Thermodynamics far from equilibrium: Reactions with multiple stationary states

Abstract

We present a thermodynamic analysis of global validity for effectively one-variable, irreversible chemical systems with multiple steady states. A hypothetical reaction chamber is held at constant temperature and volume and is connected by selectively permeable membranes to reservoirs of reactant(s) and product(s), each at constant selected pressures. An appropriate free energy function, which yields criteria of evolution to equilibrium for the composite system of reaction chamber and reservoirs, is a hybrid of Gibbs and Helmholtz free energies. The one variable in the reaction chamber is the pressure of a chemical intermediate which varies in time according to a given reaction mechanism. With the hybrid free energy, the kinetics for a given mechanism, and a concept of instantaneous indistinguishability of systems with different mechanisms, we establish a thermodynamic driving force, or species-specific affinity, for the intermediate. The species-specific affinity vanishes at steady states, and upon its differentiation we obtain necessary and sufficient conditions for the stability of steady states and for critical points. The integral of the species-specific affinity globally provides valid Liapunov functions for the evolution of the intermediate. These results are independent of the number of steady states of the system, and they hold both near to and far from equilibrium. For a large class of mechanisms with a single intermediate, the integral of the species-specific affinity appears in the irreversible part of the time-dependent transition probability of the single-variable Master equation and in its stationary solution. Hence for these mechanisms we obtain a direct interpretation of the stochastic results in terms of thermodynamic quantities. The time rate of change in the pressure of the intermediate multiplied by its species-specific affinity yields a species-specific term in the dissipation. The total system dissipation (or entropy production) is not in general a minimum at a nonequilibrium steady state, but the species-specific term is minimized at every such state. The expression of the stationary solution of the master equation in terms of the species-specific affinity provides a generalization of the Einstein relation for the probability of equilibrium fluctuations to far-from-equilibrium conditions. The functional form of the species-specific term in the dissipation parallels a form that appears in Boltzmann’s H theorem for the momentum relaxation of a dilute gas.