Pressure, Kinetic Energy and Temperature

From one molecule to $P=\frac{1}{3}\rho v_{\mathrm{rms}}^2$

1. A molecule of mass m' with x-velocity v_x bounces elastically off a wall: Δp = 2m'v_x.
2. It returns after travelling 2l: it hits every $\Delta t=\frac{2l}{v_x}$.
3. Average force: $\frac{m^{\prime }{v}_x^2}{l}$. For all $N$ molecules: $F=\frac{m^{\prime }}{l}\sum v_x^2$.
4. Random motion: $⟨v_x^2⟩=\frac{v_{\mathrm{rms}}^2}{3}$.
5. $P=\frac{F}{l^2}=\frac{1}{3}\frac{N}{V}{m}^{\prime }{v}_{\mathrm{rms}}^2=\frac{1}{3}\rho v_{\mathrm{rms}}^2$.

The same result, any shape

In unit volume there are n₀ molecules; on average a sixth head towards each face. About $\frac{n_{0}v}{6}$ hit each square metre per second, each delivering 2m'v, so $P=\frac{1}{3}{n}_0{m}^{\prime }{v}^2$. Nothing depends on the container's shape.

Same T, same average KE

Comparing $P=\frac{1}{3}\frac{N}{V}{m}^{\prime }{v}_{\mathrm{rms}}^2$ with $PV=NkT$ gives $E_K=\frac{1}{2}{m}^{\prime }{v}_{\mathrm{rms}}^2=\frac{3}{2}kT$. Temperature measures the average translational kinetic energy of the molecules, whatever the pressure or the gas: lighter molecules simply move faster.