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1.13.5: Equilibrium - Eutectics

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    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.

    This page titled 1.13.5: Equilibrium - Eutectics was last modified on Mon, 20 May 2024 07:03:17 GMT and is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer and Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform.