problem of "slow" reactions was recognized by around 1925, and Eq. (14-70) was modified by the addition of an empirical factor Ρ called the steric factor, where Ρ = A0bslAcaiie :
k = PAe~E*/RT. (14-71)
The factor was justified qualitatively on the grounds that colliding molecules might not be suitably oriented for reaction; Ρ then represents the fraction of energetically suitable collisions for which the orientation is also favorable.
The orientation explanation is perhaps acceptable for Ρ values not much below about 0.1, but as shown in the last column of Table 14-4, there are many instances of much lower values, as low as 10~6. There are in fact enough of such cases to put simple collision theory into serious difficulty. The matter is discussed further in the Commentary and Notes section.
14-7 Unimolecular Reactions
The considerable success of collision theory in the period 1910-1920 seemed to establish that elementary gas-phase reactions were always bimolecular, with the colliding molecules supplying the necessary activation energy. A puzzling difficulty developed, however, with the finding that many decomposition reactions were first-order kinetically, which implied that the reaction was unimolecular. Yet the Arrhenius equation was well obeyed, giving quite respectable activation ener
gies. How, then, could a molecule undergoing unimolecular decomposition acquire the necessary energy to react? An early proposal was that radiation absorbed and emitted between molecules provided the activation energy, but this hypothesis did not stand up to detailed inspection—molecules often would not have an absorption band in the wavelength region for which light quanta would have the requisite energy, for example.
The solution to the problem came through a recognition that there could be a lag between the time a molecule gains energy £ * and its decomposition. The decomposition reaction is essentially one of the breaking of a particular chemical bond, and if the energy E* were distributed among various vibrational degrees of freedom, only after a number of vibrational cycles might it happen to concen
trate on a particular bond vibration. The picture is difficult to present graphically, but suppose that a molecule has three bonds, each with its own potential energy
curve, as illustrated in Fig. 14-8(a), and that as a result of a collision the vibrational quantum states of the molecule are populated as shown. Bond 3 has the smallest dissociation limit, and we take this to be the one that will break first. Although bond 3 does not initially have sufficient vibrational energy for dissociation, the various vibrational modes are coupled to some extent, with the result that various redistributions of energy can occur. Eventually a situation such as shown in Fig. 14-8(b) occurs, in which bond 3 gains sufficient energy to dissociate on the next vibrational swing, and reaction occurs. During this waiting period the mole
cule has a total energy £ * or more, which is much above the average molecular energy, and if a second collision occurs, it will most probably take energy away rather than add to it. Thus if reaction is to occur, it must do so between the time of the collision which has brought the energy up to 2Γ* and that of the next collision, and if the probability of this happening is small, then in effect a Boltzmann
FIG. 14-8. A molecule with three vibrational modes, (a) As populated immediately after a collision, (b) After energy redistribution has occurred, leading to bond breaking along mode 3.
distribution of molecules of energy £ * or more is present, only slightly distorted by the disappearance of some molecules through reaction.
The kinetic scheme corresponding to this picture was proposed by F. Lindemann and J. Christiansen in 1922 and is as follows. The activating and deactivating processes are written
A + ΑΑΪ A + A*, (14-72)
where A* is a molecule which is sufficiently energized to react, and which does so with a certain probability, and hence according to the rate law
A * ^ Ρ (products). (14-73) Equations (14-72) and (14-73) may be treated according to the stationary state
approximation described in Section 14-4C to give
d(P) _ Μ :2( Α ) ' ( u 7 4,
dt fcx + fc-rfA)' K / J
If A:_2(A) > kx, then Eq. (14-74) reduces to
^ = MMA) = W A ) , (14-75) where K2 is the equilibrium constant for process (14-72), and K2(A) is therefore
the equilibrium concentration of A*. This corresponds to the situation mentioned earlier, in which the rate of decomposition of energized molecules is slow enough that their distribution is the equilibrium or Boltzmann one. Equation (14-75) is
14-7 UNIMOLECULAR REACTIONS 571 pseudo first order, and thus accounts for the existence of first-order decomposition reactions.
A crucial test of the Lindemann-Christiansen mechanism lies in its prediction that at sufficiently small (A), Eq. (14-74) becomes
^ = K(Af (14-76)
or second order. The actual test is usually made in the following way. We write Eq. (14-74) in the form
That is, the initial rate of decomposition is reported as though it were a uni-molecular reaction with rate constant kUTii. One then determines A:Uni for various pressures and plots £Uni/A:uni,oo versus pressure, where fcUni,oo is the limiting value at high pressures [given by Eq. (14-75)]. This ratio should decrease toward zero at low pressures, and such behavior has indeed been found in a number of well-substantiated cases. Some data on the isomerization of cyclopropane are shown in Fig. 14-9.
It is not necessary that A collide with a second molecule of its own kind in order to be energized, and Eq. (14-72) may be written in the more general form
A + Μ % A * + M ,
K-2
which leads to
d(P) kMW
dt kx + A:_2(M) (A), (14-78)
where Μ denotes some nonreactive gas present in large excess over A, so that A - A collisions are unimportant. The decomposition will now be first order at all pressures, since (M) in Eq. (14-78) is constant, but fcUni will vary as the pressure
I I I I I I I 1 1 1
- 2 - 1 0 1 2
log Ρ
F I G . 14-9. Variation of fcum/fcunt,« versus pressure for isomerization of cyclopropane. Ρ is in cm Hg and temperature is 490°C. [From H. Pritchard, R. Sowden, and A. Trotman-Dickenson, Proc. Roy. Soc. A217, 563 (1953).]
T A B L E 14-5. Arrhenius Parameters for Some Unimolecular Reactions a
0 Adapted from S. W. Benson, "Foundations of Chemical Kinetics."
McGraw-Hill, N e w York, 1960.
of (M) is altered in successive experiments. Most studies of the Lindemann-Christiansen mechanism are made in this manner nowadays, and collisional efficiencies as given by k2 have been studied for a large number of inert gases.
The Arrhenius parameters for the high-pressure limiting rates of some uni
molecular reactions are given in Table 14-5. In the case of the cyclopropane isomerization, investigators have compared k2 for various inert gases with that for cyclopropane itself to obtain relative efficiencies of activation. Some values are He, 0.060; N2, 0.060; CO, 0.072; H20 , 0.79; and toluene, 1.59. It appears that the more internal degrees of freedom a molecule has, the greater is its ability to transfer energy in a collision, which supports the idea that internal energy as well as translational energy transfers are important.
Example. A n illustration of the treatment presented here is as follows. T h e high-pressure limiting rate for the d e c o m p o s i t i o n of a z o m e t h a n e is f o u n d t o obey the Arrhenius equation with could, for example, assume that kx corresponds to a single vibrational frequency, or to about 1 01 8 s e c "1, in which case K2 = 2.7 x 1 0 "l e and k-2 = 5.6 x 1 01 2 T o r r "1 s e c "1.
An interesting point n o w arises. One might expect K2 to be given simply by the Boltzmann factor for the probability of a molecule of A having energy E*, or cxp(—E*/RT). The observed activa
tion energy of 51 kcal m o l e- 1 then leads to a theoretical value for K2 of 3.0 χ 10~l f l, or much smaller than the estimate based on kx = 1 01 3 s e c- 1. It is not reasonable to take kx as any larger than this estimate, s o it appears that the population of energized molecules is at least 1000 times that expected. The qualitative explanation is that the energized state is more probable than otherwise expected because of the many degrees of freedom a m o n g which E* ce.n be distributed;
that is, its entropy as well as its energy must be taken into account. S o m e further discussion of this aspect is given in the Commentary and N o t e s section.