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C O M M U N I C A T I O N S TO THE EDITOR
Comment on “Fast Reactions of Polar Molecules in Processes with No Activation Energy”
Sir: I n a recent paper, Walton’ has calculated the rates of several radical recombinations and ion-molecule reactions using a modified absolute rate theory formulation. He proposes a transition-state model in which the reacting species are confined to a square potential well. The energy of the system within this well is quantized and given by E , = n2h2/8m*RZ, where m* is the reduced mass, The corresponding expression in the well-known Gorin2 model is the centrifugal potentjal or kinetic energy of rotation EJ = J ( J l)h2/8?r2m*R2, where J is the angular momentum quantum number. Reasonably enough, Walton proceeds to maximize the total energy of the system, which is the sum of the quantized energy E , and the attractive potential, and finds in the case of radical recombinations
+
E, = .3h3/27/23S/2(m*)’/2X’/‘ where X is the coeficient of R-6 in the long-range attractive potential. Next, using this value of E,, Walton evaluates the partition function
and proceeds in the usual way3 to derive the expression for the rate constant, which differs from the Gorin result by a factor of ?r2.3 It is the evaluation of the partition function, fn, to which our comments are directed. Although Walton does not in any way discuss the degeneracy factor g, his results imply that the value g, = 2n has been used. I n our view, the problem as discussed by Walton involves a model in which a pair of particles undergoing reaction are treated as a single particle of mass equal to the reduced mass of the pair, in a square well potential along the line joining the two centers. A problem of this type may be easily reduced to a solution of the Schroedinger equation with the radial part only. I n fact, the use of the energy expression E , = (nzh2)/ (8m*R2)is rather suggestive of such an approach. It is well known from quantum mechanics* that such a problem may be visualized as a one-dimensional motion bounded on one side, for which it can be shown that the eigenvalues are nondegenerate. Hence it appears to us that the use of 2n as a degeneracy factor in the calculation of the partition function requires some further justification. It appears as well, from standard expression~~ for particles in a spherical cavity, that the
above equation for energy results in a case where 1 = 0, for which the degeneracy is unity. If higher values of 1 are used, then the expression for the energy E , would be different and the results would differ from those obtained by Walton. In the following communication,6 Walton further elaborates on his activated complex model from which we find the following. (a) With g, = 1 the evaluation of the integral for the partition function, fa, for a single particle (radical), A, leads to
which differs from Walton’s result. Furthermore, the total partition function for two particles, A and B, should be the product of the individual partition functions, fa and f b . The latter does not appear to be the case in Walton’s expression for ftotsl. (b) It is also pointed out6 that we may consider the radicals in the transition state as a single entity confined to a square potential well which exists when both species are present and it is shown that analogous expressions to those discussed in (a) result. (1) J. C. Walton, J. Phys. Chem., 71, 2763 (1967). (2) E.Gorin, Acta Physicochim. U.R.S.S., 6, 691 (1938). (3) S.Glasstone, K.J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill Book Co., Inc., New York, N . Y., 1941,
260. (4)L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958,p 109. (5) W. Kauzmann, “Quantum Chemistry,” Academic Press Inc., New York, N. Y.,1957,p 187. (6) J. C. Walton, J. Phys. Chem., 72, 375 (1968). p
DEPARTMENT OR CHEMISTRY E. TSCHUIKOW-ROUX OF CALGARY THEUNIVERSITY R. PAUL CALGARY, ALBERTA,CANADA RECEIVED SEPTEMBER 7, 1967
Reply to Comment on “Fast Reactions of Polar Molecules in Processes with No Activation Energy,” by E. Tschuikow-Roux and R. Paul
The path of a radical combination reaction proceeds from two separate species A . and B - , via the transition state, to the single product molecule AB. The high rate constant and absence of activation energy of these combinations suggests that in the transition Sir:
Volume 78, Number 1 January 1968
