1.13.4: Equilibrium- Liquid-Solid- Schroeder - van Laar Equation
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 \(\mathrm{j}\). Hence,
\[\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \,\mathrm{f}_{\mathrm{j}}(\ell)\right]=\int_{\mathrm{T}_{\mathrm{j}}^{0}}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \,\mathrm{T}^{2}} \,\mathrm{dT} \label{a} \]
Here \(x_{j}(\ell)\) is the mole fraction composition of the liquid; \(f_{j}(\ell)\) is the rational activity coefficient of substance \(j\) in the liquid mixture at mole fraction \(x_{j}(\ell)\) and temperature \(\mathrm{T}\). \(\mathrm{T}_{\mathrm{j}}^{0}\) is the melting point of pure \(j\) substance \(j\) at pressure \(\mathrm{p}\); i.e., both liquid and solid phases are pure chemical substance \(j\).
In the event that \(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) is independent of temperature [i.e. \(\Delta_{\text {trans }} C_{p j}^{0}(T, p)\) is zero] Equation \ref{a} is integrated to yield Equation \ref{b}.
\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \,\mathrm{f}_{\mathrm{j}}(\ell)\right]=\dfrac{\Delta_{\text {fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{R}} \, \left(\dfrac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{j}}^{0}}\right) \label{b} \]
The phenomenon under consideration is fusion so that \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) is the enthalpy of fusion of chemical substance \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In the event that the thermodynamic properties of the liquid-solid system are ideal, Equation \ref{b} simplifies to Equation \ref{c}.
\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell)\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{R}} \left(\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{j}}^{0}}\right) \label{c} \]
Equation \ref{c} is the Schroeder- van Laar Equation [1].
Footnote
[1] I. Prigogine and R Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Greeen, London, 1953.