Kinetic Theory and Reaction Kinetics - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 265

on the discussions we have earlier, we know that in order for the reaction to occur, the
“molecular bundle” must overcome an energy barrier. That is, there is a switch for the reac-
tion to occur, unless it is the spontaneous decaying reaction like that of nuclear reaction.
Therefore, it is expected that not every collision would lead to a successful reaction.
In the new visualization, consider a collision between hard-sphere molecules A and B with
relative velocity v r and b as an impact parameter which can be related to, but is generally not
equal to, d AB . The general view corresponds to Fig. 6.4 . Here v

are velocity compo-
nents in the parallel and perpendicular directions to the line of motion. We can also write an
energy expression in terms of these velocity components

and v t

jj

M 1

E r ¼ 1
2

þ M v jj þ v 2 t Þ ð (6.22)

A B

The second term of this expression is the energy directed along the line of centers and, in the
hard-sphere model, we assume that this is the component of energy involved in the reaction.
Let us call this energy along the line of centers E c . E c is a function of the distance between two
colliding (or interacting) spheres. From Fig. 6.4 we have

v 2 t
v jj þ

E c
E r ¼

v 2 t ¼ 0;

b > d

AB

(6.23)

v 2 t
v jj þ

b 2
d 2 AB

E c
E r ¼

v 2 t ¼ 1

; b d

AB

Now, we will further assume that there is an energy-dependent reaction probability and, in
the most simple approach, that this is just an “on

e off” switch such that the probability of

reaction is zero if E c < E a and is unity if E c > E a , where E a is the minimum energy for reaction.

If we write this out formally,

dð E c Þ¼0; E c < E a

(6.24)

dð E c Þ¼1; E c E a

FIGURE 6.4 The reactive hard-sphere model.