Mass Variables and Chemical Potentials in the Gibbs and De Donder Formulations of Chemical Equilibrium

Abstract

The classical thermodynamic treatment of chemical equilibria in closed systems is founded on theoretical formulations by J.W. Gibbs and Th. De Donder. These two theories represent mathematically equivalent energy variation problems, related to each other through a mass variable transformation analogous to coordinate transformations in mechanics. Associated with the change in mass variables, chemical potentials of reactive substances are defined differently in the two theories, and are subject to different sets of constraints. Traditionally, the two sets of constraints have been merged into one, by assuming that the chemical potential represents the same variable in both theories, an assumption that is formally inconsistent with the difference in chemical potential definitions. Merging of constraints is still possible if chemical potentials remain invariant in value as a manifestation of symmetry under a Gibbs – De Donder mass transformation, but such symmetry has not been investigated. Here we demonstrate chemical potential invariance by reformulating De Donder’s theory using Lagrange multipliers, and combining the resulting constraints on chemical potentials with the classical Gibbs constraints, in a phase equilibration thought experiment. A new form of the Gibbs–Duhem (GD) relation is derived in the De Donder mass system. Comparing this relation to the classical GD equation in Gibbs’ component framework, subject to chemical potential invariance, directly yields the linear operator mapping the De Donder mass variables onto the set of Gibbs component masses. Results are illustrated for a simple ion exchange reaction, in the process solving the longstanding problem of determining solid exchanger activities in ion exchanging mixtures. It is pointed out that, even though the Gibbs and De Donder formulations are mathematically equivalent, the formal structure of De Donder’s theory allows a more explicit and systematic treatment of constraints, and also defines chemical masses in a form more naturally adapted to typical experimental measurements.