1.13.3: Equilibirium- Solid-Liquid
A given homogeneous liquid system comprises two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at known \(\mathrm{T}\) and \(\mathrm{p}\). The temperature and/or pressure are changed. Consequently chemical substance \(\mathrm{j}\) spontaneously separates out as a solid phase but substance \(\mathrm{i}\) does not. Hence the liquid becomes richer in chemical substance \(\mathrm{i}\).
The starting point of the analysis is the following equation for the affinity for spontaneous transfer of substance \(\mathrm{j}\) from phase II to phase I [1].
\[\begin{aligned}
\delta\left(\frac{A_{j}}{T}\right)=& \frac{\left[\Delta_{\text {trans }} H_{j}^{0}(T, p)\right]}{T^{2}} \, \delta T \\
&-\frac{\left[\Delta_{\text {trans }} V_{j}^{0}(T, p)\right]}{T} \, \delta p+R \, \delta \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(\text { II }) \, f_{j}(I I)}\right]
\end{aligned} \nonumber \]
For two equilibrium states such that \(\delta\left(\mathrm{A}_{\mathrm{j}} / \mathrm{T}\right)\) is zero for the transfer of chemical substance \(\mathrm{j}\) from phase II to phase I,
\[\mathrm{R} \, \delta \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}\right]=\frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}^{2}} \, \delta \mathrm{T}-\frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}} \, \delta \mathrm{p} \nonumber \]
In this application, chemical substance \(\mathrm{i}\) cannot exist in phase I. Then the equilibrium states are determined by substance \(\mathrm{j}\). Further we consider the case where state I corresponds to pure \(\mathrm{j}\) such that \(x_{j}(I) \, f_{j}(I)\) is unity at reference temperature \(\mathrm{T}_{\text{ref}}\) and reference pressure pref. We integrate equation (b) between these two states.
\[\begin{aligned}
&\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]= \\
&\qquad \int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT}-\int_{\mathrm{p}(\mathrm{ref})}^{\mathrm{p}} \frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}} \, \mathrm{dp}
\end{aligned} \nonumber \]
In the event that the pressure is constant,
\[\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]=\int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT} \nonumber \]
Footnote
[1] By definition, for the transfer of one mole of chemical substance j from phase II to phase I, \(A_{j}=-\left[\mu_{j}(\mathrm{I})-\mu_{j}(\mathrm{II})\right] ; \mathrm{Or}, \mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{II})-\mu_{\mathrm{j}}(\mathrm{I})\)