Page 154 - Introduction to chemical reaction engineering and kinetics

136 Chapter 6: Fundamentals of Reaction Rates

transition from first-order to second-order kinetics as cM decreases. This is referred to as
the “fall-off regime,” since, although the order increases, kuni decreases as cM decreases
(from equations 6.4-20 and -2Oa).
This mechanism also illustrates the concept of a rate-determining step (rds) to desig-
nate a “slow” step (relatively low value of rate constant; as opposed to a “fast” step),
which then controls the overall rate for the purpose of constructing the rate law.
At low cM, the rate-determining step is the second-order rate of activation by col-
lision, since there is sufficient time between collisions that virtually every activated
molecule reacts; only the rate constant k, appears in the rate law (equation 6.4-22). At
high cM, the rate-determining step is the first-order disruption of A* molecules, since
both activation and deactivation are relatively rapid and at virtual equilibrium. Hence,
we have the additional concept of a rapidly established equilibrium in which an elemen-
tary process and its reverse are assumed to be at equilibrium, enabling the introduction
of an equilibrium constant to replace the ratio of two rate constants.
In equation 6.4-21, although all three rate constants appear, the ratio k,lk-, may be
considered to be a virtual equilibrium constant (but it is not usually represented as
such).
A test of the Lindemann mechanism is normally applied to observed apparent first-
order kinetics for a reaction involving a single reactant, as in A + P. The test may be
used in either a differential or an integral manner, most conveniently by using results
obtained by varying the initial concentration, c Ao (or partial pressure for a gas-phase
reaction). In the differential test, from equations 6.4-20 and -2Oa, we obtain, for an
initial concentration cAO = cM, corresponding to the initial rate rpo,

kl k2cAo
kuni = h + kelCAo
or

(6.4-23)

where k, is the asymptotic value of kuni as CA0 + 00. Thus k,&! should be a linear function
of CA:, from the intercept and slope of which k, and kl can be determined. This is
illustrated in the following example. The integral method is explored in problem 6-4.

For the gas-phase unimolecular isomerization of cyclopropane (A) to propylene (P), values
of the observed first-order rate constant, kuni, at various initial pressures, PO, at 470” C in
a batch reactor are as follows:
P&Pa 14.7 28.2 51.8 101.3
105kU,&-1 9.58 10.4 10.8 11.1
(a) Show that the results are consistent with the Lindemann mechanism.
(b) Calculate the rate constant for the energy transfer (activation) step.
(c) Calculate k,.
(d) Suggest a value of EA for the deactivation step.

SOLUTION

(a) In this example, P, is the initial pressure of cyclopropane (no other species present),
and 1s a measure of c&,. Expressing CA0 in terms of P, by means of the ideal-gas law,