Physical Chemistry Third Edition

530 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

Solution

k 4 π(6. 02 × 1023 mol−^1 )(4. 2 × 10 −^9 m^2 s−^1 )(4. 0 × 10 −^10 m)

 1. 3 × 107 m^3 mol−^1 s−^1  1. 3 × 1010 L mol−^1 s−^1

According to a theory of Debye,^4 the formula for the rate constant shown in

Eq. (12.2-2) must be multiplied by theelectrostatic factorfif the particles are ions:

f

y

ey− 1

(12.2-4)

where

y

z 2 z 3 e^2

4 πεd 23 kBT

(12.2-5)

The valences (number of proton charges) of the two ions are represented byz 2 andz 3.

The symbolestands for the charge on a proton andεstands for the permittivity of the

solvent, which was introduced in Eq. (6.4-19).

The electrostatic factorfis greater than unity ifz 2 andz 3 have opposite signs,

corresponding to an enhancement of the rate due to electrostatic attraction of the ion

pair. Ifz 2 andz 3 have the same sign, the electrostatic factor is smaller than unity,

corresponding to a decrease in the rate due to electrostatic repulsion.

EXAMPLE12.4

The diffusion coefficient of H+in water at 25◦C is equal to 9. 31 × 10 −^9 m^2 s−^1 , and that

of OH−is equal to 5. 26 × 10 −^9 m^2 s−^1. Estimate the reaction diameterd. The value of the

dielectric constant for water is equal to 78.4 at 25◦C and the permittivity of the vacuum is

equal to 8. 85 × 10 −^12 C^2 N−^1 m−^2. The dielectric constant is the ratio of the permittivity

of the medium to that of a vacuum. At this temperaturek 1. 4 × 1011 L mol−^1 s−^1.

Solution

From Eq. (12.2-5), we first obtain the relation betweenyandd:

y

(− 1 )(1. 6022 × 10 −^19 C)^2

4 π( 78. 4 )

(

8. 85 × 10 −^12 C^2 N−^1 m−^2

)(

1. 38 × 10 −^23 JK−^1

)

(298 K)d

−

7. 15 × 10 −^10 m

d

Since the reaction diameterdoccurs both in the expression forkin Eq. (12.2-2) and in the

expression foryin Eq. (12.2-5), we solve by successive approximation. We guess a value ford

and calculate the value ofkfor that value, then guess another value ofdand calculatekagain,

and so forth, until an adequate approximation forkis obtained. By this procedure the correct

value ofkis obtained with a value ofdequal to 8. 64 × 10 −^10 m 0 .864 nm 8. 64 A ̊.

(^4) P. Debye,Trans. Electrochem. Soc., 82 , 265 (1942).