1.4.5: Chemical Equilibria- Solutions- Sparingly Soluble Salt

A given aqueous solution contains a sparingly soluble 1:1 salt [e.g. \(\operatorname{AgCl}(\mathrm{s})\)] at fixed temperature and pressure. The following phase equilibrium is established.

\[\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \rightleftharpoons \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \label{a}\]

In terms of the Phase Rule, the system contains two components, water and sparingly soluble substance \(\mathrm{MX}\); \(\mathrm{C} = 2\). There are three phases: solution, vapour and solid. Then \(\mathrm{F} = 1\). Hence if we define the temperature, the vapour pressure and the equilibrium composition of the liquid phase are defined.

A thermodynamic description of equilibrium (Equation \ref{a}) is based on equality of chemical potentials of reactants and products. The key point is that the solid, \(\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s})\) is a reference state.

\[\text { Hence, } \mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=\mu^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)\]

Noting that \(\mathrm{M}^{+} \mathrm{X}^{-}\) is a 1:1 salt,

\[\begin{aligned}
&\mu^{0}(\mathrm{MX} ; \mathrm{s})= \\
&\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right)^{\mathrm{eq}}
\end{aligned}\]

The solubility product for salt \(\mathrm{M}^{+} \mathrm{X}^{-}\), \(\mathrm{K}_{\mathrm{s}}^{0}\) is defined as follows. [\(\mathrm{K}_{\mathrm{s}}^{0}\) is dimensionless.]

\[\Delta_{\mathrm{s}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]

\[\text { Hence }[1], \mathrm{K}_{\mathrm{s}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} \, \gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{2}\]

\(\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}}\) is the (equilibrium) solubility, a quantity obtained experimentally.

\[\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=-\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]

In many cases salt \(\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}))\) is so sparingly soluble that \(\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)\) can be calculated using the Debye-Huckel Limiting Law (DHLL). The DHLL relates \(\ln \left(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)^{e q}\) to the ionic strength of the solution, I. The ionic strength is controlled by adding a second soluble salt \(\mathrm{N}^{+} \mathrm{Y}^{-}\).

\[\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]

This is a classic equation [2] because in many cases \(\ln \left[\mathrm{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]\) is a linear function of \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) so that \(\ln \left(\mathrm{K}_{\mathrm{S}}^{0}\right)\) is obtained from the corresponding intercept. We understand the form of equation (g) in terms of increasing stabilisation of the ions \(\mathrm{M}^{+}(\mathrm{aq})\) and \(\mathrm{X}^{-}(\mathrm{aq})\) in solution by the ion-ion interactions in the real solution which are enhanced when the ionic strength is increased by adding soluble salt \(\mathrm{N}^{+} \mathrm{Y}^{-}\).

Footnotes

[1] \(\mathrm{K}_{\mathrm{s}}^{0}\) is dimensionless. However in many reports a quantity \(\mathrm{K}_{\mathrm{m}}^{0}\) is defined as follows.

\[\mathrm{K}_{\mathrm{m}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{e q} \, \gamma_{\pm}^{\mathrm{eq}}\right]^{2}\]

For a 1:1 salt, \(\mathrm{K}_{\mathrm{m}}^{0}\) has units, \(\left(\mathrm{mol kg}^{-1}\right)^{2}\). Or \(\mathrm{K}_{\mathrm{s}}^{0}=\mathrm{K}_{\mathrm{m}}^{0} /\left(\mathrm{m}^{0}\right)^{2}\)

[2] All three terms in equation (g) are dimensionless.