Lecture 32 : Theories Of Reaction Rates : Collision Theory
Since the cross section depends on relative energy, we need to integrate over the distribution function of energy to get the rate constant, i.e.,
d [ A] / dt = -
( ) rel L [ A] [B] becomes
(32.12)
d [ A] / dt = - {
( ) rel f ( ) d } L [A] [B]
(32.13)
and with
rel = (2 /
)1/2 from = 1/2
2rel
(32.14)
Substituting velocities by energies by using the above substitutions, we have
k c = L ( ) ( 2 / )1/2 f ( ) d
(32.15)
The primary reason for converting from rel to relative kinetic energy is that experimentally cross sections are often tabulated as functions of energy. Let us first convert the Maxwell Boltzmann velocity distribution into an energy distribution.
f ( ) d = 4 ( m / 2 kBT )3/22 exp ( - m 2 / kBT ) d
(32.16)
= 4 ( m / 2 kBT ) 3 / 2 ( 2 / m ) d / (2m ) 1/ 2
(32.17)
Where we have used that d = d/ (2 m )1/ 2 from = m2 /2
( 
) 
rel L [ A] [B] becomes 
rel = (2
)1/2 from 
d
