Kinetic Theory and Reaction Kinetics - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 264

This probably represents the first stop on the way to a collision theory of reaction rates. If,
indeed, we have the passage of a single molecule through a dilute, fixed matrix of other mole-
cules (sometimes called a “dusty gas”) with a known speed v A , and with every collision
resulting in reaction, then the reaction rate would in fact be given by Eqn (6.17) . Yet, we
know that we do not have a single molecule, we do not have a fixed speed, and we do not
have a dusty gas; so this simple theory needs some cosmetics at this point.
One of the things we need to know is the collision number between molecule A, which is
representative of a Maxwellian distribution of speed of A molecules and B molecules them-
selves possessing a Maxwellian distribution of speed. This can be done by defining a collision
volume determined not by v A but by a mean relative speed, v r between the Maxwellian pop-
ulations of A and B. This approach has been discussed by Benson [S.W. Benson, The Founda-
tions of Chemical Kinetics, McGraw-Hill Book Co., New York, NY (1960)]. The result, for
identifiable molecular species A and B, is simply that

s

8

M

RT
p

A þ M

v r ¼ (6.18)

B

The collision rate we are seeking may now be obtained by direct substitution into Eqn (6.17)
using v r instead of v A . For the total number of collisions between Maxwellian molecules A
and B per unit time, the collision rate, Z cT (A, B), is determined by substitution of v r from
Eqn (6.18) and multiplication by the concentration of A molecules, C A :

s

8

M

RT
p

þ M N AV pd 2 AB Z cT ðA; BÞ¼ C C (6.19)

A B

A B

where N AV is the Avogadro's number. A corresponding substitution for like molecules
(A ¼ B) into Eqn (6.19) gives

s

8 RT

pM

N AV C 2 A pd 2 A

Z cT ðA; AÞ¼ (6.20)

A

We have been able to show the two-molecular collision rate for two different and same
species in a system. It is also useful to find the collision rate between one molecule and
a wall (surface), which has been the starting point of kinetic theory where pressure is pre-
dicted for idea gases. This is of direct interest to reaction systems where one of the reactant
resides on a solid surface (solid phase). One can show that the total number of collisions of A
with a surface per unit time per unit surface area is given by

s

8

C RT

pM

A
4

Z cT ðA;

Þ¼ N (6.21)

surface

AV

A

6.2. COLLISION THEORY OF REACTION RATES

If we assume that every collision is effective in reaction, then the collision rate Z cT (A, B)
from Eqn (6.19) gives us the rate of the reaction A þ B / products directly. However, based