Kinetic Theory and Reaction Kinetics - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 258

Y

e E b =RT Y

Reactant Product

C n j
j

C n j
j

e E f =RT r ¼ k f0 k b0 (6.3)

j ¼1 j ¼1

which is the mathematical basis for transitional state theory of chemical kinetics. Equations
(6.2) and (6.3) are highly empirical, in line with nonequilibrium thermodynamics.
In this chapter, we provide this justification of power-law reaction rate relationship in
terms of molecular kinetic theory.

6.1. ELEMENTARY KINETIC THEORY

The simplest way in which to visualize a reaction between two chemical species is in terms
of a collision between the two. Physical proximity is obviously a necessary condition for reac-
tion, for there can be no interaction between two molecules that are well-separated from each
other. In fact, though, collisions are rather difficult to define as discrete events, since the inter-
action between two molecules extends over a distance that depends on their individual
potential energy fields. Fortunately, many useful results can be obtained by using simplified
models; for gases, the two most useful are the ideal gas (point-particle) model and the hard-
sphere model. In the ideal gas model, a molecule is pictured as a point-particle (i.e. dimension-
less) of mass equal to the molecular weight with given position and velocity coordinates. For
the hard-sphere model, the normal analogy is to a billiard ball, a rigid sphere of given diameter
and mass equal to the molecular weight. The potential energy curves for intermolecular inter-
actions according to the two models are shown in Fig. 6. la and for a representative real
system in Fig. 6.1 b. A number of more detailed models have been devised to approximate
the potentials corresponding to the interaction of real molecules; however, we shall be able
to attain our major objectives here with the use of point-particle or hard-sphere models. It
can be seen from Fig. 6.1 that the major deficiency of the models is in ignoring the attractive
forces (energy

0 on the diagrams) which exist in a certain range of intermolecular separa-
tion. However, the point-particle model will form the basis for our first try to produce
a simple theory of reaction. Before doing this, though, let us take a look at the origin of the
distribution laws that are so important in eventual application.

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6.1.1. Distribution Laws
The properties of temperature, pressure, and composition which have been used in
Chapter 3 to define rate laws refer to the averages of these quantities for the system under
consideration. To develop the idea of reaction as a result of intermolecular collision, it is
necessary to look at individual molecular events and then assemble them into the overall,
observable result. In this task, we must be concerned with what average property arises
from a distribution of individual properties. In the case of a gas, for example, the individual
molecules are in constant motion as a result of their kinetic energy and consequently are
constantly colliding with one another. The velocities of individual molecules thus change
continually, and the result is a distribution of velocities about an average value. How can
we convince ourselves more quantitatively of the existence of such a distribution? Picture