1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions
- Page ID
- 374393
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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 near ambient pressure \(\mathrm{p}\) contains a solute \(j\) at molality \(\mathrm{m}_{j}\). The chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (a).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
Then
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
By definition, the partial molar isothermal compression of solute \(j\),[1]
\[K_{T_{j}}(a q)=-\left(\frac{\partial V_{j}(a q)}{\partial p}\right)_{T} \nonumber \]
Then
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\mathrm{K}_{\mathrm{TJ}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}^{2}\right]_{\mathrm{T}} \nonumber \]
Thus by definition,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq}) \nonumber \]
Hence the difference between \(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}(\mathrm{aq})\) and \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) depends on the second differential of \(\ln \left(\gamma_{j}\right)\) with respect to pressure.
Footnotes
[1] The formal definition of \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is given by equation (a).
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{K}_{\mathrm{T}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
However,
\[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{n}(\mathrm{j})} \nonumber \]
Then,
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}(\mathrm{j})} \nonumber \]
Or,
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
In other words, equation (c) shows that \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is a Lewisian partial molar property