Generalized thermobarometry: Solution of the inverse chemical equilibrium problem using data for individual species

Abstract

The inverse chemical equilibrium problem is the determination of unknown equilibrium pressure, temperature, and chemical potentials of s species, given measurements of their thermochemical constants and the compositions of phases in which they occur. Of the s species for which free energy approximations can be made, c will be compositionally independent, i.e., form a basis of composition space and hence of chemical potential space. This means that if the equilibrium model is correct, it should be possible to express the chemical potentials of all s species as linear combinations of the chemical potentials of any c basis species. The inverse chemical equilibrium problem can then be stated: g i(p,T) − μ i = 0 i= 1…c g j(p,T)− ∑ i=1 c v ijμ i=0 j=c+1…s where g i ( p, T) is a measured approximation to the apparent free energy of formation of species i, determined from thermochemical constants and observed compositions; μ i is the unknown equilibrium chemical potential of species i; and v ij are stoichiometric coefficients relating the compositions of the s − c compositionally dependent species to the c basis species. The problem has s equations and c + 2 unknowns, hence is underdetermined, exact, or overdetermined, depending on the relative magnitudes of s and c. Because of errors in measurement, or failure to preserve equilibrium compositions, overdetermined systems will usually be inconsistent. In such cases, the ordinary least-squares solution to the problem may be found by finding p, T, and μ i that minimize ƒ Tƒ where ƒ is the s component vector obtained by evaluating (1). If desired, an error covariance matrix V can be incorporated to obtain a generalized least-squares solution at the minimum of ƒ TV −1ƒ . Confidence regions can be approximated by contouring the sum of squares and using Monte Carlo techniques. The formulation is readily extended to include data from directly calibrated equilibria as equations of form: ΔG j ( p, T) − ∑ ν ij μ i = 0. Solutions to underdetermined problems can be constrained by replacing some of the equations with inequalities such as: g i 0( p, T) − μ i ≥ 0.