Kinetic Theory and Reaction Kinetics - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 263

6.1.2. Collision Rate
With the distribution laws established, we can now attack the problem that is central to
a collision theory of reaction: the number of collisions experienced per molecule per second
in the Maxwellian gas. Clearly the magnitude of this collision number is a function of temper-
ature (through the constant a ), and if we define a total collision number, collisions of all mole-
cules per second per volume, it will also depend on molecular density (i.e. concentration).
Thus, the two independent variables of concentration and temperature used in power-law
rate equations will appear in the total collision number.
Consider the simple situation illustrated in Fig. 6.3 . Here a single, hard-sphere molecule of
A is moving through a gas composed of identical, stationary, hard-sphere B molecules. The
speed of A is v A , and in its path through the matrix of B molecules, Awill follow a randomly
directed course determined by collisions with B. Collisions are defined to occur when the
distance between centers is smaller than:

d

A þ

d

B

d

AB ¼

(6.16)

2

If the matrix of B molecules is not too dense, we can approximate the volume in which colli-
sions occur as that of a cylinder of radius d AB , developing a length of v A per second. The
number of collisions per second for the A molecule, Z(A, B), will then be the volume of
this cylinder times the number of B molecules per unit volume, C B :

d 2 AB Z ðA; B Þ¼ C

B p

v (6.17)

A

FIGURE 6.3 Trajectory of single molecule A through a stationary matrix of B.