LXXVIII. Simultaneous diffusion and reversible chemical reaction

Abstract

The problem considered is one of diffusion in which some of the diffusing substance is immobilized by chemical reaction as diffusion proceeds. The reaction is considered to be first order and reversible, the forward reaction proceeding at a rate proportional to the concentration of solute free to diffuse, and the backward reaction at a rate proportional to the concentration of immobilized solute. Mathematical solutions are derived for the diffusion of a limited amount of solute into a plane sheet, a cylinder, and a, sphere respectively. The solutions comprise two infinite series and the physical significance of the various terms is discussed briefly. Solutions involve three independent parameters ; these are the ratio of the volumes occupied by solution and sheet, cylinder or sphere respectively, the partition factor between immobilized and free solute in the final equilibrium state, and a modified rate constant expressing the relative rates of diffusion and reaction. Calculated values of M_t /M_∞ , where M_∞ is the total amount of solute present in the sheet, cylinder or sphere at time t and M_∞ the corresponding quantity after infinite time, are tabulated for the case of an infinite amount of solute and for a range of values of the other two parameters. Characteristic features of diffusion-with-reaction are shown graphically. An important general conclusion is that if the diffusion is more than a thousand times faster than the reaction, expressed in terms of the respective half-times of the two processes, the behaviour of the joint diffusion-reaction process is almost the same as if diffusion were infinitely rapid. If on the other hand the half-times for diffusion and reaction are comparable the behaviour approximates to that for an infinitely rapid reaction. It is shown that the mathematical solutions derived for absorption also describe the course of desorption under comparable conditions. Solutions for an irreversible reaction are deduced as special cases of those for a reversible reaction.