The behavior of gasses:

Matter exists in three states - solid, liquid and gaseous. Of the three the third is the less complex. The molecules that make up the gas move independently of each other except as is occasioned by collisions with each other or the walls of their container. In keeping with our tendency to look at problems that best lend themselves to analysis we look at the properties of gasses. A gas can be characterized by three parameters - the volume of the container that holds it; the pressure it is subjected to; the temperature it is held at.

Experiments by Boyle and Mariotte in the 17th century led to an empirical relation known as Boyle's law: If the temperature of a gas is held constant, the product of its pressure and its volume remain constant. As an equation,Almost a century after these
experiments were reported Jacques Charles and Joseph Gay-Lussac
independently experimented with gasses held at constant pressure. Their
temperature-volume relation is *V = K _{p} T*, where the
constant

We can combine these two "laws"
into one by writing *p V = K T*. In this form, if *T* is
held constant then *KT *becomes our earlier constant *K _{t}*,
and if the pressure is held constant we replace

This ideal gas
law, *p V = K T*, came from the empirical evidence of many
experiments. What underlying structure exists to bring it about? Such a
structure should explain the equation and why it is most general for low
gas densities and equal masses. Boyle had supported the idea that gasses
were made of discrete corpuscles. As the great success of Newton's
mechanics strongly influenced physicists of the time, it was natural that
a mechanical model was pursued.

We start with the assumption that gasses are made of accumulations of molecules, and in the simplest case these can be considered to be points of mass. These molecules move about in their containers in a random manner - the degree of their motion (i.e. their velocity or kinetic energy) depending on the state of the gas. We study their motion by applying Newton's laws of motion. In doing so we deduce that when these molecules collide with each other or the walls of their containers the collisions are perfectly elastic (otherwise the resultant energy loss would eventually cause the molecular motion to stop). By designing our molecules to be points of mass we introduce two important simplifications: As point masses they occupy essentially zero volume and so 1) cannot collide with each other; 2) do not occupy any of the volume of the container. These two simplifications become more realistic as the densities of the gasses decrease - i.e. as the gas becomes "ideal".

With this model we proceed as follows:

Take
a single molecule of our ideal gas. As one of a large number (*N*) of such
molecules we may assume its velocity is the average velocity of all the
molecules in the gas. In general the molecules move randomly in space, so one
would expect that at any time 1/3 of them would be moving in each of the three
available directions x, y and z. (If this doesn't sit well with you, note that
if an average molecule were moving in any random direction it's *components*
along the perpendicular directions would be equally distributed). Say our
molecule is moving in the x-direction with velocity *v _{x}. *When
it hits the right wall it will rebound with a momentum change

Now pressure (*p*) is force per unit
area and for our cube of sides *L* the area of a face is *L ^{2}*,
so we may write this force as

(Now we use *p* for pressure
rather than momentum - there's just not enough letters to go around - but the
context should keep things clear.)

Our gas consists of *N* molecules,
and
1/3 of these will be moving in the x-direction, contributing to the total pressure
on the wall *N/3* times that of our single molecule. Recognizing that *L ^{3 }*is just the
volume of the gas (which we move to the left side) and after inserting a factor
of 2/2 in to bring a familiar friend to the fore) we have

Conclusion: The pressure times the volume of the gas
equals two thirds the number of molecules times the average kinetic energy of a
molecule of the gas. (Or, dividing by *V* shows that the pressure is just the average
kinetic energy of a molecule times the number of molecules per unit volume -
times the factor of 2/3 - where does this pesky 2/3 come from?).

The pressure of a gas is the result of collisions of air molecules with the sides of the container - the more energetic the molecules, the greater the pressure.

We need to match our derived equation to the experimentally
arrived at ideal gas law (*p V = K T*). Recall that *K* (the
"gas constant") was a constant independent of the type, but tied to
the volume of the gas. The connection comes from a theory by the Italian
physicist Amedeo Avogadro that equal volumes of gasses contain equal numbers of
molecules. This being so, *K* must be proportional to *N* and we may
write *K = k N* where our new constant *k* is the gas constant
per molecule, known as Boltzmann's constant. Using this and equating the
right hand side of the ideal gas law to the right hand side of the results of
our model yields

and the ideal gas law becomes (now
referring to a gas with *N *molecules)

The end result is the gratifying
conclusion that completely supports the (at the time controversial) molecular
nature of gasses, arrived at by a purely mechanical application of Newton's
laws. It also
clearly shows that temperature is a result of molecular motion, contained in the
kinetic energy of the moving molecules. Since kinetic energy cannot be negative
it follows that if the molecules of a gas somehow can slow down and come to
rest, the resulting temperature must reach a minimum attainable value of zero.
The temperature in the gas law is an absolute reference to the energy of the gas
- its absolute temperature *T*. This is the limiting value of the graph of *V*
vs. *T* experimentally arrived at by Charles and Gay-Lussac.

As an example of the consequences of these relations, from

it follows that the average velocity (more properly, speed) of a gas molecule is

Boltzmann's constant has the value *k*
= 1.38x10^{-23} , room temperature is about 300 K. A Nitrogen molecule
(of which air is mostly made) has an atomic weight of 14 AMU (atomic mass
units), and 1 AMU = 1.66x10^{-27} kg. Inserting these into the velocity
formula provides *v* = 730 m/sec. This is about as fast as a bullet from a high
powered rifle.