1.13.5: Equilibrium - Eutectics
- Page ID
- 375584
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given homogeneous binary liquid system (at pressure \(\mathrm{p}\)) contains two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at temperature \(\mathrm{T}\). The liquid system is cooled and only substance \(\mathrm{j}\) separates out as the pure solid substance \(\mathrm{j}\) leaving the liquid richer in chemical substance \(\mathrm{i}\). The mole fraction composition of the liquid is given by the Schroeder-van Laar equation written in the following form.
\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \mathrm{f}_{\mathrm{j}}(\ell)\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus; } \mathrm{j}}^{0}}\right] \label{a} \]
In many cases as the mole fraction composition of substance \(\mathrm{i}\) in the liquid increases the equilibrium temperature \(\mathrm{T}\) decreases until at the eutectic temperature \(\mathrm{T}_{\mathrm{e}}\) and mole fraction \(\left(\mathbf{X}_{\mathrm{j}}\right)_{\mathrm{e}}\) the system comprises a solid, the eutectic mixture. In the event that the thermodynamic properties of the system can be described as ideal, Equation \ref{a} simplifies to Equation \ref{b} where it is assumed that \(\mathrm{f}_{\mathrm{j}}(\ell)\) is unity at all temperatures. Then
\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus; } ; \mathrm{j}}^{0}}\right] \label{b} \]
For the other component \(\mathrm{i}\), a corresponding plot is obtained when on cooling the liquid mixture only pure solid \(\mathrm{i}\) separates out.
\[-\ln \left[\mathrm{x}_{\mathrm{i}}(\ell) \mathrm{f}_{\mathrm{i}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus }, \mathrm{i}}^{0}}\right] \nonumber \]
If the thermodynamic properties of the system are ideal then the analogue of equation (b) is equation (d).
\[-\ln \left[\mathrm{x}_{\mathrm{i}}(\ell)\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{i}}^{0}}\right] \nonumber \]
The two curves described by equations (a) and (c) meet at the eutectic temperature. \(\mathrm{T}_{\mathrm{e}}\). Granted that the thermodynamic properties of the system are ideal, the following two equations follow from equations (b) and (d).
\[-\ln \left[\mathrm{x}_{\mathrm{j}}^{\mathrm{e}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fus } ; j}^{0}}\right] \nonumber \]
\[-\ln \left[\mathrm{x}_{\mathrm{i}}^{\mathrm{e}}(\ell)\right]=-\ln \left[1-\mathrm{x}_{\mathrm{j}}^{\mathrm{e}}\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss; } \mathrm{i}}^{0}}\right] \nonumber \]
In the event that
\[\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{j}}^{0}}\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{i}}^{0}}\right] \nonumber \]
then \(x_{i}^{e}=x_{j}^{e}=0.5\). The impact of the non-ideal thermodynamic properties can be explored using equation (a) and (c) in conjunction with empirical equations relating, for example, \(\mathrm{f}_{\mathrm{j}}(\ell)\) and \(\mathrm{x}_{\mathrm{j}}(\ell)\); e.g. equation (g).
\[\ln \left[\mathrm{f}_{\mathrm{j}}(\ell)\right]=\alpha \left[1-\mathrm{x}_{\mathrm{j}}(\ell)\right]^{2} \nonumber \]
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, tranls. D. H. Everett, Longmans Greeen, London, 1953.