Prove that, quite generally, [ C_{P}-C_{V}=-k frac{left[frac{partial}{partial T}left{Tleft(frac{partial ln Q}{partial V}ight)_{T}ight}ight]_{V}^{2}}{left(frac{partial^{2} ln Q}{partial V^{2}}ight)_{T}}>0 . ] Verify

Question:

Prove that, quite generally,

\[
C_{P}-C_{V}=-k \frac{\left[\frac{\partial}{\partial T}\left\{T\left(\frac{\partial \ln Q}{\partial V}ight)_{T}ight\}ight]_{V}^{2}}{\left(\frac{\partial^{2} \ln Q}{\partial V^{2}}ight)_{T}}>0 .
\]

Verify that the value of this quantity for a classical ideal classical gas is $N k$.

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