## Reactions In Solution

of the x axis, for example. The rate of diffusion, J = dn/dt (in molecules s"1), is given, according to Fick's first law, by dn S[AU

J = — = -DE-, dt 8x where dn is the amount of A crossing the area E (cm2) in time dt, D is the diffusion coefficient (in units of cm2 s"1), and <5[_/VA]/Sx is the gradient (in units of molecules cm"4) of the concentration of A in the x direction.

Starting with Fick's first law, one can calculate for a solution of two reactants A and B the frequency of A-B encounters, which is in effect the reaction rate constant for diffusion-controlled reactions. This is given by the following, in units of L mol"1 s"1:

rAB is the distance between the centers of the molecules when they react and DAB = (DA + DB), where DA and Du are the diffusion coefficients of A and B, respectively. For typical values of rAU = 0.4 nm and DA= DB = 2 X 10"5 cm2 s"1, a rate constant of ~10l() L mol"1 s"1 is obtained. In solution then, diffusion-controlled bimolecular reactions between uncharged species occur with rate constants ~10'° L mol"1 s"1. For reactions with significant activation energies and/or steric requirements, the rate constants are correspondingly lower.

In this case of uncharged, nonpolar reactions, there is little interaction between the reactants and the solvent. As a result, the solvent does not play an important role in the kinetics per se, except through its role in determining the solubility of reactive species and cage effects. The rate constants for such reactions therefore tend to be similar to those for the same reactions occurring in the gas phase. Thus, as we saw earlier, diffusion-controlled reactions in the gas phase have rate constants of ~ 10"1(1 cm3 molecule-1 s"1, which in units of L mol-1 s"1 corresponds to ~6 X 10"' L mol-1 s"1, about equal to (usually slightly greater than) that for diffusion-controlled reactions in solution.

### 3. Reactions of Charged Species in Solution

If the reactants are ionic with opposite charges, the rate constant can be greater than 1010 L mol-1 s"1 due to the favorable attractive forces. For example, the rate constant for the reaction of H+ with OH- in aqueous solutions at 25°C is ~10" L mol"1 s"1. On the other hand, the electrostatic repulsion between ions of like sign can significantly slow their reaction. Similarly, if the reactants are polar molecules, electrostatic forces between them and the solvent may come into play.

For ions and polar molecules, the nature of the solvent is an important factor in solution-phase reactions. Following the derivation of Laidler and Meiser (1982), we first consider the reaction between two ions A and B with charges ZAe and Zue, respectively, where e is unit electronic charge and ZA and Zu are the number of unit charges on the ions, i.e., are whole positive or negative numbers. The electrostatic force (F) between these two ions separated by a distance r in a vacuum is given by Coulomb's law,

Aire^r2

where e0 = 8.85 X 10"12 C2 N"1 m"2, the permittivity of a vacuum. However, if the ions are immersed in a solvent, having a dielectric constant s, the electrostatic force between them is modified by the properties of the solvent. Equation (II) thus becomes

4ire()er2

The higher the solvent dielectric constant s, the more the electrostatic force between the ions is reduced. From this expression for the force between two ions, one can calculate the work done to bring the two ions from infinite distance to the distance necessary to react, r = dAB; this is equal to the change in free energy due to the electrostatic forces as the ions approach each other, AGcs. The total free energy change in bringing the ions together is the sum of this electrostatic term and a nonelectrostatic one, AG():

(6.02 X 10 )ZAZBe 4 TTs0edAli

(Avogadro's number is included in the electrostatic term to convert to units of per mole rather than per molecule.)

As seen in Section A.3.b, the transition state form of the rate constant is given by k T

The free energy of activation AG0# for bringing two ions to the necessary distance dA13 in order to react is given by equation (KK). Thus the natural logarithm of the rate constant becomes k T AG,,

4TrsnsdAnkT

where

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