Concepts from Chemical Reaction Kinetics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 214

Consider the following activated complex formation in solution: A + B

( AB ) † P.The equilibrium constant for activated complex formation in the solution

phase must account for nonidealities due to the solvent phase. Hence

→

AB † [ ] γ AB †

γ A γ B

K sol = (5.80)

[ A ][ B ]

and the rate in solution is

k B T
h ·[

k B T

] γ A γ B

h K sol [

AB † r sol = ]= A ][ B

γ AB † .

(5.81)

The rate constant for the reaction in solution is therefore given by

k B T

h K sol γ A γ B

k sol =

γ AB † .

(5.82)

The activity coefficients are referred to the standard state of infinite dilution for
solutes. The ratio of rate constants for solution and gas-phase reactions is

V 0
RT

γ A γ B
γ AB † .

k sol
k gas =

K HA K HB
K H,AB

(5.83)

†
AB are similar if the reactant and activated
complex are similar in nature, and hence k sol ≈

γ A and γ

For a unimolecular reaction both

k gas . Examples of these cases abound

in the environmental engineering literature.
The above discussion presupposes that the solvent merely modifies the interactions
between the species. In these cases, since the solvent concentration is in excess of
the reactants, it provides a medium for reaction. Hence it will not appear in the rate
expression. If the solvent molecule participates directly in the reaction, its concen-
tration will appear in the rate equation. It can also play a role in catalyzing reactions.
In solution, unlike the gas phase, the reaction must proceed in steps: (i) diffusion of
reactants toward each other, (ii) actual chemical reaction, and (iii) diffusion of prod-
ucts away from one another. In most cases, steps (i) and (iii) have activation energies
of the order of 20 kJ, which is much smaller than the activation energy for step (ii).
Hence, diffusion is rarely the rate-limiting step in solution reactions. If the rate is
dependent on either step (i) or (iii), then reaction will show an effect on the solvent
viscosity.
We can rewrite the equation for k sol as

k 0 γ A γ B k sol =

γ AB † ,

(5.84)

where k 0 is the rate constant when γ → 1 (ideal solution). The dependence of γ on

solvent type is best represented by the Scatchard-Hildebrand equation

δ i − δ s ) 2 , RT ln γ i = V i ( (5.85)