1.4.4: Chemical Equilibria- Solutions- Ion Association
- Page ID
- 352526
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given 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.