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6.4 Simple Collision Theory of Reaction Rates 129

The notion of a collision implies at least two collision partners, but collision-based the-
ories are applicable for theories of unimolecular reactions as well.

6.4.1 Simple Collision Theory (SCT) of Bimolecular Gas-Phase Reactions

6.4.1.1 Frequency of Binary Molecular Collisions
In this section, we consider the total rate of molecular collisions without considering
whether they result in reaction. This treatment introduces many of the concepts used
in collision-based theories; the criteria for success are included in succeeding sections.
Consider a volume containing CL molecules of A (mass m,J and cn molecules of B
(mass mn) per unit volume. A simple estimate of the frequency of A-B collisions can
be obtained by assuming that the molecules are hard spheres with a finite size, and
that, like billiard balls, a collision occurs if the center of the B molecule is within the
“collision diameter” d,, of the center of A. This distance is the arithmetic mean of the
two molecular diameters dA and dB:

d AB = @A + dB)/2 (6.4-3) 1

and is shown in Figure 6.9(a). The area of the circle of radius dAB, u = rdi,, is the
collision target area (known as the collision “cross-section”). If the A molecules move at
average velocity ii (equation 6.3-11) and the B molecules are assumed to be stationary,
then each A sweeps out a volume c+ii per unit time (Figure 6.9(b)) such that every B
molecule inside is hit. The frequency of A-B collisions for each A molecule is then a&~;.
By multiplying by the concentration of A, we obtain the frequency of A-B collisions per
unit volume:

Z AB = ~iid& (6.4-4)

This simple calculation gives a result close to that obtained by integrating over the three-
dimensional Maxwell velocity distributions for both A and B. In this case, the same
expression is obtained with the characteristic velocity of approach between A and B
given by

ii = (8k,Tl,rrp)1’2 (6.4-5)

C--
_--- a
/-
_--- _/-- -\\/I
__-- _---
*--- _--- - - - - ,y
\ _--- _---
&.--- _/--
il /--
8
\A-----
Figure 6.9 (a) Collision diameter d*B; (b) simplified basis for calculating fre-
quency of A-B collisions