Page 150 - Introduction to chemical reaction engineering and kinetics

132 Chapter 6: Fundamentals of Reaction Rates

available, since other configurations around the transition state (at higher energy) can
be reached, and the geometric requirements of the collision are not as precise. There-
fore, the best representation of the “necessary” amount of energy is somewhat higher
than the barrier height. Because the Boltzmann factor decreases rapidly with increasing
energy, this difference is not great. Nevertheless, in the simplified theory, we call this
“necessary” energy E * to distinguish it from the barrier height. The simplest model
for the collision theory of rates assumes that the molecules are hard spheres and that
only the component of kinetic energy between the molecular centers is effective. As
illustrated in Figure 6.10, in a head-on collision (b = 0), all of the translational energy
of approach is available for internal changes, whereas in a grazing collision (b = dAB)
none is. By counting only collisions where the intermolecular component at the moment
of collision exceeds the “necessary” energy E *, we obtain a simple expression from the
tedious, but straightforward, integration over the joint Maxwell velocity distributions
and b (Steinfeld et al., 1989, pp. 248-250). Thus, for the reaction A + B + products, if
there are no steric requirements, the rate of reaction is
r c (-rA) 7 q&-E”‘RT (6.4-11)

that is, the function f(E) in equation 6.4-9 (in molar units) is exp( -E*IRT).
Similarly, for the reaction 2A -+ products,

r = ( -rA)/2 = ZAAe-E*‘RT (6.4-12)

6.4.1.4 Orientation or Steric Factors
The third factor in equation 6.4-9, p, contains any criteria other than energy that the
reactants must satisfy to form products. Consider a hydrogen atom and an ethyl radical
colliding in the fifth step in the mechanism in Section 6.1.2. If the hydrogen atom collides
with the wrong (CH,) end of the ethyl radical, the new C-H bond in ethane cannot be
formed; a fraction of the collisions is thus ineffective. Calculation of the real distribution
of successful collisions is complex, but for simplicity, we use the steric factor approach,
where all orientational effects are represented by p as a constant. This factor can be
estimated if enough is known about the reaction coordinate: in the case above, an esti-
mate of the fraction of directions given by the H-CH,-CH, bond angle which can form
a C-H bond. A reasonable, but uncertain, estimate forp in this case is 0.2. Alternatively,
if the value of the rate constant is known, the value of p, and therefore some informa-
tion about the reaction coordinate, can be estimated by comparing the measured value
to that given by theory. In this case p(derived) = r(observed)/r(theory). Reasonable
values ofp are equal to or less than 1; however, in some cases the observed rate is much
greater than expected (p >> 1); in such cases a chain mechanism is probably involved
(Chapter 7), and the reaction is not an elementary step.

6.4.1.5 SCT Rate Expression
We obtain the SCT rate expression by incorporating the steric factor p in equation
6.4-11 or -12. Thus,

rscrlmolecules mP3 s-l = PZ~-~*‘~~ (6.4-13)

where Z = Z,, for A +B + products, or Z = Z,, for A +A + products. We develop
the latter case in more detail at this point; a similar treatment for A + B + products is
left to problem 6-3.