1.12.8: Expansions- Solutions- Isobaric- Partial and Apparent Molar
- Page ID
- 377791
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The volume of a given aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is related to the composition by equation (a).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively. The (equilibrium) isobaric thermal expansion of the solution (at fixed pressure) \(\mathrm{E}_{\mathrm{p}}\) characterises the differential dependence of \(\mathrm{V}(\mathrm{aq})\) on temperature.
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
\(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})\) is an extensive property of the solution [1]. Two partial molar isobaric thermal expansions are defined, characteristic of solute and solvent [2].
\[\mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})=\left(\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\left(\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
From equation (a),
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) \nonumber \]
In the treatment of volumetric properties of solutions we define an apparent molar volume of the solute, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). By analogy we rewrite equation (e) in a form which defines the apparent molar isobaric expansion of the solute, \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)\). Thus,
\[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
Here [3],
\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
For the pure solvent,
\[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Footnotes
[1] \(\mathrm{E}_{\mathrm{p}}\) is an extensive property; the larger the volume \(\mathrm{V}\) the larger the change in volume for a given increase in temperature.
[2] \(E_{p}=\left[m^{3} K^{-1}\right] \quad E_{p 1}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \quad E_{p j}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)
[3] \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)