Consider a cubical box, L on each side. The box contains N molecules of ideal gas, each of mass m.

All collisions are elastic. The force exerted by one molecule when it collides with a wall of the box that is perpendicular to the x-axis is, from the impulse equation:

F = 2mv_x/Δt

Our single molecule collides with this wall once every:

Δt = 2L/v_x

This gives an average force of:

F = mv_x²/L

This is the force from a single molecule. The total force on the wall is the sum over all the molecules:

F = Σmv_x²/L = (mN/L) [ Σv_x²/N ]

The term in square brackets represents the average value of v_x². The square root of this average is known as the root-mean-square (rms) average of v_x, so:

F = (mN/L) v_{x_rms}²

By symmetry, all directions in the box are equivalent, so:

v² = v_x² + v_y² + v_z² = 3v_x²

This gives:

F = (mN/3L) v_rms²

Dividing by the area, L², of the wall gives the pressure:

P = (mN/3L³) v_rms²

which is:

PV = (N/3)[mv_rms²] = (2N/3) [½ mv_rms²]

The term in square brackets is K_av, the average translational kinetic energy of the molecules in the box.

Therefore PV = (2N/3)K_av

Compare our result:

PV = (2N/3)K_av

to the ideal gas law

PV = NkT

The two equations agree when the average translational kinetic energy of the molecules is:

K_av = (3/2)kT

Here we have a fundamental connection between temperature and the average translational kinetic energy of the atoms - they are directly proportional to one another. Temperature is a measure of the average kinetic energy of the atoms.

The internal energy is the total of all the energy associated with the motion of the atoms or molecules in the system. This includes energy associated with translation, rotation, and vibration.

Equipartition of Energy: Each contribution to the internal energy contributes an equal amount of energy.

For a monatomic ideal gas, the only contribution to the energy comes from translational kinetic energy. The internal energy is therefore:

Monatomic ideal gas: E_int = (3/2)NkT = (3/2)nRT

Each direction (x, y, and z) contributes (1/2)NkT to the energy. This is where the equipartition of energy idea comes in - any other contribution to the energy must also contribute (1/2)NkT.

For a diatomic molecule there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

Diatomic ideal gas: E_int = (5/2)NkT

This actually applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.