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ChemActivity 0
5.
A light year is the distance that light travels in one year. Determine the number
of miles, meters, and kilometers in one light year.
6.
The density of mercury is 13.55 g/mL, and the density of gold is 19.32 g/mL.
a) What is the density of mercury in kg/L?
b) A 10-mL graduated cylinder is filled to 5.00 mL. A ring is placed in the
graduated cylinder, and the water level rises to 5.15 mL. The ring is then
dried and placed on a balance, and its mass is 2.8315 g. Find the density of
the ring.
c) Is the ring pure gold? Explain how you arrived at your conclusion.
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7.
A single layer of gold atoms forms a surface whose dimensions are 1.0 x 10 3
angstroms by 1.0 x 103 angstroms. 1 angstrom x 10–10 meter.
a) What is the area of this surface in square angstroms?
b) What is the surface area in square centimeters?
8.
The units of the chain system of measure, used by surveyors, are as follows:
7.92 inches = 1 link
10 chains = 1 furlong
100 links = 1 chain
80 chains = 1 mile
The distance of the Kentucky Derby, a classic horse race, is 1.25 miles. How is
this distance expressed in furlongs?
9.
The displacement (total volume of the cylinders) of the engine in a Ford Mustang
is 5.0 L. Convert this to cubic inches.
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ChemActivity 0
10.
A cube that has a length of 1 cm on each side has a volume of 1 cm 3. How many
cubic centimeters are in 1 cubic meter? (Hint: The answer is not 100.)
The Challenge Problem:
Mercury (Hg) poisoning is a debilitating disease that is often fatal. In the human body,
mercury reacts with essential enzymes leading to irreversible inactivity of these
enzymes. If the amount of mercury in a polluted lake is 0.4 g Hg/mL, what is the total
mass in kilograms of mercury in the lake? (The lake has a surface area of 1.0 x 10 2 mi2
and an average depth of 20.0 ft.) Be sure to use the correct number of significant
digits in your answer.
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