6.2.1 ‘Ijpes of Elementary Reactions
6.4.2 Collision Theory of Unimolecular Reactions
For a unimolecular reaction, such as I, -+ 21’, there are apparently no collisions nec- essary, but the overwhelming majority of molecules do not have the energy required for this dissociation. For those that have enough energy (> 149 kJ mol-l), the reaction occurs in the time for energy to become concentrated into motion along the reaction coordinate, and for the rearrangement to occur (about the time of a molecular vibra- tion, lOPi3 s). The internal energy can be distributed among all the internal modes, and so the time required for the energy to become concentrated in the critical reaction co- ordinate is greater for complex molecules than for smaller ones. Those that do not have
6.4 Simple Collision Theory of Reaction Rates 135 enough energy must wait until sufficient energy is transferred by collision, as in Section 6.4.1.6. Therefore, as Lindemann (1922) recognized, three separate basic processes are involved in this reaction:
(1) Collisions which transfer the critical amount of energy:
I, + M (any molecule in the mixture) 3 I;(energized molecule) + M (4 (2) The removal of this energy (deactivation) by subsequent collisions (reverse of
(A)):
I;+MbI,+M (B)
(3) The dissociation reaction:
I;. -Z 21’ (0
Steps (A), W, and CC> constitute a reaction mechanism from which a rate law may be deduced for the overall reaction. Thus, if, in a generic sense, we replace I, by the reactant A, I; by A*, and 21’ by the product P, the rate of formation of A* is
?-A. = -k2cA* + klcAcM - kelcA*cM and the rate of reaction to form product P, r,, is:
(6.4-18)
rP = k2cA. =
kZ(hcAcM - IA”)
k2 + k-,c,
(6.4-19)
if we use equation 6.4-18 to eliminate cA.. Equation 6.4-19 contains the unknown rA*.
To eliminate this we use the stationary-state hypothesis (SSH): an approximation used to simplify the derivation of a rate law from a reaction mechanism by eliminating the concentration of a reactive intermediate (RI) on the assumption that its rate of forma- tion and rate of disappearance are equal (i.e., net rate r,, = 0).
By considering A* as a reactive intermediate, we set rA* = 0 in equations 6.4-18 and -19, and the latter may be rewritten as
(6.4-20)
(6*4-20a3 where kuni is an effective first-order rate constant that depends on CM. There are two limiting cases of equation 6.4-20, corresponding to relatively high CM (“high pressure”
for a gas-phase-reaction), k-,cM >> b, and low CM (“low pressure”), k2 >> k-,c,:
rp = (klk21k-l)cA (“high-pressure” limit)
(6.4-21)
T-, = klCMC/, (“low-pressure” limit)
(6.4-22)
Thus, according to this (Lindemann) mechanism, a unimolecular reaction is first-order at relatively high concentration (cM) and second-order at low concentration. There is a
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,
6.4 Simple Collision Theory of Reaction Rates 137
8.5
8.Oi 0 0.01 0.02 0.03 0.04 0.05 0.06 0
P,-'/kPa-'
Figure 6.11 Test of Lindemann mechanism in Example 6-3
equation 4.2.3a, we rewrite equation 6.4-23 as:
-=1 $+L!Z
kuni m kl po
(6.4-23a) The linear relation is shown in Figure 6.11.
(b)
From the slope of the fitted linear form, k, = 0.253 L mol-’ s-l.(c) Similarly, from the intercept, km = 11.4 X lop5 s-l