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KINETIC THEORY OF GASES
In an attempt to explain the compressibility of
gases, Bernoulli proposed the hypothesis that is
accepted as the kinetic theory
of gases. by
continual bombardment of the walls by molecules
of the gas.
Consider the container shown in figure 11-2 as
containing a gas. At any given time, some molecules
are moving in one direction, some are traveling
in other directions; some are traveling fast,
some slow, and some may even be in a state of
rest. The average effect of the molecules bombarding
each container wall corresponds to the
pressure of the gas.
As more gas is pumped into the container, more
molecules are available to bombard the walls;
thus the pressure in the container increases. The
gas pressure in a container can also be increased
by increasing the speed with which the molecules
hit the walls. If the temperature of the gas
is raised, the molecules move faster causing an
increase in pressure. This can be shown by considering
the automobile tire. When you take a
long drive on a hot day, the pressure in the tires increases
and a tire which appeared to be somewhat
"soft" in cool morning temperature may
appear normal at a higher midday temperature.
Figure 11-2.—Molecular bombardment creating pressure.
BOYLE’S LAW
When the automobile tire is initially inflated, air
which normally occupies a specific volume is compressed
into a smaller volume inside the tire. This
increases the pressure on the inside of the tire. Charles
Boyle, an English scientist, was among the
first to experiment with the pressure-volume relationship
of gas. During an experiment when he
compressed a volume of air he found that the volume
decreased as the pressure increased, and by
doubling the force exerted on the air he could decrease
the volume of the air by half. See figure 11-3.
Recall from the example of the automobile tire
that changes in temperature of a gas also change
the pressure and volume. Therefore, the experiment
must be performed at a constant temperature.
The relationship between pressure and
volume is known as Boyle’s
law. It states: When
the temperature of a gas is kept constant, the
volume of an enclosed gas varies inversely with its
pressure.
In equation form, this relationship may be expressed
as either
or 
Equation 11-6
where V1 and
P1 are
the original volume and pressure,
and V2 and
P2 are
the final volume and pressure (P1
and P2
are absolute pressures).
Figure 11-3.-Gas compressed to half its original volume by
a
doubled force.
Example of Boyle’s law: 4 cubic feet of nitrogen
are under a pressure of 100 psi (gauge).
The
nitrogen is allowed to expand to a volume of
6 cubic feet. What is the new gauge pressure? Remember
to convert gauge pressure to absolute pressure
by adding 14.7. Using equation
11-6, V1P1
= V2P2,
where V1 is
4 ft3,
V2 is
6 ft, and P1 is
100 psig:
CHARLES’S LAW
Boyle’s law assumes conditions of constant temperature.
In actual situations this is rarely the case.
Temperature changes continually and affects the
volume of a given mass of gas. Jacques
Charles, a French physicist, provided much
of the foundation for the modern kinetic theory
of gases. Through experiments, he found that
all gases expand and contract proportionally to
the change in the absolute temperature, providing
the pressure remains constant. The relationship
between volume and temperature is known
as Charles’s law. It
states: The volume of
a gas is proportional to its absolute
temperature, if constant
pressure is maintained. In
equation form, this relationship
may be expressed as
Equation 11-7
where V1 and
V2 are
the original and final volumes, and
T1 and
T2 are
the original and final absolute
temperatures.
Since an increase in the temperature of a gas causes
it to expand if the pressure is kept constant, it
is reasonable to expect that if a given sample is
heated within a closed container and its volume remains
constant, the pressure of the gas will increase.
Experiments have proven this to be true. In
equation form, this becomes
 or 
Equation 11-8
This equation states that for a constant volume, the
absolute pressure of a gas varies directly with the
absolute temperature.
Example: A cylinder of gas under a pressure of
1800 psig at 70°F is left out in the sun in the tropics
and heats up to a temperature of 130°F. What
is the new pressure within the cylinder? (Remember
that both pressure and temperature must
be converted to absolute pressure and absolute
temperature.)
Converting absolute pressure to gauge
pressure:
GENERAL GAS LAW
We have learned that Boyle’s law pertains to situations
in which the temperature remains constant
(fig. 11-4), and that Charles’s law pertains
to situations in which pressure remains constant
(fig. 11-4). It is usually not possible to control
pressure or temperature in tanks or bottles of
gas subject to the weather and shipboard demands.
Boyle’s and Charles’s laws are combined to
form the general gas law. This
law states: The product of the
initial pressure, initial volume, and
new temperature (absolute scale) of an enclosed
gas is equal to the product of the new pressure,
new volume, and initial temperature. It
is a mathematical statement which allows
many gas problems to be solved by
using the principles of Boyle’s
law and/or Charles’s law. The equation is
expressed as
or
(P and T represent absolute pressure and absolute temperature,
respectively.) You can see by
examining figure 11-4 that the three
equations are special cases of the general equation.
Thus, if the temperature remains constant,
T1 equals
T2 and
both can be eliminated from the
general formula, which then reduces to the
form shown in part A. When the volume remains
constant, V1 equals
V2,
thereby reducing
Figure 11-4.—The general gas law.
the general equation to the form given in part B. Similarly,
P1 is
equated to P2 for
constant pressure, and the equation
then takes the form given in part
C.
The general gas law applies with exactness only to
"ideal" gases in which the molecules are assumed
to be perfectly elastic. However, it describes
the behavior of actual gases with sufficient
accuracy for most practical purposes. Two
examples of the general equation follow: 1.
Two cubic feet of a gas at 75 psig and 80°F are
compressed to a volume of 1 cubic foot and then
heated to a temperature of 300°F. What is the
new gauge pressure? Using equation
11-9, P1V1T2
= P2V2T1,
where V1 is
2 ft3,
P1 is
75 psig, T1 is
80°F, V2 is
1 ft3 and
T2 is
300°F:
Solution:
Substituting:
Converting absolute pressure to gauge
pressure:
2. Four cubic feet of a gas at 75 psig and 80°F are
compressed to 237.8 psig and heated to a temperature
of 300°F. What is the volume of the gas
resulting from these changes? Using equation 11-9,
P1V1T2
= P2V2T1,
where V1 is
4 ft3,
P2 is
T1 is
800, P1 is
237.8 psig, and T2 is
300°F:
Solution:
Substituting:
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