1.4.4: Chemical Equilibria- Solutions- Ion Association
A given equilibrium in aqueous solution involves association of two ions to form a neutral solute.
\[\text { Thus, } \quad \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \rightleftharpoons \mathrm{Z}(\mathrm{aq}) \nonumber \]
The chemical equilibrium is described in terms of chemical potentials using the following equation in which we recognise that the reactant is a 1:1 salt.
\[\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{\mathrm{eq}} = \mu^{0}(Z ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}\right]^{\mathrm{cq}} \nonumber \]
In the latter equation, \(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\) is the mean ionic activity for salt \(\mathrm{M}^{+} \mathrm{X}^{-}\) in the aqueous solution.
By definition, at fixed temperature \(\mathrm{T}\) and pressure \(p\) where this pressure is ambient and hence close to the standard pressure \(p^{0}\),
\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=\mu^{0}(\mathrm{Z} ; \mathrm{aq})-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{0} \nonumber \]
where,
\[\mathrm{K}^{0}=\left[\frac{\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}}{\mathrm{~m}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) /\left(\mathrm{m}^{0}\right)^{2}}\right]^{e q} \nonumber \]
In many cases, particularly for dilute solutions \(\gamma(\mathrm{Z})\) is approximately unity but rarely can one ignore the term \(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\).
\[\text { By definition, } \mathrm{K}^{0}(\mathrm{app})=\left\{\mathrm{m}(\mathrm{Z}) \, \mathrm{m}^{0} /\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right]^{2}\right\}^{\mathrm{eq}} \nonumber \]
\[\text { Then, } \ln \mathrm{K}^{0}(\text { app })=\ln \mathrm{K}^{0}+2 \ln \left[\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right] \nonumber \]
The solution may be so dilute that the mean ionic activity coefficient can be calculated using the Debye-Huckel Limiting Law (DHLL).
\[\text { Hence } \quad \ln \mathrm{K}^{0}(\mathrm{app})=\ln \mathrm{K}^{0}-2 \, \mathrm{S}_{\gamma} \,\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right] 11 / 2 \nonumber \]
In this equation the negative sign signals that in real solutions the extent of ion association to form \(\mathrm{Z}(\mathrm{aq})\) is less than in the corresponding ideal solution because charge - charge interactions in real solutions stabilise the ions.