Diverging Thermodynamic Derivatives Associated with Heterogeneous Chemical Equilibrium in a Binary Liquid Mixture with a Consolute Point

Abstract

The solubilities of tin(II) oxide, copper(II) oxide, and cobalt(II) oxide have been determined in the liquid mixture, isobutyric acid + water, along the critical isopleth. When plotted in van’t Hoff form with \(\ln s\) versus \(1/T\), the solubility measurements, \(s\), lie on a straight line for values of the temperature, \(T\), which are sufficiently in excess of the critical solution temperature, \(T_\mathrm{c}.\) In the case of SnO, the dissolution reaction is exothermic, and the slope of the van’t Hoff plot diverges toward positive infinity as \(T\rightarrow T_\mathrm{c} .\) In the case of both CuO and CoO, the dissolution reaction is endothermic, and the slope of the van’t Hoff plot diverges toward negative infinity as \(T\rightarrow T_\mathrm{c} .\) Analysis of these ternary, heterogeneous equilibria using finite dimensional vector space stoichiometry theory shows that each contains two linearly independent components. According to the Gibbs phase rule, two-phase equilibria of this type can be described by two fixed, intensive variables, which are accounted for by the temperature and the pressure, respectively. The Gibbs–Helmholtz equation and the principle of critical-point universality can be combined to predict under conditions of fixed temperature and pressure that when dissolution is exothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward positive infinity in the critical region, while when dissolution is endothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward negative infinity. Our experiments include examples confirming both these predictions.