Corrections to the Law of Mass Action and Properties of the Asymptotic $t = \infty$ State for Reversible Diffusion-Limited Reactions

Abstract

On example of diffusion-limited reversible $A+A \rightleftharpoons B$ reactions we re-examine two fundamental concepts of classical chemical kinetics - the notion of "Chemical Equilibrium" and the "Law of Mass Action". We consider a general model with distance-dependent reaction rates, such that any pair of $A$ particles, performing standard random walks on sites of a $d$-dimensional lattice and being at a distance $μ$ apart of each other at time moment $t$, may associate forming a $B$ particle at the rate $k_+(μ)$. In turn, any randomly moving $B$ particle may spontaneously dissociate at the rate $k_-(λ)$ into a geminate pair of $A$s "born" at a distance $λ$ apart of each other. Within a formally exact approach based on Gardiner's Poisson representation method we show that the asymptotic $t = \infty$ state attained by such diffusion-limited reactions is generally \textit{not a true thermodynamic equilibrium}, but rather a non-equilibrium steady-state, and that the Law of Mass Action is invalid. The classical concepts hold \text{only} in case when the ratio $k_+(μ)/k_-(μ)$ does not depend on $μ$ for any $μ$.