Wolfram Alpha:

```Kinetic Theory of Gases
-----------------------

An ideal gas is defined as one in which all
collisions between atoms or molecules are
perfectly elastic and in which there are no
intermolecular attractive forces. One can
visualize it as a collection of perfectly
hard spheres which collide but which otherwise
do not interact with each other. In such a gas,
all the internal energy is in the form of
kinetic energy and any change in internal
energy is accompanied by a change in
temperature.

Consider a molecule striking the wall of a box.
Assume the collision with the wall is perfectly
elastic so that the change in the x momentum is
given by:

mvx - (-mvx) = 2mvx

Now Fx = max = 2mvx/Δt

Now, if we assume the molecule makes a
'round trip' we get:

Δt = 2L/vx

So,

Fx = 2mvx/(2L/vx) = mvx2/L

The force due to N molecules is Nmvx2/L

Now v2 = vx2 + vy2 + vz2 = 3vx2

∴ vx2 = v2/3

P = F/A =  Nmv2/3LA = Nmv2/3V

∴ PV = Nmv2/3 = nRT ... the Ideal Gas Law

Now,

n = N/NA where NA is Avogadro's Number

∴ Nmv2/3 = NRT/NA

mv2/3 = RT/NA

mv2 = 3kBT where kB =  R/NA = BOLTZMANN'S CONSTANT

∴ (1/2)mv2 = 3kBT/2

The total INTERNAL ENERGY of the gas is thus:

U =  3NkBT/2 = 3nRT/2
```