On algorithm for the calculation of the equilibrium composition of water-salt systems on the basis of the Pitzer model

Abstract

Calculating the equilibrium composition of systems under given conditions (usually, temperature and pressure) is a core in the majority of the existing computer models of geochemical processes. Recently, Shvarov [1] investigated the thermodynamic consistency of the models of nonideal aqueous solutions that are used in geochemical calculations. He distinguished necessary and sufficient conditions for the thermodynamic consistency of the physicochemical models of real (nonideal) solutions and presented particular examples of errors appearing when these criteria are not met. In addition, Shvarov [1] claimed that the models that are currently used for concentrated electrolyte solutions are thermodynamically inconsistent. Such a pessimistic conclusion is related to a large extent to the fact that Shvarov’s [1] analysis was limited to the Debye‐Huckel model, which is applicable to diluted aqueous solutions, and its empirical modifications aimed at extending to concentrated aqueous solutions. Indeed, such models are thermodynamically inconsistent. However, models have been developed allowing adequate description of the behavior of concentrated aqueous solutions. One of them is the wellknown Pitzer model [2, 3] proposed for the description of the properties of dense electrolyte solutions in a wide range of temperatures, pressures, and compositions. The Pitzer model was developed on the basis of the Gibbs method and is, consequently, thermodynamically sound. If correct numerical algorithms are employed, the Pitzer model allows for the calculation of chemical equilibria in water‐ salt systems within a wide concentration range. The Pitzer model is an example of the rigorous statistical mechanical description of a multicomponent fluid system and the calculation of free energy by the Gibbs method. The derivation of an expression for free energy in the Pitzer model was described in detail in [2, 3], and only its basic principles are shortly discussed here. It is based on the approach developed by McMillan and Mayer [4], who described interactions between dissolved species within the mean field approximation and ignored the detailed description of interactions between individual particles. It should be emphasized that these approximations concern interactions between particles, i.e., a microscopic Hamiltonian. The free energy function is further derived in accordance with the statistical method of Gibbs using the methods of group expansion of Mayer [5, 6] and divergence elimination [6, 7] for the calculation of the configuration integrals of the long-range potentials of electrostatic interactions. Eventually, the excess Gibbs free energy of solution is given in the Pitzer model, after transition from volume concentrations to molalities, as [2]