The chemical reaction critical point exponent

Abstract

The principle of critical point universality is thought to govern critical phenomena in systems as disparate as ferromagnets, pure fluids, and binary liquid mixtures exhibiting a miscibility gap ending in a critical point of solution. The goal of any critical point theory is to determine how a given thermophysical property of interest depends upon the reduced temperature, t=(T−Tc)/Tc, where T is the thermostat temperature, and Tc is the critical temperature. Two theoretical formulations are available. The Landau mean field theory ignores fluctuations in composition in the critical region, while the mathematically more complicated Ising model takes fluctuations into account. The Landau and Ising theories agree, however, that certain of the thermophysical properties, ω, diverge in the critical region according to ω ∝ t−x as t → 0, where the value of x depends upon the property. In the Landau model, x assumes rational values, whereas in the Ising model, x assumes irrational values. In the majority of cases, the Ising model has been in better agreement with the experiment. Binary liquid mixtures with immiscibility gaps ending in a critical point of solution can be used as solvents in order to determine the critical effect in the extent, ξ, of a chemical reaction. With ξc serving as the critical value of ξ, some consensus exists in support of ξ−ξc∝tx, as t → 0. Otherwise, speculation prevails as to the value of x. Consistent with the universality principle, we find that the critical effect in the extent of the reaction, such as the shape of the liquid–liquid coexistence curve in the critical region, has its basis in the failure of phase stability. Pursuing this analogy, we note that the exponent governing the temperature dependence of ξ is x = 1/2 in the Landau model, whereas it is x = 0.3265 in the Ising model. Thermodynamic theory is exploited to distinguish the Landau/Ising limiting law, ξ−ξc∝tx, which prevails in the critical region, from the van’t Hoff background, which applies at temperatures removed from critical. The resulting equations are converted to dimensionless form and compared with data from homogeneous and heterogeneous chemical equilibria involving binary liquid mixtures as solvents.