1.14.21: Donnan Membrane Equilibria
- Page ID
- 374485
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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 experimental system comprises two compartments, I and II, separated by a membrane. The two compartments contain aqueous solutions at common temperature and pressure. The experimental system is set up by placing in compartment I an aqueous salt solution; e.g. \(\mathrm{NaCl}(\mathrm{aq})\) having concentration \(\mathrm{c}_{1} \mathrm{~mol dm}^{-3}\). However compartment II contains a salt, \(\mathrm{R}^{+}\mathrm{Cl}^{-} (\mathrm{aq})\), concentration \(\mathrm{c}_{2} \mathrm{~mol dm}^{-3}\). The membrane is permeable to both \(\mathrm{Na}^{+} and \(\mathrm{Cl}^{-}\) ions but not to \(\mathrm{R}^{+}\) cations. The sodium and chloride ions spontaneously diffuse across the membrane until the two solutions are in thermodynamic equilibrium. We represent the equilibrium system as follows where | | represents the membrane.
[I]
\[ \mathrm{Na}^{+}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{cq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{eq}}||[\mathrm{II}] \mathrm{Na}^{+}(\alpha)^{\mathrm{eq}} \mathrm{R}^{+}\left(\mathrm{c}_{2}\right)^{\mathrm{eq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}} \nonumber \]
The solutions on both sides are electrically neutral. A thermodynamic analysis is somewhat complicated if account is taken of the role of ion-ion interactions. However the essential features of the argument are revealed if we identify the activities of the ions as equal to their concentrations.
Hence at equilibrium at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\begin{aligned}
&{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Na}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}} \\
&+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}= \\
&{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\alpha_{\mathrm{Na+}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}}} \\
&+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{2}+\alpha\right)_{\mathrm{Cl-}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}}
\end{aligned} \nonumber \]
But
\[\left(c_{1}-\alpha\right)_{\mathrm{Na+}}^{\mathrm{eq}}=\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}-}^{\mathrm{eq}} \nonumber \]
Or,
\[\left[\left(c_{1}-\alpha\right)_{\mathrm{Na}+}^{\mathrm{eq}}\right]^{2}=\alpha_{\mathrm{Na}+}^{\mathrm{eq}} \,\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}} \mathrm{CI}^{-} \nonumber \]
Then [1],
\[\frac{\alpha^{\mathrm{eq}}}{\mathrm{c}_{1}}=\frac{\mathrm{c}_{1}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}} \nonumber \]
The latter is Donnan’s Equation. The ratio \(\left(\alpha^{\mathrm{eq}} / \mathrm{c}_{1}\right)\) tends to be smaller the larger is \(\mathrm{c}_{2}\). This conclusion is confirmed by experiment.
We have simplified the algebra by writing \(\mathrm{R}^{+} \mathrm{Cl}^{-}\) as the salt in compartment II. In practice the Donnan equilibrium finds major application where salt \(\mathrm{RCl}\) is a macromolecule [2-4].
Footnotes
[1] \(\mathrm{c}_{1}^{2}-2 \, \alpha \, \mathrm{c}_{1}+\alpha^{2}=\alpha \, \mathrm{c}_{2}+\alpha^{2}\)
Then, \(c_{1}^{2}-2 \, \alpha \, c_{1}=\alpha \, c_{2}\)
Or, \(\alpha=\frac{\mathrm{c}_{1}^{2}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}}\)
[2] F. G. Donnan et al, J. Chem. Soc.,1911, 1554; 1914,1941.
[3] F. G. Donnan, Chem. Rev.,1924, 1,73.
[4] F. G. Donnan and E. A. Guggenheim, Z. Physik Chem. A, 1932,162,346.