1.11.3: Gibbs-Duhem Equation- Solvent and Solutes- Aqueous Solutions
- Page ID
- 379665
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \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 aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to the standard pressure) is prepared using water (\(1 \mathrm{~kg}\)) and \(\mathrm{m}_{\mathrm{j}}\) moles of solute-\(\mathrm{j}\). The chemical potential of solute–j is given by equation (a) where \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of solute-\(\mathrm{j}\) in an aqueous solutions where \(\mathrm{m}_{\mathrm{j}} = 1 \mathrm{~mol kg}^{-1}\) and the thermodynamic properties of the solution are ideal.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
At all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1 \nonumber \]
The chemical potential of solvent, water \(\mu_{1}(\mathrm{aq})\) is given by equation (c) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the molal osmotic coefficient.
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
At all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1 \nonumber \]
For a solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the Gibbs-Duhem equation relates \(\mu_{1}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\) using equation (e).
\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
Therefore
\[\begin{aligned}
&\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\
&+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0
\end{aligned} \nonumber \]
Thus,
\[\mathrm{d}\left(-\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}}\right)=0 \nonumber \]
\[-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d \ln \left(m_{j}\right)+m_{j} \, d \ln \left(\gamma_{j}\right)=0 \nonumber \]
From equation (h) (dividing by \(\mathrm{m}_{\mathrm{j}}\))
\[-\phi \, d \ln \left(m_{j}\right)-d \phi+d \ln \left(m_{j}\right)+d \ln \left(\gamma_{j}\right)=0 \nonumber \]
\[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{d} \phi-(1-\phi) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
The latter equation is integrated between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\); equivalent to limits ‘\(\phi =1\)’ and \(\phi\).
\[\int_{0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{\phi=1}^{\phi} \mathrm{d} \phi-\int_{0}^{m(j)}(1-\phi) \, d \ln \left(m_{j}\right) \nonumber \]
Then,
\[-\ln \left(\gamma_{\mathrm{j}}\right)=(1-\phi)+\int_{0}^{\mathrm{m}(\mathrm{j})}(1-\phi) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
From equation (g)
\[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
or,
\[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}^{0} \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right] \nonumber \]
Following integration from ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ to \(\mathrm{m}_{\mathrm{j}}\),
\[\phi \, \mathrm{m}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}}+\int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
or,
\[\phi=1+\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
Hence,
\[\phi-1=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
In other words, \((\phi-1)\) is related to the integral of \(\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \gamma_{\mathrm{j}}\) between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\). Equation (q) marks the limit of the thermodynamics analysis. However we explore the significance of the equation by adopting an equation relating \(\phi\) and \(\mathrm{m}_{\mathrm{j}}\). Equation (r) signals one assumption in which \(\mathrm{r}\) is a parameter characteristic of the solution under examination. Thus
\[\phi-1=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \nonumber \]
Then [1]
\[-\ln \left(\gamma_{\mathrm{j}}\right)=(1-\phi) \,(1+\mathrm{r}) / \mathrm{r} \nonumber \]
Or,
\[(1-\phi)=[r /(1+r)] \,\left\{-\ln \left(\gamma_{j}\right)\right\} \nonumber \]
From equation (m),
\[\phi=1-\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \nonumber \]
If for example, \(\alpha > 1\), then \(\phi < 1\) for all solutions. According to equation (c),
\[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=-\left[1-\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{t}}\right] \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
If therefore \(\phi>1, \mu_{1}(\mathrm{aq})<\mu_{1}^{*}(\ell)\); relative to the chemical potential of the pure solvent, the solvent in the solution is stabilised. For the solute according to equation (n),
\[-\ln \left(\gamma_{\mathrm{j}}\right)=[(1+\mathrm{r}) / \mathrm{r}] \,\left[\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}\right] \nonumber \]
Or, [Bjerrum’s Equation]
\[-\ln \left(\gamma_{\mathrm{j}}\right)=[(1+\mathrm{r}) / \mathrm{r}] \,[1-\phi] \nonumber \]
Footnotes
[1] From equations (q) and (s),
\[-\ln \left(\gamma_{\mathrm{j}}\right)=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}+\alpha \, \int_{0}^{\mathrm{m}(\mathrm{j})}\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}-1} \, d \mathrm{~m}_{\mathrm{j}} \nonumber \]
Then,
\[\begin{aligned}
-\ln \left(\gamma_{\mathrm{j}}\right) &=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}+\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} / \mathrm{r} \\
&=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \,[1+(1 / \mathrm{r})]
\end{aligned} \nonumber \]
Hence, \(-\ln \left(\gamma_{\mathrm{j}}\right)=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \,[1+\mathrm{r}] / \mathrm{r}\)