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).[1] 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.
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.
The following derivation is adapted from Foundations of Chemical Kinetics.[2] This derivation assumes the reaction
A+B → C
RA
RAB
If we assume a local steady state, then the rate at which B reaches
RAB
Therefore, the steady state condition becomes
1.
k[B]=-4\pir2JB
where
JB
2.
JB=-DAB(
dB(r) | + | |
dr |
[B] | |
kBT |
dU | |
dr |
)
where
DAB
Inserting 2 into 1 results in
3.
k[B]=4\pir2DAB(
dB(r) | + | |
dr |
[B](r) | |
kBT |
dU | |
dr |
)
It is convenient at this point to use the identity
\exp(-U(r)/kBT) ⋅
d | |
dr |
([B](r)\exp(U(r)/kBT)=(
dB(r) | + | |
dr |
[B](r) | |
kBT |
dU | |
dr |
)
4.
k[B]=4\pir2DAB\exp(-U(r)/kBT) ⋅
d | |
dr |
([B](r)\exp(U(r)/kBT)
Rearranging 4 allows us to write
5.
k[B]\exp(U(r)/kBT) | |
4\pir2DAB |
=
d | |
dr |
([B](r)\exp(U(r)/kBT)
Using the boundary conditions that
[B](r) → [B]
U(r) → 0
r → infty
6.
infty | |
\int | |
RAB |
dr
k[B]\exp(U(r)/kBT) | |
4\pir2DAB |
=
infty | |
\int | |
RAB |
d([B](r)\exp(U(r)/kBT)
7.
k[B] | |
4\piDAB\beta |
=[B]-[B](RAB)\exp(U(RAB)/kBT)
\beta-1=
infty | |
\int | |
RAB |
1 | \exp( | |
r2 |
U(r) | |
kBT |
dr)
For the reaction between A and B, there is an inherent reaction constant
kr
[B](RAB)=k[B]/kr
8.
k=
4\piDAB\betakr | |||||||||
|
Suppose
kr
kD=4\piDAB\beta
9.
k=
kDkr | |||||||||
|
If the forces that bind A and B together are weak, ie
U(r) ≈ 0
\beta-1 ≈
1 | |
RAB |
10.
k=
kDkr | |
kr+kD |
The Stokes-Einstein equation describes a frictional force on a sphere of diameter
RA
DA=
kBT | |
3\piRAη |
η
kD
8RT | |
3η |
η
kD
Solvent | Viscosity (centipoise) | kD(
) | |||
---|---|---|---|---|---|
0.24 | 27 | ||||
Hexadecane | 3.34 | 1.9 | |||
0.55 | 11.8 | ||||
0.89 | 7.42 | ||||
0.59 | 11 |