Physical Chemistry Third Edition

12.3 The Temperature Dependence of Rate Constants 535

The Collision Theory of Bimolecular Elementary

Gas-Phase Reactions

The principal assumption of the theory is that the initiation of the reaction requires

an inelastic collision in which at least a certain amount of energy is transferred from

translational motion to internal motions (rotations, vibrations, or electronic motion)

that can lead to reaction. We first assume that the probability of reaction equals zero if

the relative speed of two colliding particles is smaller than a certain critical value and

equals unity for relative speeds larger than this value. This probability is represented

in Figure 12.1a.

Consider a gas-phase bimolecular elementary reaction between molecules of sub-

stance 1 and substance 2. Equation (2.8-31) gives the total rate of collisions per unit

volume between molecules of type 1 and type 2:

Z 12 

√

8 kBT

πμ 12

πd 122 N 1 N 2 〈v 12 〉πd 122 N 1 N 2 (12.3-5)

where〈v 12 〉is the mean relative speed,N 1 is the number density of molecules of type 1,

N 2 is the number density of molecules of type 2, andd 12 is the collision diameter of

the molecule pair. We now want to include only molecules of type 1 whose velocities

lie in the infinitesimal ranged^3 v 1 and molecules of type 2 whose velocities lie in the

infinitesimal ranged^3 v 2. We replace the mean relative speed by our particular relative

speed,v, the magnitude of the difference between the two velocities:

v|v||v 2 −v 1 | (12.3-6)

We multiply by the probability that the first molecule has its velocityv 1 in the infini-

tesimal ranged^3 v 1 and that the second molecule has its velocityv 2 in the infinitesimal

ranged^3 v 2. We call this collision ratedZ 12 :

dZ 12 |v|πd^212 N 1 N 2 g(v 1 )d^3 v 1 g(v 2 )d^3 v 2 (12.3-7)

wheregis the probability distribution of Eq. (2.3-40).

In order to obtain the total rate of collisions that leads to reaction, we integrate over

all velocities that satisfy the condition that the relative speed exceedsvc, thecritical

1

vc

Probability Probability

Relative speed

(a)(b)

(^) c
Relative energy
Figure 12.1 Probability of Reaction as a Function of Relative Speed.(a) First assump-
tion. (b) Assumed probability of reaction as a function of relative kinetic energy according
to Eq. (12.3-16).