1.14.3: Excess Thermodynamic Properties- Aqueous Solutions
- Page ID
- 392425
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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 aqueous solution, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong \mathrm{p}^{0}\)), contains \(\mathrm{i}\)-solutes, with \(\mathrm{n}_{\mathrm{j}}\) moles of each solute \(\mathrm{j}\), and \(\mathrm{n}_{1}\) moles of water(\(\ell\)).The Gibbs energy of the solution is given by equation (a).
\[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
For a solution prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)), in vast molar excess,
\[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\sum_{\mathrm{j}=1}^{\mathrm{s}} \mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
We assert that the system is at thermodynamic equilibrium. For each solute \(\mathrm{j}\), \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{\mathrm{j}}\) and the reference chemical potential for solute \(\mathrm{j}\) in a solution where \(\mathrm{m}_{\mathrm{j}} = 1 \mathrm{~mol kg}^{-1}\) and the thermodynamic properties of the solute are ideal. Then,
\[\left\{\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right\} \quad \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
where \(\operatorname{limit}\left(m_{j} \rightarrow 0\right) \gamma_{j}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\).
For the solvent we express the properties in terms of a practical osmotic coefficient, \(\phi\).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \nonumber \]
At all \(\mathrm{T}\) and \(\mathrm{p}\), \(\operatorname{limit}\left(\sum_{j=1}^{j=i} m_{j} \rightarrow 0\right) \phi=1.0\)
For the solution,
\[\begin{aligned}
\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}\right] \\
&+\sum_{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
If the thermodynamic properties of the solution are ideal,
\[\begin{aligned}
\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}\right] \\
&+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]
\end{aligned} \nonumber \]
By definition the solution excess Gibbs energy of the solution,
\[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) \nonumber \]
\(\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) \text { is expressed in }\left[\mathrm{J} \mathrm{kg}^{-1}\right]\).
Then
\[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,(1-\phi) \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
\[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) / \mathrm{R} \, \mathrm{T}=(1-\phi) \, \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
For a solution containing a single solute \(\mathrm{j}\),
\[\mathrm{G}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right) / \mathrm{R} \, \mathrm{T}=\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \, \mathrm{m}_{\mathrm{j}} \nonumber \]
If the thermodynamic properties of the solution are ideal, the chemical potential of the solute is given by equation (k).
\[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Equation (c) describes the properties of solute \(\mathrm{j}\) in a real solution. By definition the excess chemical potential \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is given by equation (l).
\[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mu_{\mathrm{j}}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id}) \nonumber \]
Then,
\[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
Often an excess chemical potential \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is written in the form \(\mathrm{G}_{\mathrm{j}}^{\mathrm{E}}\). In the case of the solvent, water(\(\ell\)) the corresponding equations for the chemical potentials in solutions having either real or ideal thermodynamic properties are given by equations (n) and (o).
\[\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
\[\mu_{1}^{\mathrm{E}}(\mathrm{aq})=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnotes
[1] For further comments see—
- M. I. Davis and G. Douheret, Thermochim. Acta, 1991,190,267.
- H. L. Friedman, J. Chem.Phys.,1969,32,1351.