Kinetic theory

Consider a cube-shaped box, each side of length L, filled with molecules of an ideal gas. A molecule of ideal gas is like a bouncy rubber ball; whenever it's involved in a collision with a wall of the box, it rebounds with the same kinetic energy it had before hitting the wall. Similarly, if ideal gas molecules collide, the collisions are elastic, so no kinetic energy is lost.
Now consider one of these ideal gas molecules in this box, with a mass m and velocity v. If this molecule bounces off one of the walls perpendicular to the x-direction, the y and z components of the molecule's velocity are unaffected, and the x-component of velocity reverses. The molecule maintains the same speed, because the collsion is elastic. How much force, on average, does it exert on the wall?
To answer this we just have to think about momentum, and impulse. Momentum is a vector, so if the particle reverses its x-component of momentum there is a net change in momentum of 2 m vx. The magnitude of the average force exerted by the wall to produce this change in momentum can be found from the impulse equation:

 
If we're dealing with the average force, the time interval is just the time between collisions, which is the time it takes for the molecule to make a round trip in the box and come back to hit the same wall. This time simply depends on the distance travelled in the x-direction (2L), and the x-component of velocity; the time between collisions with that wall is 2L / vx.
Solving for the average force gives:

 
This is the magnitude of the average force exerted by the wall on the molecule, as well as the magnitude of the average force exerted by the molecule on the wall; the forces are equal and opposite.
This is just the average force exerted on one wall by one molecule selected at random. To find the total force, we need to include all the other molecules, which travel at a wide range of speeds. The distribution of speeds follows a curve that looks a lot like a Bell curve, with the peak of the distribution increasing as the temperature increases. The shape of the speed distribution is known as a Maxwellian distribution curve, named after James Clerk Maxwell, the Scottish physicist who first worked it out.

To find the total force on the wall, then, we can just add up all the individual forces:

 
Multiplying and dividing the right-hand side by N, the number of molecules in the box, gives:

 
The term in square brackets is simply an average...we're adding up the square of the x-component of velocity for all the molecules and dividing by the number of molecules. This average is known as the root-mean-square average, symbolized by rms.

 
Therefore:

 
Consider now how this average x velocity compares to the average velocity. For any molecule, the velocity can be found from its components using the Pythagorean theorem in three dimensions:

 
For a randomly-chosen molecule, the x, y, and z components of velocity may be similar but they don't have to be. If we take an average over all the molecules in the box, though, then the average x, y, and z speeds should be equal, because there's no reason for one direction to be preferred over another. So:

 
The force exerted by the molecules on a wall of the box can then be expressed in terms of the average velocity, rather than the average x component of velocity.

 
The pressure exerted by the gas on each wall is simply the force divided by the area of a wall. Therefore:

 
Rearranging things a little gives:

This equation has many of the same variables as the ideal gas law, PV = NkT. This means that:

 
This is a very important result, because it tells us something fundamental about temperature. The absolute temperature of an ideal gas is proportional to the average kinetic energy per gas molecule. If changes in pressure and/or volume result in changes in temperature, it means the average kinetic energy of the molecules has been changed.