1.12.18: Expansions- Solutions- Isentropic Dependence of Apparent Molar Volume of Solute on Temperature and Pressure
- Page ID
- 377815
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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 starting point is the following calculus operation.
\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]
Also
\[\left(\frac{\partial \phi\left(V_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]
Or,
\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \, \frac{\mathrm{V}}{\mathrm{V}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]
Hence,
\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]
But \(\frac{\sigma}{\mathrm{T}}=\frac{\left[\alpha_{p}\right]^{2}}{\delta}\) Then
\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\alpha_{\mathrm{p}}}{\delta} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}} \nonumber \]
Also,
\[\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}} \nonumber \]
Then \(\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}}=-\frac{\alpha_{\mathrm{p}}}{\delta}\) Hence
\[\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{s}}}{\alpha_{\mathrm{s}}}=-\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \nonumber \]