1.12.22: Expansions and Compressions- Solutions- Isentropic Dependence of Volume on Temperature and Pressure
The starting point is the calculus operation for a double differential.
\[\frac{\partial^{2} U}{\partial S \, \partial V}=\frac{\partial^{2} U}{\partial V \, \partial S} \nonumber \]
Then, \(\left(\frac{\partial T}{\partial V}\right)_{s}=-\left(\frac{\partial p}{\partial S}\right)_{v}\) Or,
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \nonumber \]
But,
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \nonumber \]
Also we note that
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Then,
\[\left(\frac{\partial V}{\partial T}\right)_{s}=\left(\frac{\partial V}{\partial p}\right)_{s} \,\left(\frac{\partial S}{\partial T}\right)_{p} \,\left(\frac{\partial T}{\partial V}\right)_{p} \nonumber \]
However from the Gibbs - Helmholtz Equation, \(\left(\frac{\partial S}{\partial T}\right)_{p}=\frac{C_{p}}{T}\)
Then \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \, \frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}}\) Or,
\[\mathrm{E}_{\mathrm{s}}=-\frac{\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}} \nonumber \]
We divide both sides of equation (f) by volume \(\mathrm{V}\). Hence
\[\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}} \nonumber \]