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.

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

A + B → C

and occurs at rate ${\textstyle k_{r}}$ when molecules of A and B touch. For a bulk system, the observed reaction rate ${\textstyle k}$ is depressed, because molecules of A and B must diffuse towards each other before reacting. At very large values of ${\textstyle k_{r}}$, the bulk reaction occurs at a rate ${\textstyle k_{D}}$ which is relatively independent of the properties of the reaction itself. The following derivation is adapted from Foundations of Chemical Kinetics. ^[2]

Consider sphere of radius ${\textstyle R_{A}}$, centered at a spherical molecule A, with reactant B flowing in and out of it; molecules A and B touch when the distance between the two molecules is ${\textstyle R_{AB}}$ apart. Thus ${\textstyle [B](R_{AB})k_{r}=[B]k}$, where ${\textstyle [B](r)}$ is the smoothed "local concentration" of B at position ${\textstyle r}$.

If we assume a local steady state, then the average rate at which B reaches ${\displaystyle R_{AB}}$ corresponds to the observed reaction rate ${\textstyle k}$. This can be written as:

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

1

where ${\textstyle J_{B}}$ is the flux of B into the sphere. By Fick's law of diffusion,

$${\displaystyle J_{B}=-D_{AB}\left({\frac {d[B](r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}}\right){\text{,}}}$$

2

where ${\textstyle D_{AB}}$ is the diffusion coefficient, obtained by the Stokes-Einstein equation. The second term is the positional gradient of the chemical potential.

Inserting ( 2 ) into ( 1 ) gives

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

3

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

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

4

Thus

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

5

which is an ordinary differential equation in ${\textstyle [B](r)}$.

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

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

6

Defining $${\displaystyle \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp \left({\frac {U(r)}{k_{B}T}}\right)\,dr{\text{,}}}$$( 6 ) simplifies to

$${\displaystyle {\frac {[B]k}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp \left({\frac {U(R_{AB})}{k_{B}T}}\right)}$$

7

From the definition of ${\textstyle k_{r}}$, we have . Substituting this into ( 7 ) and rearranging yields

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

8

Taking ${\textstyle k_{r}}$ very large gives the diffusion-limited reaction rate $${\displaystyle k_{D}=4\pi D_{AB}\beta {\text{.}}}$$( 8 ) can then be re-written as the "diffusion influenced rate constant"

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

9

If the forces that bind A and B together are weak, i.e. ${\textstyle U(r)\approx 0}$ for all ${\textstyle r>R_{AB}}$, then $${\displaystyle \beta ^{-1}\approx {\frac {1}{R_{AB}}}}$$ In that case, ( 9 ) simplifies even further to

$${\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}$$

10

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:

Solvents and ${\displaystyle k_{D}}$ ^[3]

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

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.