Diffusion-controlled (or diffusion-limited) reactions are reactions in which the reaction rate is equal to the rate of transport of the reactants through the reaction medium (usually a solution).

The process of chemical reaction can be considered as involving the diffusion of reactants until they encounter each other in the right stoichiometry and form an activated complex which can form the product species. The observed rate of chemical reactions is, generally speaking, the rate of the slowest or "rate determining" step. In diffusion controlled reactions the formation of products from the activated complex is much faster than the diffusion of reactants and thus the rate is governed by collision frequency.

Occurrence

Diffusion control is rare in the gas phase, where rates of diffusion of molecules are generally very high. Diffusion control is more likely in solution where diffusion of reactants is slower due to the greater number of collisions with solvent molecules. Reactions where the activated complex forms easily and the products form rapidly are most likely to be limited by diffusion control. Examples are those involving catalysis and enzymatic reactions. Heterogeneous reactions where reactants are in different phases are also candidates for diffusion control.

One classical test for diffusion control of a heterogeneous reaction is to observe whether the rate of reaction is affected by stirring or agitation; if so then the reaction is almost certainly diffusion controlled under those conditions.

Diffusion limit

Consider a reaction, in which the rate-limiting elementary reaction step is of the form

1. A+ B C

which is relatively independent of the properties of the reaction itself. The following derivation is adapted from Foundations of Chemical Kinetics .

.

. This can be written as: is the flux of B into the sphere. By Fick's law of diffusion, is the diffusion coefficient, obtained by the Stokes-Einstein equation. The second term is the positional gradient of the chemical potential.

Inserting into gives and rewrite as Thus which is an ordinary differential equation.

, we can solve by separation of variables. Namely: simplifies to

, we have . Substituting this into and rearranging yields

can then be re-written as the "diffusion influenced rate constant"

In that case, simplifies even further to This equation is true for a very large proportion of industrially relevant reactions in solution.

Viscosity dependence

The Stokes-Einstein is the viscosity of the solution. Inserting this into is given in centipoise:

See also

• Diffusion limited enzyme

References