Exact thermodynamic formulation of chemical association

The thermodynamics of interacting systems of two species of particles, A and B, may be specified in "physical" terms using only the two densities ρ_a and ρ_b or, alternatively, in a "chemical picture" using three densities ρ_a, ρ_b, and ρ_c related by a mass-action law corresponding to the "reaction" A+B⇌C, where C denotes a "compound," "complex," "cluster," or "associated pair." We present exact methods for generating associative or "chemical" thermodynamics from an arbitrary physical specification. Both explicit order-by-order matching conditions and a variety of thermodynamically stable, closed-form solutions are derived. The analysis elucidates precisely the freedom available to choose the association constant, the definition of a cluster, and the interactions of a cluster (or "pair") with other clusters and with unpaired (i.e., dissociated, or "free") species. A single-species system of, say, X particles described chemically by 2X⇌Y is analyzed similarly. Various examples, including purely hard-core liquids and the van der Waals fluid, demonstrate applications of the theory, which should aid in improving approximate treatments for electrolytes and other systems. The precautions necessary in selecting a physically acceptable association constant are discussed.