Diffusion-controlled reaction

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 ${\displaystyle A+B\rightarrow C}$. Consider a sphere of radius ${\displaystyle R_{A}}$, 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 ${\displaystyle R_{AB}}$ apart.

If we assume a local steady state, then the rate at which B reaches ${\displaystyle R_{AB}}$ is the limiting factor and balances the reaction.

Therefore, the steady state condition becomes

1.${\displaystyle k[B]=-4\pi r^{2}J_{B}}$

where

${\displaystyle J_{B}}$ is the flux of B, as given by Fick's law of diffusion,

2. ${\displaystyle J_{B}=-D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B]}{k_{B}T}}{\frac {dU}{dr}})}$,

where ${\displaystyle D_{AB}}$ 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.${\displaystyle k[B]=4\pi r^{2}D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}$.

It is convenient at this point to use the identity ${\displaystyle \exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)=({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}$ allowing us to rewrite 3 as

4. ${\displaystyle k[B]=4\pi r^{2}D_{AB}\exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}$.

Rearranging 4 allows us to write

5. ${\displaystyle {\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}={\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}$

Using the boundary conditions that ${\displaystyle [B](r)\rightarrow [B]}$, ie the local concentration of B approaches that of the solution at large distances, and consequently ${\displaystyle U(r)\rightarrow 0}$, as ${\displaystyle r\rightarrow \infty }$, we can solve 5 by separation of variables, we get

6. ${\displaystyle \int _{R_{AB}}^{\infty }dr{\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}=\int _{R_{AB}}^{\infty }d([B](r)\exp(U(r)/k_{B}T)}$ or

7. ${\displaystyle {\frac {k[B]}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T)}$ (where : ${\displaystyle \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp({\frac {U(r)}{k_{B}T}}dr)}$)

For the reaction between A and B, there is an inherent reaction constant ${\displaystyle k_{r}}$, so ${\displaystyle [B](R_{AB})=k[B]/k_{r}}$. Substituting this into 7 and rearranging yields

8.${\displaystyle k={\frac {4\pi D_{AB}\beta k_{r}}{k_{r}+4\pi D_{AB}\beta \exp({\frac {U(R_{AB})}{k_{B}T}})}}}$

Limiting conditions

Very fast intrinsic reaction

Suppose ${\displaystyle k_{r}}$ 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 ${\displaystyle k_{D}=4\pi D_{AB}\beta }$. 8 can then be re-written as the "diffusion influenced rate constant" as

9.${\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}\exp({\frac {U(R_{AB})}{k_{B}T}})}}}$

Weak intermolecular forces

If the forces that bind A and B together are weak, ie ${\displaystyle U(r)\approx 0}$ for all r except very small r, ${\displaystyle \beta ^{-1}\approx {\frac {1}{R_{AB}}}}$. The reaction rate 9 simplifies even further to

10. ${\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}$ 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 ${\displaystyle R_{A}}$ as ${\displaystyle D_{A}={\frac {k_{B}T}{3\pi R_{A}\eta }}}$ where ${\displaystyle \eta }$ is the viscosity of the solution. Inserting this into 9 gives an estimate for ${\displaystyle k_{D}}$ as ${\displaystyle {\frac {8RT}{3\eta }}}$, where R is the gas constant, and ${\displaystyle \eta }$ is given in centipoise. For the following molecules, an estimate for ${\displaystyle k_{D}}$ is given:

Solvents and ${\displaystyle k_{D}}$

Solvent	Viscosity (centipoise)	${\displaystyle k_{D}({\frac {\times 1e9}{M\cdot s}})}$
n-Pentane	0.24	27
Hexadecane	3.34	1.9
Methanol	0.55	11.8
Water	0.89	7.42
Toluene	0.59	11

^[3]

See also

• Diffusion limited enzyme

References

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