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 distance d 23 is called the reaction
diameter. The concentration of type 2 molecules at distances smaller than d 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 distance d 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 radius d 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 4πN Av D 2 d 23 (approximate equation)
(12.2-1)
where N Av is Avogadro’s constant and D 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 4πN Av (D 2 + D 3 )d 23
(diffusion-limited second-order
reaction with two reactants)
(12.2-2)
where D 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
4πN Av (2D 2 )d 22 4πN Av D 2 d 22
(diffusion-limited second-order
reaction with one reactant)
(12.2-3)
where d 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.
E X A M P L E 12.3
Assume that the following reaction in carbon tetrachloride is elementary and diffusioncontrolled:
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.
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 distance d 23 is called the reaction
diameter. The concentration of type 2 molecules at distances smaller than d 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 distance d 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 radius d 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 4πN Av D 2 d 23 (approximate equation)
(12.2-1)
where N Av is Avogadro’s constant and D 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 4πN Av (D 2 + D 3 )d 23
(diffusion-limited second-order
reaction with two reactants)
(12.2-2)
where D 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
4πN Av (2D 2 )d 22 4πN Av D 2 d 22
(diffusion-limited second-order
reaction with one reactant)
(12.2-3)
where d 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.
E X A M P L E 12.3
Assume that the following reaction in carbon tetrachloride is elementary and diffusioncontrolled:
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.