Page 26 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 2] MATHEMATICAL METHODS IN CHEMISTRY 15

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EXAMPLE 2.10. A rectangle having an area of 22.0 m is 4.00 m wide. How long is it?
Ans. A = l × w
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22.0m = l(4.00 m)
l = 5.50 m
Note that the length has a unit of distance (meter).

EXAMPLE 2.11. What happens to the area of a square when the length of each side is doubled?
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2
Ans. Let l = original length of side; then l = original area; 2l = new length of side; so (2l) = new area. The area has
2
2
increased from l to 4l ; it has increased by a factor of 4. (See Problem 2.24.)
Volume
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The SI unit of volume is the cubic meter, m . Just as area is derived from length, so can volume be derived
from length. Volume is length × length × length. Volume also can be regarded as area × length. The cubic meter
3
3
is a rather large unit; a cement mixer usually can carry between 2 and 3 m of cement. Smaller units are dm ,
3
3
cm , and mm ; the first two of these are reasonable sizes to be useful in the laboratory.
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The older version of the metric system uses the liter as the unit of volume. It is defined as 1 dm . Chemists
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often use the liter in preference to m because it is about the magnitude of the quantities with which they deal.
The student has to know both units and the relationship between them.
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Often it is necessary to multiply by a factor raised to a power. Consider the problem of changing 5.00 m to
cubic centimeters:

100 cm
3
5.00 m
1m
2
If we multiply by the ratio of 100 cm to 1 m, we will still be left with m (and cm) in our answer. We must
multiply by (100 cm/m) three times;
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100 cm 3
5.00 m 3 = 5 000 000 cm
1m
100 cm 100 cm 100 cm 100 cm
3
means
1m 1m 1m 1m
3
3
3
and includes 100 cm in the numerator and 1 m in the denominator.
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EXAMPLE 2.12. How many liters are there in 1 m ?
10 dm 1L
3
Ans. 1m 3 = 1000 L
1m 1dm 3
One cubic meter is 1000 L. The liter can have prefixes just as any other unit can. Thus 1 mL is 0.001 L, and 1 kL is
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1000 L = 1m .
Mass
Mass is a measure of the quantity of material in a sample. We can measure that mass by its weight—the
attraction of the sample to the earth or by its inertia—the resistance to change in its motion. Since weight and
mass are directly proportional as long as we stay on the surface of the earth, chemists sometimes use these terms
interchangeably. (Physicists do not do that.)
The unit of mass is the gram. [Since 1 g is a very small mass, the legal standard of mass in the United
States is the kilogram. A standard for a type of measurement is an easily measured quantity that is chosen for