The collision theory for gas-phase reactions presents a very direct, indis
putable physical picture of how molecules react—they must collide and must do so with the proper orientation and with a certain minimum energy E*. One difficulty is in treating the so-called steric factor P. Not only does the simple
COMMENTARY AND NOTES, SECTION 2 581 geometric interpretation become implausible in the case of very small Ρ values such as 10"6, but this interpretation provides virtually no way to make calculational estimates or correlations.
A major further development of collision theory took place as a result of some observations of the rates of unimolecular reactions. According to the Lindemann mechanism, the maximum possible rate for a unimolecular reaction would be observed if each energized molecule decomposed on the next vibrational swing, which means that
kr
r*-> p-E*IRT
h
Thus the maximum possible frequency factor for a unimolecular reaction should be about 101 3 s e c- 1. However, the observed Arrhenius A parameter is over 101 5 for cyclopropane isomerization (see Table 14-5) and is 9.23 χ 101 5 for the decom
position of azomethane (from the numerical illustration in Section 14-7). It appeared at first that something was seriously wrong.
The explanation that was developed by O. Rice, H. Ramsperger, and L. Kassel around 1930 is the following. G. Lewis had pointed out several years previously that it was not necessarily true that E* came only from the kinetic energy of relative motion of the colliding molecules, since it would also be supplied by their internal energy. The resulting modification of Eq. (14-70) gives
(F*/RTMf β)-1
where / is the number of degrees of freedom available to supply E*. The equation reduces to the usual one if / = 2, but has the effect of increasing the apparent A value for larger values of / . It should be mentioned that / is actually the number of so-called square terms, that is, terms contributing to the equipartition energy, so that each vibrational degree of freedom contributes 2 toward / . Thus the cases of very high preexponential or A factors could easily be explained—in the cyclo
propane case it would only be necessary to say that about one-third of the molecu
lar vibrations contribute to £ * in addition to the impact kinetic energy.
There remained some discrepancies between the Lindemann mechanism and the experimentally observed variation of &Uni/&uni,ao with pressure, and one of the contributions of Rice et al was to further suppose that kx of Eq. (14-73) depended on the actual energy possessed by A*. That is, the greater it was in excess of the minimum amount E*, the greater the probability of the necessary energy localizing on the particular bond that was to break before a deactivating collision occurred.
Thus if energy Ε was distributed among s vibrational degrees of freedom, the con
clusion was that for that particular molecule
An elaborate integration procedure is necessary to sum this expression over all possible Ε values, that is, from E* to infinity, but the results have given reasonably good detailed agreement with experiment.
There appear to remain no fundamental difficulties with collision theory—
only the problem that it is rather difficult to estimate steric factors, that is, very low Arrhenius A values.
Transition state theory draws its justification from the type of absolute rate calculations that can be made on simple reactions such as that of H' + H2 - » H ' H + H. In actual practice one is dealing with reactions for which the exact structure of the activated complex can only be guessed. There is generally sufficient flexibility to account for any observed A§ot, and it is therefore difficult to say that the theory has really been verified. On the other hand, its formalism is very convenient and allows for a pleasing structural approach to the detailed reaction path.
One or two ambiguities might be mentioned. If we have a simple reaction A + Β = C + D,
then for the forward rate
A + Β ^ [AB]* i C + D, Rt = j £ ([AB]*).
Similarly, for the reverse rate,
C + D *= [CD]* % A + B,
R* = ^Y
([CD]i)*
Yet, according to the principle of microscopic reversibility (Section 7-CN-l), the forward and back reactions should follow the same detailed path, which seems to imply that [AB]* is the same as [CD]*, yet if this is so, Rt should always equal Rh , which cannot be true. It has been argued that [AB]* and [CD]* are perhaps the same structurally but differ in their directions of motion along the reaction path. The analogy can be made to two billiard balls in collision. If ball A has been propelled to a head-on collision with stationary ball B, the two balls at the moment of collision would look the same as if ball Β had been propelled against stationary ball A. Yet in the first case A remains stationary after the collision, whereas in the second case Β remains stationary, as illustrated in Fig. 14-13.
Θ Θ φ Θ 0
-F I G . 14-13.
A second question is that of whether an activated complex can really be in thermodynamic equilibrium with its surroundings. Collision theory certainly implies that the energized molecules which react do not necessarily have an equilibrium distribution of energy among the internal degrees of freedom, although they can proceed to react. It has been suggested that a species [AB] of energy just below £ * cannot react, so that its population will be given by equilibrium thermodynamics, and that the perturbation from equilibrium when [AB] gains a further small increment of energy to become [AB] is negligibly small. To decide
COMMENTARY AND NOTES, SECTION 3 583 this question we must find some independent means of measuring the free energy of [AB]*, and this does not seem possible to do. One other ambiguity is mentioned in the Special Topics section.
On the whole, it appears that because of the desirable features of each, both theories will remain in active use. Very likely, as each is developed further the differences between them will diminish. Thus the introduction of Eq. (14-89) adds to collision theory some of the flavor of the statistical mechanical calculation of AS*ih , while for the reaction Η ' + H2 Η Ή + Η absolute rate theory is not really very different from collision theory.