1.14.61: Solubility Products
A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient) contains an aqueous solution of a sparingly soluble salt \(\mathrm{MX}\); e.g. \(\mathrm{AgCl}\). The system also contains solid salt \(\mathrm{MX}\). When a soluble salt (e.g. \(\mathrm{KNO}_{3}\)) is added the solubility of salt \(\mathrm{MX}\) increases. This remarkable observation is readily accounted for. The equilibrium involving the sparingly soluble salt is represented as follows.
| \(\mathrm{MX}(\mathrm{s})\) | \(\Leftrightarrow\) | \(\mathrm{M}^{+}\mathrm{~X}^{-}(\mathrm{aq})\) |
| solid | solution |
We represent the salt \(\mathrm{MX}\) by the symbol \(j\). At equilibrium,
\[\mu_{\mathrm{j}}^{\prime \prime}(\mathrm{s})=\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \nonumber \]
In terms of the solubility \(\mathrm{S}_{j}\) of the salt \(\mathrm{MX}\), a 1:1 salt,
\[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{\mathrm{o}}\right) \nonumber \]
By definition
\[\Delta_{\text {sol }} G^{0}=-R \, T \, \ln K_{s}=\mu_{j}^{0}(a q)-\mu_{j}^{*}(s) \nonumber \]
\(\mathrm{K}_{\mathrm{S}}\) is the solubility product, a characteristic property of salt \(\mathrm{MX}\) (at defined \(\mathrm{T}\) and \(\mathrm{p}\)).
\[\mathrm{K}_{\mathrm{S}}=\left[\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]^{2} \nonumber \]
Or,
\[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)-\ln \left(\gamma_{\pm}\right) \nonumber \]
According to the DHLL,
\[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
\(\mathrm{I}\) is the ionic strength of the solution which can be changed by adding a soluble salt. From equations (e) and (f),
\[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)+\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
The key point to note is the positive sign in equation (g) showing that the theory accounts for the observed salting–in of the sparingly soluble salt. Further a plot of \(\ln \left(S_{j} / m^{0}\right)\) against \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) is linear yielding an estimate for \(\mathrm{K}_{\mathrm{S}}\) from the intercept.