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).<ref>{{Cite book |last=Atkins |first=Peter |title=Physical Chemistry |location=New York |publisher=Freeman |year=1998 |edition=6th |pages=825–8}}</ref>

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 <math display=inline>k_r</math> when molecules of A and B touch. For a bulk system, the observed reaction rate <math display=inline>k</math> is depressed, because molecules of A and B must diffuse towards each other before reacting. At very large values of <math display=inline>k_r</math>, the bulk reaction occurs at a rate <math display=inline>k_D</math> which is relatively independent of the properties of the reaction itself. The following derivation is adapted from ''Foundations of Chemical Kinetics''.<ref>{{cite web |last1=Roussel |first1=Marc R. |title=Lecture 28:Diffusion-influenced reactions, Part I |url=http://people.uleth.ca/~roussel/C4000foundations/slides/28diffusion_influencedI.pdf |website=Foundations of Chemical Kinetics |publisher=University of Lethbridge (Canada) |access-date=19 February 2021}}</ref>

Consider sphere of radius <math display=inline>R_A</math>, 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 <math display=inline>R_{AB}</math> apart. Thus <math display=inline>[B](R_{AB})k_r = [B]k</math>, where <math display=inline>[B](r)</math> is the smoothed "local concentration" of B at position <math display=inline>r</math>.

If we assume a local steady state, then the average rate at which B reaches <math>R_{AB}</math> corresponds to the observed reaction rate <math display=inline>k</math>. This can be written as: {{NumBlk||<math display=block> [B]k=-4\pi r^2 J_{B}\text{,}</math>|{{EquationRef|1}}}} where <math display=inline>J_{B}</math> is the flux of B into the sphere. By Fick's law of diffusion, {{NumBlk||<math display=block> J_{B} = -D_{AB} \left(\frac{d[B](r)}{dr} +\frac{[B](r)}{k_{B}T} \frac{dU}{dr}\right)\text{,}</math>|{{EquationRef|2}}}} where <math display=inline>D_{AB}</math> is the diffusion coefficient, obtained by the Stokes-Einstein equation. The second term is the positional gradient of the chemical potential.

Inserting {{EqNote|2}} into {{EqNote|1}} gives {{NumBlk||<math display=block>[B]k = 4\pi r^2 D_{AB}\left(\frac{dB(r)}{dr}+\frac{[B](r)}{k_{B}T} \frac{dU}{dr}\right)\text{.}</math>|{{EquationRef|3}}}} It is convenient at this point to use the identity <math display=block> \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} </math> and rewrite {{EqNote|3}} as {{NumBlk||<math display=block>[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)</math>|{{EquationRef|4}}}} Thus {{NumBlk||<math display=block>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)</math>|{{EquationRef|5}}}} which is an ordinary differential equation in <math display=inline>[B](r)</math>.

Using the boundary conditions that <math display=inline>[B](r)\rightarrow [B]</math>, ie the local concentration of B approaches that of the solution at large distances, and consequently <math display=inline>U(r) \rightarrow 0 </math> as <math display=inline> r \rightarrow \infty </math>, we can solve {{EqNote|5}} by separation of variables. Namely: {{NumBlk||<math display=block> \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)</math>|{{EquationRef|6}}}} Defining <math display=block>\beta^{-1} = \int_{R_{AB}}^{\infty} \frac{1}{r^2}\exp\left(\frac{U(r)}{k_B T}\right)\,dr\text{,}</math> {{EqNote|6}} simplifies to {{NumBlk||<math display=block> \frac{[B]k}{4\pi D_{AB}\beta }= [B]-[B](R_{AB})\exp\left(\frac{U(R_{AB})}{k_{B}T}\right)</math>|{{EquationRef|7}}}}

From the definition of <math display=inline>k_r</math>, we have . Substituting this into {{EqNote|7}} and rearranging yields {{NumBlk||<math display=block> 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) } </math>|{{EquationRef|8}}}}

Taking <math display=inline>k_r</math> very large gives the diffusion-limited reaction rate <math display=block>k_D = 4\pi D_{AB} \beta\text{.}</math> {{EqNote|8}} can then be re-written as the "diffusion influenced rate constant" {{NumBlk||<math display=block> k= \frac{k_D k_r}{k_r + k_D \exp\left(\frac{U(R_{AB} )}{k_B T}\right)} </math>|{{EquationRef|9}}}}

If the forces that bind A and B together are weak, i.e. <math display=inline>U(r) \approx 0</math> for all <math display=inline>r>R_{AB}</math>, then <math display=block>\beta^{-1} \approx \frac{1}{R_{AB}}</math> In that case, {{EqNote|9}} simplifies even further to {{NumBlk||<math display=block> k = \frac{k_D k_r}{k_r + k_D} </math>|{{EquationRef|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 <math>R_A</math> as <math>D_A = \frac{k_BT}{3\pi R_A \eta}</math> where <math>\eta</math> is the viscosity of the solution. Inserting this into {{EqNote|9}} gives an estimate for <math>k_D</math> as <math>\frac{8 RT}{3\eta} </math>, where R is the gas constant, and <math>\eta</math> is given in centipoise: {| class="wikitable sortable" |+ Solvents and <math>k_D</math><ref name ="NIST">{{Cite book |last=Berg |first=Howard, C |title=Random Walks in Biology |pages=145–148}} </ref> |- ! Solvent !! Viscosity (centipoise) !! <math>k_D (\frac{\times 1e9}{M\cdot s})</math> |- | 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 == {{reflist}}

Category:Chemical reactions Category:Chemical reaction engineering Category:Chemical kinetics