Physical Chemistry Third Edition

12.2 Elementary Processes in Liquid Solutions 529

constitutes a diffusion flux that obeys Fick’s law of diffusion. It is assumed that the

reaction occurs as soon as the center of the type 2 molecule reaches a critical distance

d 23 from the center of the type 3 molecule. The distanced 23 is called thereaction

diameter. The concentration of type 2 molecules at distances smaller thand 23 vanishes

because of the immediate reaction. A concentration gradient is set up, with the value

of the concentration ranging from its “bulk” value at large distances down to zero at

a distanced 23. A solution to Fick’s law of diffusion was sought that matched these

conditions. The diffusion flux gives the rate of reaction, since all molecules that diffuse

up to the sphere of radiusd 23 react immediately. We do not discuss the theory, but the

result is that the reaction is second order overall, first order with respect to 2, and first

order with respect to 3. The second-order rate constant is found to be

k 4 πNAvD 2 d 23 (approximate equation) (12.2-1)

whereNAvis Avogadro’s constant andD 2 is the diffusion coefficient of substance 2.

When the fact that the type 3 molecules are also moving is included, the result is^3

k 4 πNAv(D 2 +D 3 )d 23

(diffusion-limited second-order

reaction with two reactants)

(12.2-2)

whereD 3 is the diffusion coefficient of substance 3. Equation (12.2-2) is used in

preference to Eq. (12.2-1) for calculations.

If two molecules of substance 2 react in a diffusion-limited reaction, the reaction

is second order in that substance. By an analysis that is analogous to that leading to

Eq. (12.2-2),

k

1

2

4 πNAv( 2 D 2 )d 22  4 πNAvD 2 d 22

(diffusion-limited second-order

reaction with one reactant)

(12.2-3)

whered 22 is the reaction diameter for two type 2 molecules. The factor 1/2 in

Eq. (12.2-3) is included because of the factor of 1/2 in the definition of the rate in

Eq. (11.1-6) for a substance with stoichiometric coefficient equal to 2.

EXAMPLE12.3

Assume that the following reaction in carbon tetrachloride is elementary and diffusion-

controlled:

2I−→I 2

Calculate the rate constant at 298 K. Assume that the reaction diameter is 4. 0 ×

10 −^10 m. The diffusion coefficient of iodine atoms in CCl 4 at this temperature is equal to

4. 2 × 10 −^9 m^2 s−^1.

(^3) K. J. Laidler,Chemical Kinetics, Harper and Row, New York, 1987, p. 212ff.