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 In chemistry an activated complex is defined by the International Union of Pure and Applied Chemistry (IUPAC) as "that assembly of atoms which corresponds to an arbitrary infinitesimally small region at or near the col (saddle point) of a potential ...
is much faster than the diffusion of reactants and thus the rate is governed by
collision frequency
Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of specie ...
.
Diffusion control is rare in the gas phase, where rates of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemica ...
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 In chemistry an activated complex is defined by the International Union of Pure and Applied Chemistry (IUPAC) as "that assembly of atoms which corresponds to an arbitrary infinitesimally small region at or near the col (saddle point) of a potential ...
forms easily and the products form rapidly are most likely to be limited by diffusion control. Examples are those involving
catalysis
Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
and
enzymatic
Enzymes () are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products. ...
reactions.
Heterogeneous reactions where reactants are in different phases are also candidates for diffusion control.
One classical test for diffusion control 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.
Derivation
The following derivation is adapted from ''Foundations of Chemical Kinetics''.
This derivation assumes the reaction
. Consider a sphere of radius
, centered at a spherical molecule A, with reactant B flowing in and out of it. A reaction is considered to occur if molecules A and B touch, that is, when the distance between the two molecules is
apart.
If we assume a local steady state, then the rate at which B reaches
is the limiting factor and balances the reaction.
Therefore, the steady state condition becomes
1.
where
is the flux of B, as given by
Fick's law of diffusion
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion e ...
,
2.
,
where
is the diffusion coefficient and can be obtained by the
Stokes-Einstein equation, and the second term is the gradient of the chemical potential with respect to position. Note that
refers to the average concentration of B in the solution, while
r) is the "local concentration" of B at position r.
Inserting 2 into 1 results in
3.
.
It is convenient at this point to use the identity
allowing us to rewrite 3 as
4.
.
Rearranging 4 allows us to write
5.
Using the boundary conditions that