## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 14. Some Special Topics: Reactions in Solutions

### 14.4. Diffusion Control

The main task in this section is to estimate the rate constants for diffusion *k*_{D} and *k*_{−D}. While the reactants may be either neutral or ionic, we start with ionic reactions.

The *flux* *j*_{A} – that is, the number of moles of ions crossing a unit area per unit time – of ions A with molarity *c*_{A} across a surface normal to the mass velocity *v*_{A} is given by

(14.32)

with the mobility *u*_{A} and electric field *E*. The factor *z*_{A}/|*z*_{A}| is introduced to fix the sign of the charge *z*_{A}*e* of the ion A, since a positive ion has a positive flux in the direction of *E*. For this expression it is assumed that the flux is parallel to the electric field. With an additional concentration gradient, also assumed to be parallel to the electric field, the total flux in the direction of decreasing concentration becomes

(14.33)

with *D*_{A} the diffusion coefficient relative to the solvent. For the rate with which the reactant ions A and B approach each other we must replace *D*_{A} by *D*_{A} + *D*_{B}.

Consider now the first step of the scheme in __Eq. (14.25)__, where the labels A and B represent ions with charge *z*_{A} and *z*_{B}, respectively. We assume that a gradient in *c*_{B} is present around each A ion, and *vice versa*. Using the *Nernst–Einstein relation* (Eq. 12.67)

(14.34)

and the relation between electric field *E _{z}* = −∂

*ϕ*/∂

*z*, with

*ϕ*the electric potential, and potential energy

*Φ*(

*r*)

(14.35)

we obtain for the flux of B ions across a sphere of radius *r* around each ion A

__(14.36)__

where the overall negative sign disappears because the flux is now taken in the direction of increasing concentration. To illustrate the following argument more clearly, we ignore the d*Φ*_{B}(*r*)/d*r* contribution for the moment and add it again later.

The *flow* is *J*_{B} = 4π*r*^{2}*j*_{B}, and we obtain for the molarity *c*_{B}(*r*)

__(14.37)__

The flow *J*_{B} can be taken outside the integral because under steady-state conditions the value of the flow is independent of the radius of the sphere. Because *c*_{B}(∞) represents the bulk molarity [B], the total result is

(14.38)

We now assume that, if ion A approaches ion B within a critical radius *r**, a reaction occurs and *c*_{B}(*r*) = 0 for *r* < *r** (__Figure 14.3__). This implies that the concentration gradient does not extend to *r* = 0 since all ions are lost reaching *r**. The flow thus becomes

(14.39)

so that the rate of reaction reads

(14.40)

** Figure 14.3** The behavior of

*c*

_{B}(

*r*) and

*B*(

*r*) as a function of

*r*.

The rate constant *k*_{D} is thus proportional to the diffusion constants, as expected, and a critical length *r**.

Let us now introduce the d*Φ*_{B}(*r*)/d*r* contribution again. To solve the complete __Eq. (14.36)__ we use a somewhat unexpected “Ansatz,” and note that if we differentiate the function

(14.41)

we obtain

(14.42)

From __Eq. (14.36)__ we thus have

__(14.43)__

The integration corresponding to __Eq. (14.37)__ becomes

(14.44)

where *B*(∞) is simply *c*_{B}(∞) = [B] and *λ* has the dimensions of length. Making the same assumption of a critical radius *r** where *c*_{B}(*r*) = 0 and hence *B*(*r*) = 0, the total result is

(14.45)

(14.46)

Finally, we must link *λ* with *r**, and this is done by assuming that the potential energy between two ions is given by

(14.47)

where the subscripts are omitted since the expression is symmetric in A and B. As usual, *ε*_{0} and *ε*_{r} are the permittivity of vacuum and the relative permittivity, respectively. Substituting in __Eq. (14.43)__ for *λ* and integrating results in

(14.48)

which, as it should, results in *λ* = *r** for *Φ*(*r*) = 0 for *r* > *r**. The behavior of the concentration profile *c*_{B}(*r*) and function *B*(*r*) as a function of *r* is sketched in __Figure 14.3__.

For the dimension of *k _{D}* we have to consider the dimension of

*D*which is m

^{2}s

^{−1}; hence,

*k*

_{D}will have dimension m

^{3}s

^{−1}. Conversion to molar quantities is achieved by multiplying with Avogadro's number

*N*

_{A}.

The rate constant for the reverse diffusion *k*_{−D} in the scheme of __Eq. (14.25)__ is the reciprocal of the time in which A and B remain nearest neighbors. If specific interactions are absent, this quantity can be estimated from random walk diffusion *d* = (6*Dt*)^{1/2}, where *d* is the diffusion distance in time *t*. Hence, in this case *k*_{−D} is given by

(14.49)

Clearly, the controlling parameters for the controlled reactions are the charges *z*_{A} and *z*_{B}, the permittivity *ε*_{r}, and the diffusion coefficients *D*_{A} and *D*_{B}. Moreover, we need an estimate for *r**, which is essentially an empirical parameter. We discuss the agreement with experiment after we have dealt with reaction control.