1.14.61: Solubility Products
- Page ID
- 390924
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.