Physical Chemistry Third Edition

12.3 The Temperature Dependence of Rate Constants 537

EXAMPLE12.7

For the reaction of Example 12.6, calculate the fractional change in the exponential factor

and in the preexponential factor in Eq. (12.3-14) ifTis changed from 20◦Cto30◦C.

Solution

The value ofEc/Ris 2. 045 × 104 K. The ratio of the exponential factors is

exp(− 2. 045 × 104 K/ 303 .15 K)

exp(− 2. 045 × 104 K/ 293 .15 K)

 10. 0

The ratio of the preexponential factors is

( 303 .15 K)^1 /^2 (1+ 2. 045 × 104 K/ 303 .15 K)

( 293 .15 K)^1 /^2

(

1 + 2. 045 × 104 K/ 293 .15 K

) 0. 984

The change in the preexponential factor is negligible compared with the change in the expo-

nential factor.

Equation (12.3-14) corresponds to the probability of reaction shown in Figure 12.1a.

A more realistic assumption is that the probability of reaction is given by

probability

{

0ifεr<εc

1 −εc/εr ifεr>εc

(12.3-16)

whereεris the relative kinetic energy of the pair of molecules,

εr

1

2

μv^2

and whereεcis the minimum relative kinetic energy that can lead to reaction. This

probability function is represented in Figure 12.1b. The dependence shown in this

figure is more reasonable than that of Figure 12.1a because a collision with a rel-

ative kinetic energy barely great enough to initiate a reaction should have a lower

probability of producing a reaction than collisions with a much larger relative kinetic

energy.

We do not present the mathematics, but when the probability of Eq. (12.3-16) is

introduced into the integration of Eq. (12.3-11), the result is^6

kNAv〈v 12 〉πd 122 e−Ec/RTNAvπd^212

(

8 kBT

πμ

) 1 / 2

e−Ec/RT (12.3-17)

We will use this equation in preference over Eq. (12.3-14). It does not exactly agree

with the Arrhenius formula since the preexponential factor in this formula depends on

temperature. However, the exponential factor depends much more strongly on temper-

ature than does the preexponential factor. The difference between the two equations is

numerically small over a limited range of temperature, andEccan be approximately

identified withEa, the Arrhenius activation energy.

(^6) K. J. Laidler,op. cit., p. 85ff. (note 3).