1.7.14: Compressions- Ratio- Isentropic and Isothermal
Using a calculus operation, we obtain equations relating isothermal and isentropic dependencies of volume on pressure.
Thus,
\[\begin{aligned}
(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \\ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}
\end{aligned} \nonumber \]
and,
\[\begin{aligned}
(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=&-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \\
&=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}
\end{aligned} \nonumber \]
Then,
\[(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}} \nonumber \]
The Gibbs-Helmholtz equation requires that
\[\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}} \nonumber \]
Also
\[(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} \nonumber \]
Similarly,
\[(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}} \nonumber \]
Hence,
\[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{s}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}} \nonumber \]