Diffusion-controlled reaction explained

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.

Derivation

The following derivation is adapted from Foundations of Chemical Kinetics.[2] This derivation assumes the reaction

A+BC

. Consider a sphere of radius

RA

, 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

RAB

apart.

If we assume a local steady state, then the rate at which B reaches

RAB

is the limiting factor and balances the reaction.

Therefore, the steady state condition becomes

1.

k[B]=-4\pir2JB

where

JB

is the flux of B, as given by Fick's law of diffusion,

2.

JB=-DAB(

dB(r)+
dr
[B]
kBT
dU
dr

)

,

where

DAB

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 [B] refers to the average concentration of B in the solution, while [B](r) is the "local concentration" of B at position r.

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

)

allowing us to rewrite 3 as

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]

, ie the local concentration of B approaches that of the solution at large distances, and consequently

U(r)0

, as

rinfty

, we can solve 5 by separation of variables, we get

6.

infty
\int
RAB

dr

k[B]\exp(U(r)/kBT)
4\pir2DAB

=

infty
\int
RAB

d([B](r)\exp(U(r)/kBT)

or

7.

k[B]
4\piDAB\beta

=[B]-[B](RAB)\exp(U(RAB)/kBT)

(where :

\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

, so

[B](RAB)=k[B]/kr

. Substituting this into 7 and rearranging yields

8.

k=

4\piDAB\betakr
k+4\piDAB\beta
\exp(U(RAB)
kBT
)
r

Limiting conditions

Very fast intrinsic reaction

Suppose

kr

is very large compared to the diffusion process, so A and B react immediately. This is the classic diffusion limited reaction, and the corresponding diffusion limited rate constant, can be obtained from 8 as

kD=4\piDAB\beta

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

9.

k=

kDkr
k+kD
\exp(U(RAB)
kBT
)
r

Weak intermolecular forces

If the forces that bind A and B together are weak, ie

U(r)0

for all r except very small r,

\beta-1

1
RAB
. The reaction rate 9 simplifies even further to

10.

k=

kDkr
kr+kD

This equation is true for a very large proportion of industrially relevant reactions in solution.

Viscosity dependence

The Stokes-Einstein equation describes a frictional force on a sphere of diameter

RA

as

DA=

kBT
3\piRAη
where

η

is the viscosity of the solution. Inserting this into 9 gives an estimate for

kD

as
8RT

, where R is the gas constant, and

η

is given in centipoise. For the following molecules, an estimate for

kD

is given:
Solvents and

kD

Solvent Viscosity (centipoise)

kD(

x 1e9
Ms

)

0.24 27
Hexadecane 3.34 1.9
0.55 11.8
0.89 7.42
0.59 11
[3]

See also

Notes and References

  1. Book: Atkins, Peter . Physical Chemistry . New York . Freeman . 1998 . 6th . 825–8.
  2. Web site: Roussel . Marc R. . Lecture 28:Diffusion-influenced reactions, Part I . Foundations of Chemical Kinetics . University of Lethbridge (Canada) . 19 February 2021.
  3. Book: Berg, Howard, C . Random Walks in Biology . 145-148.