1.13.3: Equilibirium- Solid-Liquid
- Page ID
- 375582
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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 homogeneous liquid system comprises two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at known \(\mathrm{T}\) and \(\mathrm{p}\). The temperature and/or pressure are changed. Consequently chemical substance \(\mathrm{j}\) spontaneously separates out as a solid phase but substance \(\mathrm{i}\) does not. Hence the liquid becomes richer in chemical substance \(\mathrm{i}\).
The starting point of the analysis is the following equation for the affinity for spontaneous transfer of substance \(\mathrm{j}\) from phase II to phase I [1].
\[\begin{aligned}
\delta\left(\frac{A_{j}}{T}\right)=& \frac{\left[\Delta_{\text {trans }} H_{j}^{0}(T, p)\right]}{T^{2}} \, \delta T \\
&-\frac{\left[\Delta_{\text {trans }} V_{j}^{0}(T, p)\right]}{T} \, \delta p+R \, \delta \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(\text { II }) \, f_{j}(I I)}\right]
\end{aligned} \nonumber \]
For two equilibrium states such that \(\delta\left(\mathrm{A}_{\mathrm{j}} / \mathrm{T}\right)\) is zero for the transfer of chemical substance \(\mathrm{j}\) from phase II to phase I,
\[\mathrm{R} \, \delta \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}\right]=\frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}^{2}} \, \delta \mathrm{T}-\frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}} \, \delta \mathrm{p} \nonumber \]
In this application, chemical substance \(\mathrm{i}\) cannot exist in phase I. Then the equilibrium states are determined by substance \(\mathrm{j}\). Further we consider the case where state I corresponds to pure \(\mathrm{j}\) such that \(x_{j}(I) \, f_{j}(I)\) is unity at reference temperature \(\mathrm{T}_{\text{ref}}\) and reference pressure pref. We integrate equation (b) between these two states.
\[\begin{aligned}
&\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]= \\
&\qquad \int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT}-\int_{\mathrm{p}(\mathrm{ref})}^{\mathrm{p}} \frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}} \, \mathrm{dp}
\end{aligned} \nonumber \]
In the event that the pressure is constant,
\[\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]=\int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT} \nonumber \]
Footnote
[1] By definition, for the transfer of one mole of chemical substance j from phase II to phase I, \(A_{j}=-\left[\mu_{j}(\mathrm{I})-\mu_{j}(\mathrm{II})\right] ; \mathrm{Or}, \mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{II})-\mu_{\mathrm{j}}(\mathrm{I})\)