Lecture 32 : Theories Of Reaction Rates : Collision Theory
If the collision is between molecules of type A with molecules of type B, the factor of 1/2 in eq. (32.4) drops out. If two molecules of A collide it is counted as one collision. When molecules of A collide with molecules of B, there are distinct AB collisions, AA and BB collisions. The total number of collisions between A and B molecules in unit time and unit volume is given by
Z AB =
(8 kBT /)1/ 2 L2 [A] [B]
(32.6)
Substitution of the values of all the factors in eq (32.5) and (32.6) gives values of Z AA or Z AB in the range of 10 34 collisions in a volume of 1 m3 in a second.
If we assume that each collision leads to the formation of the products then the rate of the reaction r = - d [A] / dt will equal ZAB/ L. The division by L is to get the correct molar unit of (moles / volume) / s . Since all collisions are not effective but only those collisions with energy greater than some E'a are effective, the rate may be expressed as
r = - d [ A ] / dt = -( Z AB / L ) e - E'a/ RT
(32.7)
E'a is slightly different from the activation energy Ea (see below)
Expanding eq. (32.7) we get,
- d [A] / dt = - (8 kBT / )1/ 2 L [A] [B] e - E 'a/ RT
(32.8)
= - kc [ A] [ B] where
kc = (8 kBT / )1/ 2 L e - E 'a/ RT
(32.9)
Where kc is the rate constant in the collision theory. Notice that unlike the Arrhenius theory, the prefactor depends on
. If we insist on rewriting d [A] /dt = - k [A] [B] where k = A e - Ea/ RT with A independent of T, then the activation energy Ea of the Arrhenius equation is given by Ea = RT2 d ln k / dT. Applying this formula to eq. (32.9) we get
(8 kBT /
)1/ 2 L2 [A] [B] 
. If we insist on rewriting d [A] /dt = - k [A] [B] where k = A e - E
