Kinetic Theory

The kinetic theory of an ideal gas assumes: the gas contains a very large number of molecules moving randomly; the volume of the molecules is negligible compared with the volume of the container; there are no forces between molecules except during collisions; collisions are perfectly elastic; and the duration of a collision is negligible compared with the time between collisions.

These assumptions are why ideal behaviour is approached at low pressure and high temperature. At high pressure molecules take up a significant fraction of the volume; at low temperature intermolecular attractions become important.

Consider one molecule of mass $m$ moving with velocity component $u$ towards a wall. It rebounds elastically with $-u$, so its momentum change is $2mu$.

It travels to the opposite wall and back, a distance $2l$, before hitting the same wall again, so it collides every $2l/u$ seconds. The average force it exerts is $\dfrac{2mu}{2l/u} = \dfrac{mu^{2}}{l}$.

Summing over $N$ molecules, and noting that on average the motion is shared equally between three dimensions, gives $\langle u^{2}\rangle = \tfrac13\langle c^{2}\rangle$. Dividing force by wall area $l^{2}$ gives the result.

Molecules have a range of speeds, so the model uses the $\textbf{mean square speed}$ $\langle c^{2}\rangle$: square every speed, then take the average.

Its square root is the $\textbf{root-mean-square}$ speed, $c_{\mathrm{rms}} = \sqrt{\langle c^{2}\rangle}$. It is not the same as the average speed, and is always slightly larger.

Since $Nm$ is the total mass and $Nm/V$ is density, the equation can also be written $p = \tfrac13\rho\langle c^{2}\rangle$.

Equating $\tfrac13 Nm\langle c^{2}\rangle$ with $NkT$ gives $\tfrac13 m\langle c^{2}\rangle = kT$. Multiplying by $\tfrac32$ gives the average kinetic energy of one molecule.

So the average molecular kinetic energy is proportional to absolute temperature and $\textbf{independent of the type of gas}$. At the same temperature, hydrogen and oxygen molecules have the same average kinetic energy — but hydrogen molecules, being lighter, move faster.

At absolute zero the average kinetic energy would be zero, which matches its definition. For an ideal gas, with no potential energy between molecules, the internal energy is entirely this kinetic energy: $\tfrac32 NkT$.