12.3 The Temperature Dependence of Rate Constants
537
E X A M P L E 12.7
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) if T is changed from 20 ◦ C to 30 ◦ C.
Solution
The value of E c /R is 2.045 × 10 4 K. The ratio of the exponential factors is
exp(−2.045 × 10 4 K/303.15 K)
exp(−2.045 × 10 4 K/293.15 K)
10.0
The ratio of the preexponential factors is
(303.15 K) 1/2 (1 + 2.045 × 10 4 K/303.15 K)
(293.15 K) 1/2
1 + 2.045 × 10 4 K/293.15 K
0.984
The change in the preexponential factor is negligible compared with the change in the exponential 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
0
i fε r < ε c
1 − ε c /ε r if ε r > ε c
(12.3-16)
where ε r is the relative kinetic energy of the pair of molecules,
ε r
1
2
µv
2
and where ε c is 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 relative 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
k N Av v 12 πd
2
12 e
−E c /RT
N Av πd
2
12
8k B T
πµ
1/2
e
−E c /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 temperature than does the preexponential factor. The difference between the two equations is
numerically small over a limited range of temperature, and E c can be approximately
identified with E a , the Arrhenius activation energy.
6 K. J. Laidler, op. cit., p. 85ff. (note 3).
537
E X A M P L E 12.7
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) if T is changed from 20 ◦ C to 30 ◦ C.
Solution
The value of E c /R is 2.045 × 10 4 K. The ratio of the exponential factors is
exp(−2.045 × 10 4 K/303.15 K)
exp(−2.045 × 10 4 K/293.15 K)
10.0
The ratio of the preexponential factors is
(303.15 K) 1/2 (1 + 2.045 × 10 4 K/303.15 K)
(293.15 K) 1/2
1 + 2.045 × 10 4 K/293.15 K
0.984
The change in the preexponential factor is negligible compared with the change in the exponential 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
0
i fε r < ε c
1 − ε c /ε r if ε r > ε c
(12.3-16)
where ε r is the relative kinetic energy of the pair of molecules,
ε r
1
2
µv
2
and where ε c is 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 relative 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
k N Av v 12 πd
2
12 e
−E c /RT
N Av πd
2
12
8k B T
πµ
1/2
e
−E c /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 temperature than does the preexponential factor. The difference between the two equations is
numerically small over a limited range of temperature, and E c can be approximately
identified with E a , the Arrhenius activation energy.
6 K. J. Laidler, op. cit., p. 85ff. (note 3).