Consider air from outside at $T_0$ effusing into an evacuated and thermally isolated chamber.
By thermodynamic potentials, the temperature inside the tank is given by:
$$T_1 = \gamma T_0 = \frac{5}{3} T_0$$
However, by using Maxwell-Boltzmann statistics, the distribution of particles effusing is $\propto v^3 exp(-\alpha v^2)$. Calculating the average KE, we find that
$$T_1 = \frac{4}{3}T_0$$
Why is there a difference between the two?
