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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?

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