1.14.63: Solubilities of Solids in Liquids
This very large subject can be divided into two groups. The first group concerns the solubility of a given solid substance \(j\) in a given solvent, liquid \(\ell_{1}\). The second group involves comparison of the solubilities of a given solid in two liquids, \(\ell_{1}\) and \(\ell_{2}\).
A closed system (at defined \(\mathrm{T}\) and \(\mathrm{p}\), the latter being close to the standard pressure) contains solid substance \(j\) in equilibrium with an aqueous solution containing solute \(j\). The system is characterized by the (equilibrium) solubility, \(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\). At equilibrium,
\[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right] \nonumber \]
Then
\[\Delta \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{*}(\mathrm{~s})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right] \nonumber \]
If the aqueous solution is dilute and the solubility is low, it can often be assumed that the properties of the solution are ideal. Hence,
\[\Delta \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{\mathrm{*}}(\mathrm{s})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) / \mathrm{m}^{0}\right] \nonumber \]
It should be noted that the sign of \(\Delta \mu_{j}^{0}\) depends on whether or not \(m_{j}^{e q}(a q)\) is larger or less than unity.
We illustrate the second approach by considering a combination of the experiment described above and an experiment where the solvent is a binary aqueous mixture, mole fraction composition \(\mathrm{x}_{2}\). At equilibrium,
\[\mu_{j}^{*}(s)=\mu_{j}^{0}\left(s \ln ; x_{2}\right)+R \, T \, \ln \left[m_{j}^{\mathrm{eq}}\left(s \ln ; x_{2}\right) \, \gamma_{j}^{\mathrm{eq}}\left(s \ln ; x_{2}\right) / m^{0}\right] \nonumber \]
\[\begin{aligned}
\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{x}_{2}\right)-\mu_{\mathrm{j}}^{0}(\mathrm{aq}) \\
=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right]
\end{aligned} \nonumber \]
If both solutions are dilute in substance \(j\), the ratio, \(\gamma_{j}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \gamma_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\) can be assumed to be close to unity. In fact this is a better approximation than assuming both activity coefficients are unity. Then
\[\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right] \nonumber \]
In other words if the solubility of substance \(j\) increases on adding solvent component 2 then \(\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{\mathrm{c}}\) is negative. This stabilization is a consequence of a difference in solute-solvent interactions.