1.12.19: Expansions- Solutions- Isentropic Dependence of Partial Molar Volume on Temperature
We switch the condition on a derivative expressing the dependence of partial molar volume on temperature.
\[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}} \nonumber \]
Then
\[\left(\frac{\partial V_{j}}{\partial T}\right)_{\mathrm{s}}=E_{p j}-\left(\frac{\partial V_{j}}{\partial p}\right)_{T} \, \frac{(\partial S / \partial T)_{p}}{(\partial S / \partial p)_{T}} \nonumber \]
Or,
\[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{S}}=\mathrm{E}_{\mathrm{pj}}+\frac{(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}}{(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \nonumber \]
But
\[\frac{(\partial S / \partial T)_{p}}{(\partial S / \partial p)_{T}}=-\frac{C_{p}}{T \,(\partial V / \partial T)_{p}}=-\frac{C_{p}}{T \, V \, \alpha_{p}} \nonumber \]
Or,
\[\frac{(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}}{(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}}=-\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \nonumber \]
Hence,
\[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\mathrm{E}_{\mathrm{pj}}-\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \nonumber \]