Entropy Generation Rate in a Chemically Reacting System

Abstract

For a nonchemical-equilibrium state of an isolated system A that has r constituents with initial amounts n a = {n1a , n2a , [[ellipsis]], nra } , and that is subject to τ chemical reaction mechanisms, temperature, pressure, and chemical potentials cannot be defined. As time evolves, the values of the amounts of constitutents vary according to the stoichiometric relations ni (t) = nia + Σj=1 τ νi (j) εj (t) , where νi (j) is the stoichiometric coefficient of the i th constituent in the j -reaction mechanism and εj (t) the reaction coordinate of the j th reaction at time t . For such a state, we approximate the values of all the properties at time t with the corresponding properties of the stable equilibrium state of a surrogate system B consisting of the same constituents as A with amounts equal to ni (t) for i = 1, 2, [[ellipsis]], r , but experiencing no chemical reactions. Under this approximation, the rate of entropy generation is given by the expression Ṡ irr = ε̇ · Y , where ε̇ is the row vector of the τ rates of change of the reaction coordinates, ε̇ = { ε̇1 , [[ellipsis]], ε̇τ }, Y the column vector of the τ ratios a j /T off for j = 1, 2, [[ellipsis]], τ, a j = −Σi=1 r νi (j) μ i ,off , that is, the j th affinity of the stable equilibrium state of the surrogate system B , and μ i ,off , and T off are the chemical potential of the i th constituent and the temperature of the stable equilibrium state of the surrogate system. Under the same approximation, by further assuming that ε̇ can be represented as a function of Y only that is, ε̇(Y ) , with ε̇(0) = 0 for chemical equilibrium, we show that ε̇ = L·Y + (higher order terms in Y ), where L is a τ × τ matrix that must be non-negative definite and symmetric, that is, such that the matrix elements Lij satisfy the Onsager reciprocal relations, Lij = Lji . It is noteworthy that, for the first time, the Onsager relations are proven without reference to microscopic reversibility. In our view, if a process is irreversible, microscopic reversibility does not exist.