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Consider two ideal gases of differing heat capacities. They occupy separate compartments of the same total volume. Assume their initial temperatures and that the amount of substance for both are the same. Assume that there is a pressure/volume imbalance such that the system is not in equilibrium, and that the two compartments are separated by an adiabatic and impermeable wall that allows for movement.

Now, at least according to the solutions to my problem set, the final temperatures for both gases are equal and also equal to the initial temperature. But if the gases are expanding and contracting, wouldn't this imply a change in said temperatures?

Is it simply that the process evolves such that $PV$ is always constant and (since $nR$ is constant, too), $T$ therefore remains fixed? If that's the case, how can we infer this from the setup? Is it because the system is initially in thermal equilibrium and therefore is not going to evolve in such a way that "negates" that?

agaminon
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1 Answers1

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It is a consequence of the first law of thermodynamics applied to the whole system. Since the whole system is isolated $dU=dU_1+dU_2=0$, also at equilibrium the final temperature of the two compartments is the same, combining this two conditions you can obtain that $T_f=T_0$.

Also, the final pressure must be the same to ensure mechanical equilibrium.

(Remember that for an ideal gas $dU=C_vdT$)

Lucky Charms
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