We have seen in Section 5.2.1 that the solvent cage can have a substantial influence on reaction rates in solution. When the reaction within the cage is very fast com- pared to the escape rate, then the rate of the overall reaction is controlled by the rate at which the reactants encounter one another. Under such conditions, the reaction rate is called diffusion controlled, and it is now of interest to develop an expression for the overall rate of such reactions. Consider a simple model for this diffusion- controlled limiting rate constant in which species A reacts with species B every time the two approach one another to within their contact distance R, the sum of their two radii, as shown in Figure 5.2.
We now briefly generalize to three dimensions the treatment of diffusion in Chapter 4. In three dimensions, the flux is related to the gradient of the concentra- tion: J = - DVc, where J is a three-dimensional vector, and V is called the gradi- ent operator. In Cartesian coordinates, V = i(dldx) + j(d/ay) + k(d/dz), with i, j, and k as unit vectors in the x, y, and z directions. In spherical coordinates, V = a,(dldr) + a,(llr)(dld8) + a,(llr sin 8)(dld+), with a,, a,, and a, as unit vectors in the c 8, and + directions.
Section 5.2 The Cage Effect, Friction, and Diffusion Control
II Figure 5.2
In a diffusion-controlled reaction every reactant B that approaches to within a radius R of A will react. Reaction causes a gradient in [B] that gives rise to a flux of B toward A.
Returning to Figure 5.2, since reaction will deplete the concentration of B around each A, the reaction itself will establish a concentration gradient, and this gradient will cause molecules of type B to flow toward those of type A. Let A(r,8,+) represent the spatially dependent concentration of A, and let B(r,8,4) represent the spatially dependent concentration of B. The three-dimensional vector representing the flux JA-B of reactants A and B toward one another is equal to the flux of B toward A due to concentration gradient,
plus the flux of A toward B due to concentration gradient,
-JA = DAVA(Y;@,$), (5.13)
so that
JA-B = -(JA + J e ) = (DA + D B ) V B ( ~ @ , + ) - (5.14) In this last equation, we have assumed quite reasonably that VA(r,8,+) = VB(r,B,+), since the gradients in the two reactants are caused by the same effect, namely, the fact that A molecules around B are depleted by reaction and vice versa.
Note that the gradient VB(r,B,+) is positive, so that our choice of sign gives that the flux JA-e is also positive, as it should be.
Suppose that we have a mixture of reactants but that we prevent them from reacting. Their concentrations will then be the bulk, equilibrium concentrations;
that is, B(r,8,+) = [B] and A(r,8,+) = [A], where B(r,8,$) and A(r,8,+) are the spa- tially dependent concentrations and [B] and [A] are the bulk ones, which we will assume to be constants. Now we imagine that the reaction is suddenly turned on.
The concentration of A in the vicinity of B will decrease, and vice versa, so that a concentration gradient is formed. But after a short time, steady state will be approached, so that the flux of A and B toward one another will be constant. The
concentrations A(r,@,+) and B(r,0,+) at any position will also be constant. Under the assumption that every encounter leads to reaction, the flux of A and B toward one another will equal to the flux of products. Thus, at steady state
Jmn = JA-B = (DA + D B ) V B ( ~ , ~ , + )
= constant. (5.15)
Now let us consider the steady-state mathematical solution. Following argu- ments that exactly parallel those in Section 4.7 (see equation 4.40), we find that
We wish a solution to the rate constant for which the concentrations of A and B at any position are fixed in time, so that, at equilibrium, aB(r,O,+)ldt = 0.
The coordinates most appropriate for considering the motion of the A-B pair are spherical coordinates, since the gradient in concentration and (we assume) the gradient in electrical potential depend only on the distance between the pair. Thus, B(r,8,4) = B(r). In spherical coordinates the operator V2, called the Laplacian, is
given by
Substituting equation 5.17 into equation 5.16 and recognizing that B(r) depends only on r and that dB(r)ldt = 0, we find that the solution for B(r) needs to obey the equation
The solution to this equation, as may be readily verified by substitution, is
where the constants c, and c, must still be determined from the boundary condi- tions. When the distance between A and B is sufficiently large, i.e., as r + w, B(r) must approach its bulk concentration, [B]. Thus, we find that c, = [B].
To evaluate c,, let us calculate the concentration of B at the distance R equal to the sum of the two radii. To do this, we examine in more detail the gradient caused by the reaction. Consider the flux of B into of a sphere of radius r centered on a particular reactant of type A, as shown in Figure 5.2. Every B flowing into the sphere eventually reacts with A at a distance R, where the concentration of B is B(R). Thus, the rate of the reaction is then just the flux of B, in number per time per area, times the area of the sphere times concentration of A:
where [PI is the concentration of products, k, is the phenomenological rate constant we wish to determine (see Section 5.2.3), and 4n-13 is the area of the sphere around A at a distance z Combination of equations 5.15 and 5.20 leads to
Section 5.2 The Cage Effect, Friction, and Diffusion Control In this equation [A] is constant (equal to the. macroscopic concentration), B(R) is constant (to be determined) and B(r) denotes how the microscopic concentration of B varies with distance. Division of both sides of the equation by [A16 and multi- plication by dr yields
Finally, integration of both sides over dr from r = R to r = a gives
In this last equation, we have used the fact that B(r = a ) is just the bulk concen- tration [B]. Comparison of the last line of this equation with equation 5.19 leads, after some algebra, to the conclusion that
but we will have little use for this equation now that the last line of equation 5.23 gives us an expression for B(R). Substitution of this solution for B(R) into equation 5.20 gives
Thus, when kr >> 4r(DA + DB)R, the overall rate constant for the reaction is given by
We see that the rate constant, k, is then completely controlled by the encounter rate, k,,,, defined in Section 5.2.3; i.e., it is controlled by diffusion. In this limiting case, the reaction occurs instantly when the reactants approach to within their aver- age diameter R. Example 5.1 illustrates the utility of equation 5.26.
example 5.1
Calculating Diffusion-Controlled Rate Constants
Objective Given that the diffusion coefficient of many species in aqueous solu- tion is on the order of lop9 m2/s, calculate the diffusion limited rate constant for a pair of reactants whose average diameter is 2.0 nm.
Method Use equation 5.26.
Solution The rate constant should be k = 4v(DA + DB)R, with DA = DB and R = 2.0 nm. Thus, k = 4v(2 X lop9 m2/s)(2.0 X m/molecule)(106 cm3/m3)(6.02 X molecule/mol) = 3.0 X 1013 cm3 mol-l s-I = (3.0 X 1013 cm3 mol-l s-I) (1 LIlOOO cm3) =
3.0 X 101° L mol-I s-l.
Equation 5.26 provides the diffusion-controlled rate constant in the case when the reactants are uncharged. When the reactants are ionic, the situation is somewhat more complicated because, in addition to the concentration gradient caused by reaction, there is also a concentration gradient caused by the attraction or repulsion of charged particles. A detailed examination, discussed in Appendix 5.2, shows that
where
and U(r) is the potential of interaction between the ionic reactants. Note that when U(r) = 0, the integration in equation 5.28 can be performed to yield /3 = R, and we recover equation 5.26 from equation 5.27. In general, this potential will be given for charged particles by U(r) = zAzBe2/er; where E here refers to the dielec- tric constant of the solution, zA and zB are the integer charges on the ions, and e is the magnitude of the charge on an electron. The integration in equation 5.28 then shows that
where ro = e2/ekT and is equal to about 0.7 nm in water at 25OC.
5.3 REACTIONS OF CHARGED SPECIES IN SOLUTION:
IONIC STRENGTH AND ELECTRON TRANSFER
As discussed in the Introduction to this chapter, most rate constants in solution are similar to those for the corresponding reaction in the gas phase, except when the rate of the reaction is limited by how fast the reactants can diffuse through the solu- tion. Another situation for which the solution-phase rate constant can differ sub- stantially from the gas-phase rate is when the reactants or the activated complex
Section 5.3 Reactions of Charged Species in Solution interact strongly with the solvent. An example of such interaction is the electro- static stabilization of ionic reactants or complexes by the solvent. Two situations will be considered. In the first, we will examine the effect on the rate constant of additional ions in the solution, and we will find that the rate constant is influenced by the ionic strength of the solution. In the second situation we will see that even a neutral solvent can influence the reaction rate if the energy of solvation is substan- tially different for the reactants and products. An example of this second effect is when the reaction involves an electron transfer, either from one molecule to another or between two different sites on the same molecule. As the reaction proceeds, a dielectric solvent must rearrange its structure to attain the minimum energy, and this solvent reorganization will have an influence on the rate constant. How the rate constant varies with the solvent reorganization energy is the subject of Marcus the- ory, which we will briefly develop.
We consider first the influence on reactions of solutions with high ionic strength.