Phase rule classification of physical and chemical critical effects in liquid mixtures

Abstract

The isobaric heat capacity and the coefficient of thermal expansion of a binary liquid mixture with a critical point of solution diverge as functions of temperature in the critical region. When the same liquids are used as solvents for chemical reactions, the van’t Hoff plot of the extent of reaction diverges in the critical region. According to the phase rule, the physical and chemical properties are isomorphic in the sense that in each case the critical behavior is determined by three fixed thermodynamic intensive variables.

Introduction

Griffiths and Wheeler were the first to formulate a set of phenomenological rules for predicting the existence of critical effects in liquid mixtures [1]. Their predictions are based upon a division of the intensive thermodynamic variables into two classes, termed “densities” and “fields”. A density is defined as an intensive variable whose value depends upon the phase in which it is measured. Examples include the mole fraction, $X$, the entropy density, $\overline{S}$, and the molar volume, $\overline{V}$ . By contrast, a field variable is defined as an intensive variable whose value is uniform across all coexisting phases. Examples include the temperature, $T$, the pressure, $P$, and the chemical potential, $\mu$. According to Griffiths and Wheeler, a thermodynamic property that can be represented as the derivative of a density with respect to a field, should diverge toward infinity as the critical point is approached, so long as the direction of approach involves no more than one fixed density variable.

In the case of a non-chemically reactive binary liquid mixture with a critical point of solution, such as carbon disulfide + methanol, the equilibrium state is completely specified when $T$ , $P$ , and the mole fraction, $X$, of one of the components are known. As only one of these three variables is a density, the thermophysical properties, such as isobaric heat capacity, $C_{P,X}=T{(\partial \overline{S}/\partial T)}_{P.X}$, and the coefficient of thermal expansion, ${\beta }_{P,X}=(1/\overline{V}){\left(\partial \overline{V}/\partial T\right)}_{P,X}$ diverge as $T$ approaches the critical solution temperature along the critical isopleth, $X=X_c$ [2].

The Griffiths-Wheeler rules are very successful in systematizing the critical effects in the physical properties, such as $C_{P,X}$ and ${\beta }_{P,X}$ [3]. Nonetheless, it was clear from the outset that these rules, or something similar, might also be used to guide the search for critical effects associated with chemical reactions in binary liquid mixtures [4], [5], [6]. The choice was wide, there being more than 1000 pairs of liquids known to exhibit critical solution behavior [7]. The presence of reaction products complicates the Griffiths-Wheeler analysis, however, by increasing the number of mole fractions associated with the description of the system. For example, in the case of isobutyric acid + water, which ionizes to produce the hydronium and the isobutyrate ion, there are four mole fractions. Yet, the heat capacity, $C_{P,X}$, is observed to diverge in the critical region [2]. Citing the reaction stoichiometry, Wheeler reasoned intuitively that only one of these mole fractions was actually fixed. Relations, which he derived linking the other three made them functions of the temperature [8].

The first direct experimental search for a critical effect in the position of chemical equilibrium involved the dimerization of $N{O}_2$ to form $N_2{O}_4$ in a mixture of carbon tetrachloride + perfluoromethylcyclohexane [9]. Although the effect was noticeably above the background, its interpretation in terms of the Griffiths-Wheeler rules was obscured by the absence of a systematic method for enumerating the number of fixed density variables. As a consequence, there was a long delay before the search for chemical equilibrium critical effects was resumed [10], [11], [12], [13], [14], [15], [16], [17], [18]. Although the applicability of the Gibbs phase rule to critical effects was suggested quite early, [19] there was a similar delay before the connection between this rule and the concepts of Griffiths and Wheeler were finally realized [15], [16], [17]. Below, we generalize these relations [15], [16], [17] and propose the phase rule as the general principle capable of unifying the physical and chemical effects observed in binary liquid mixtures with a critical point of solution.