1.12.18: Expansions- Solutions- Isentropic Dependence of Apparent Molar Volume of Solute on Temperature and Pressure

The starting point is the following calculus operation.

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]

Also

\[\left(\frac{\partial \phi\left(V_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]

Or,

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \, \frac{\mathrm{V}}{\mathrm{V}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]

Hence,

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]

But \(\frac{\sigma}{\mathrm{T}}=\frac{\left[\alpha_{p}\right]^{2}}{\delta}\) Then

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\alpha_{\mathrm{p}}}{\delta} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]

Also,

\[\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}} \nonumber \]

Then \(\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}}=-\frac{\alpha_{\mathrm{p}}}{\delta}\) Hence

\[\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{s}}}{\alpha_{\mathrm{s}}}=-\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \nonumber \]

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