Below it is proved that the internal energy of an ideal gas depends only on its temperature. Therefore,
The process is adiabatic, so from the first law of thermodynamics
which implies that
Using integration by parts then gives that is constant. Therefore, answer (C) is correct.
Proof that the internal energy of an ideal gas depends only on its temperature:
and the equality of mixed partial derivatives it follows that
Therefore, equation 1 becomes
which equals for an ideal gas. This completes the proof.
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