1.13.7: Equilibrium - Liquid-Solids - Hildebrand Rules
A given homogeneous liquid system (at pressure \(\mathrm{p}\)) contains two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at temperature \(\mathrm{T}\). Chemical substance \(\mathrm{j}\) at temperature \(\mathrm{T}\) and \(\mathrm{p}\) is a liquid which being in vast excess in this system is the solvent. The system is cooled and pure solid substance \(\mathrm{i}\) separates out leaving the system less concentrated in the solute \(\mathrm{i}\). The solution is dilute and we assume that the thermodynamic properties of the solution are ideal. Then, from the Schroeder–van Laar Equation
\[-\ln \left[\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )\right]=\frac{\left[\Delta_{\mathrm{fius}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R}} \,\left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{i}}^{0}}\right] \label{a} \]
\(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}\) is the molar enthalpy of fusion of chemical substance \(\mathrm{i}\), melting point \(\mathrm{T}_{\mathrm{i}}^{0}\). Mole fraction \(\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )\) is the composition of the saturated solution at temperature \(\mathrm{T}\); i.e. the solubility of substance \(\mathrm{i}\). From Equation \ref{a},
\[\ln \left[\frac{1}{\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )}\right]^{\mathrm{eq}}=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R}} \,\left[\frac{\mathrm{T}_{\mathrm{i}}^{0}-\mathrm{T}}{\mathrm{T} \, \mathrm{T}_{\mathrm{i}}^{0}}\right] \label{b} \]
Equation \ref{b} forms the background to several generalisations concerning solubilities; i.e. Hildebrand Rules [1]. We note that \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\mathrm{T}_{\mathrm{i}}^{0}\) characterise the solute .
- Solubilities increase with increase in temperature.
- For two solutes with equal \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\), the solute with lower \(\mathrm{T}_{\mathrm{i}}^{0}\) will be more soluble at a common temperature \(\mathrm{T}\).
- For two solutes with the same \(\mathrm{T}_{\mathrm{i}}^{0}\), the solid with lower \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\) will be more soluble.
Footnote
[1] see I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Greeen, London, 1953.