


The Kinetic Theory of Gases & the
Ideal Gas Law
We said earlier that temperature is a measure of the kinetic
energy of the molecules in a material, but we didn’t elaborate on
that remark. Because individual molecules are so small, and because
there are so many molecules in most substances, it would be impossible to
study their behavior individually. However, if we know the basic
rules that govern the behavior of individual molecules, we can make
statistical calculations that tell us roughly how a collection of
millions of molecules would behave. This, essentially, is what thermal physics
is: the study of the macroscopic effects of the microscopic molecules
that make up the world of everyday things.
The kinetic theory of gases makes the transition
between the microscopic world of molecules and the macroscopic world
of quantities like temperature and pressure. It starts out with
a few basic postulates regarding molecular behavior, and infers
how this behavior manifests itself on a macroscopic level. One of
the most important results of the kinetic theory is the derivation
of the ideal gas law, which not only is very useful
and important, it’s also almost certain to be tested on SAT II Physics.
The Kinetic Theory of Gases
We can summarize the kinetic theory of gases with four
basic postulates:
 Gases are made up of molecules: We can treat molecules as point masses that are perfect spheres. Molecules in a gas are very far apart, so that the space between each individual molecule is many orders of magnitude greater than the diameter of the molecule.
 Molecules are in constant random motion: There is no general pattern governing either the magnitude or direction of the velocity of the molecules in a gas. At any given time, molecules are moving in many different directions at many different speeds.
 The movement of molecules is governed by Newton’s Laws: In accordance with Newton’s First Law, each molecule moves in a straight line at a steady velocity, not interacting with any of the other molecules except in a collision. In a collision, molecules exert equal and opposite forces on one another.
 Molecular collisions are perfectly elastic: Molecules do not lose any kinetic energy when they collide with one another.
The kinetic theory projects a picture of gases as tiny
balls that bounce off one another whenever they come into contact.
This is, of course, only an approximation, but it turns out to be
a remarkably accurate approximation for how gases behave in the
real world.
These assumptions allow us to build definitions of temperature
and pressure that are based on the mass movement of molecules.
Temperature
The kinetic theory explains why temperature should be
a measure of the average kinetic energy of molecules. According
to the kinetic theory, any given molecule has a certain mass, m;
a certain velocity, v; and a kinetic
energy of ^{1}/
_{2 } mv^{2}.
As we said, molecules in any system move at a wide variety of different
velocities, but the average of these velocities reflects the total
amount of energy in that system.
We know from experience that substances are solids at
lower temperatures and liquids and gases at higher temperatures.
This accords with our definition of temperature as average kinetic
energy: since the molecules in gases and liquids have more freedom
of movement, they have a higher average velocity.
Pressure
In physics, pressure, P,
is the measure of the force exerted over a certain area. We generally say
something exerts a lot of pressure on an object if it exerts a great
amount of force on that object, and if that force is exerted over
a small area. Mathematically:
Pressure is measured in units of pascals (Pa),
where 1 Pa = 1 N/m^{2}.
Pressure comes into play whenever force is exerted on
a certain area, but it plays a particularly important role with
regard to gases. The kinetic theory tells us that gas molecules obey
Newton’s Laws: they travel with a constant velocity until they collide,
exerting a force on the object with which they collide. If we imagine
gas molecules in a closed container, the molecules will collide
with the walls of the container with some frequency, each time exerting
a small force on the walls of the container. The more frequently
these molecules collide with the walls of the container, the greater
the net force and hence the greater the pressure they exert on the
walls of the container.
Balloons provide an example of how pressure works. By
forcing more and more air into an enclosed space, a great deal of
pressure builds up inside the balloon. In the meantime, the rubber
walls of the balloon stretch out more and more, becoming increasingly
weak. The balloon will pop when the force of pressure exerted on
the rubber walls is greater than the walls can withstand.
The Ideal Gas Law
The ideal gas law relates temperature, volume,
and pressure, so that we can calculate any one of these quantities
in terms of the others. This law stands in relation to gases in
the same way that Newton’s Second Law stands in relation to dynamics:
if you master this, you’ve mastered all the math you’re going to
need to know. Ready for it? Here it is:
Effectively, this equation tells us that temperature, T,
is directly proportional to volume, V, and
pressure, P. In metric units, volume
is measured in m^{3}, where 1m^{3} =
10^{6}cm^{2}.
The n stands for the number
of moles of gas molecules. One mole (mol) is just a
big number— to be precise—that, conveniently, is the
number of hydrogen atoms in a gram of hydrogen. Because we deal
with a huge number of gas molecules at any given time, it is usually
a lot easier to count them in moles rather than counting them individually.
The R in the law is a constant
of proportionality called the universal gas constant,
set at 8.31 J/mol · K. This constant effectively relates
temperature to kinetic energy. If we think of RT as
the kinetic energy of an average molecule, then nRT is
the total kinetic energy of all the gas molecules put together.
Deriving the Ideal Gas Law
Imagine a gas in a cylinder of base A,
with one moving wall. The pressure of the gas exerts a force of F =
PA on the moving wall of the cylinder. This force
is sufficient to move the cylinder’s wall back a distance L,
meaning that the volume of the cylinder increases by = AL. In
terms of A, this equation reads A = /L. If we
now substitute in /L for A in the
equation F = PA,
we get F = P/L, or
If you recall in the chapter on work, energy, and power,
we defined work as force multiplied by displacement. By pushing
the movable wall of the container a distance L by
exerting a force F, the
gas molecules have done an amount of work equal to FL,
which in turn is equal to P.
The work done by a gas signifies a change in energy: as
the gas increases in energy, it does a certain amount of work on
the cylinder. If a change in the value of PV signifies
a change in energy, then PV itself
should signify the total energy of the gas. In other words, both PV and nRT are
expressions for the total kinetic energy of the molecules of a gas.
Boyle’s Law and Charles’s Law
SAT II Physics will not expect you to plug a series of
numbers into the ideal gas law equation. The value of n is
usually constant, and the value of R is
always constant. In most problems, either T, P,
or V will also be held constant, so
that you will only need to consider how changes in one of those
values affects another of those values. There are a couple of simplifications
of the ideal gas law that deal with just these situations.
Boyle’s Law
Boyle’s Law deals with gases at a constant
temperature. It tells us that an increase in pressure is accompanied
by a decrease in volume, and vice versa: . Aerosol canisters contain compressed (i.e.,
lowvolume) gases, which is why they are marked with highpressure
warning labels. When you spray a substance out of an aerosol container,
the substance expands and the pressure upon it decreases.
Charles’s Law
Charles’s Law deals with gases at a constant
pressure. In such cases, volume and temperature are directly proportional: . This is how hotair balloons work: the balloon
expands when the air inside of it is heated.
Gases in a Closed Container
You may also encounter problems that deal with “gases
in a closed container,” which is another way of saying that the
volume remains constant. For such problems, pressure and temperature
are directly proportional: . This relationship, however, apparently
does not deserve a name.
Example 1

Questions like this come up all the time on SAT II Physics.
Answering it is a simple matter of applying Boyle’s Law, or remembering
that pressure and volume are inversely proportional in the ideal
gas law. If volume is halved, pressure is doubled.
Example 2

First, we have to remember that in the ideal gas law,
temperature is measured in Kelvins. In those terms, the temperature
goes from 273 K to 546 K; in other words, the temperature doubles.
Because we are dealing with a closed container, we know the volume
remains constant. Because pressure and temperature are directly
proportional, we know that if the temperature is doubled, then the
pressure is doubled as well. This is why it’s a really bad idea
to heat an aerosol canister.
