Spontaneity and Equilibrium II: Multireaction Systems

The thermodynamic criteria for spontaneity and equilibrium in multireaction systems are developed and discussed. When N reactions are occurring simultaneously, it is shown that G and A will depend upon N independent reaction coordinates, ξα (α = 1,2, ..., N), in addition to T and p for G or T and V for A. The general criteria for spontaneity and equilibrium are the same as those for a single-reaction system, dG ≤ 0 (T and p constant) or dA ≤ 0 (T and V constant). However, dG and dA are now the sum of N terms. It is shown that this sum has the form dG = dA = ∑α=1 N Δrμα dξα, where Δrμα is the reaction chemical potential for reaction α, (∂G/∂ξα) T,p,ξj . Consequently, the result that dG < 0 at a particular composition provides no information about which of the N reactions are proceeding spontaneously and which are nonspontaneous. At most compositions, there exist an infinite number of both spontaneous and nonspontaneous pathways for the system. Equilibrium in a multireaction system exists at a single composition point, as is the case for a single-reaction system. Determination of this point requires the simultaneous solution of N algebraic equations, which are often nonlinear. Although there exists an infinite number of possible, spontaneous paths, the thermodynamically most probable path is shown to be along the negative gradient of the thermodynamic potential. This path corresponds to that for the maximum chemical force. It is also shown to correspond to the statistically most-probable reaction path. The analysis demonstrates that the direction cosines for the negative gradient vector at each composition point are given by the reaction chemical potentials. The equations yielding the gradient vector comprise a set of N, coupled, first-order differential equations involving the reaction chemical potentials whose solution depends upon the initial state of the system. The calculation is, therefore, analogous to the computation of a trajectory in a molecular dynamics study. The analysis clearly shows that the reaction chemical potentials determine spontaneity, equilibrium, and the thermodynamically expected reaction path, not G, A, ΔG, or ΔA. The principles resulting from the analysis are illustrated by application to a simple reaction system involving unimolecular isomerization along two pathways.