Geometric constraints at triple points

Abstract

A class of rules concerning the disposition of phase boundaries at triple and multiple points in phase diagrams of single and multicomponent systems is discussed from a unified point of view. These rules, which may be referred to collectively as ``The 180° Rule,'' state that in a phase diagram with thermodynamically proper independent variables, no phase occupies more than 180° of angle at a triple point. A purely thermodynamic proof of the 180° rule is presented which relies only upon the convexity properties of thermodynamic potentials required by the second law of thermodynamics, and upon two explicit assumptions concerning the behavior of properties such as the entropy and number density as the triple point is approached. These auxiliary assumptions are shown to be essential: if either is violated, a model thermodynamic potential can be constructed which satisfies all of the requirements of convexity, yet violates the 180° rule. A distinction is emphasized between two fundamentally different types of intensive variables: ``fields'' such as temperature and pressure which must be the same in coexisting phases, and ``densities'' such as the entropy per unit volume and the molar volume which generally take on different values in coexisting phases. The phase diagram of a ``simple'' or one-component system is quite different in appearance according to whether both, one, or none of the independent variables are fields rather than densities. The 180° rule is proved in each of these cases. For the case when two densities are used as independent variables, the 180° rule takes on a stronger form which is known, for ternary systems, as Schreinemakers' rule. Generalizations of the 180° rule are discussed: to multicomponent systems, to quadruple and multiple invariant points, and to higher order phase transitions.