2.7.Rate Constants of the Elementary Steps
(a) Estimation of k−1 and k3
The rate constants k−1 and k3 are associated with the unimolecular reactions of dissociation of X∗ to reform the reactants and of X to form the products, respectively. M. considers a model for X∗ in which the collision complex in solution is made up of the two reactants contained in a solvent cage [26, pp. 156, 403–404]. The reactants are supposed to vibrate in the solvent cage striking the walls about 1013 times a second. The chance of one of them escaping from the cage is α per collision with the walls of the cage, with α < 1. In a first mode of dissociation of X∗, this reaction happens every time one of the reactants escapes from the cage, and so the unimolecular rate constant for this mode of X∗ decomposition is 1013α sec−1.
A second mode of decomposition is “a disorganizing motion of the solvent destroying the polarization appropriate to the intermediate state.” We have seen that the relaxation times for orientation and atomic polarizations are from 10−11 to 10−13 seconds. The atomic polarization is an appreciable fraction of the total U-polarization Pu(r) so that if it reverts to an unsuitable value, this is enough to destroy the X∗ state. The unimolecular rate constant for this mode of decomposition would then be 10−13 sec−1. This is no less than the value for the solvent escape mechanism and therefore it is to be considered a prevalent mode of decomposition. What has been said for X∗ is also valid for X, so that:
(b) Estimation of k1
The equilibrium constant of reaction (2.1) is k1/k−1. M. calculates the equilibrium constant using the recipe given by Statistical Mechanics, using in this case elementary collision theory and partition functions (see, e.g., Ref. [27], Ref. [17, p. 382], and Ref. [7, p. 93]) and, knowing k−1 from the previous section, obtains for k1 the formula:
where Z is the collision number in the gas phase, see Ref. [7, p. 93, 130], approximately the number of collisions occurring between two neutral species in unit volume in unit time at the mean separation distance in the TS [9]. Z is about 1011 liter mole−1 sec−1, which is adequate also for the majority of reactions in solution, see Ref. [7, p. 130]. Marcus will later correct this number, using the more precise value 1012 he obtained in Ref. [28].
∗ Proof of Eq. (2.36)
Each of the two reactants A and B has three translational degrees of freedom. In the intermediate state X∗, they become three translational degrees of freedom of the center of gravity of the two reactants, two rotational degrees of freedom about this center and one degree of freedom of the reactants vibrating with respect to each other in the solvent cage. The partition function of the three translational degrees of freedom of reactant 1, a rigid sphere A say, in the gas phase would be
The corresponding factors for reactant 2 (B), and those for the state X∗, are obtained from the preceding formula replacing m1 by m2 and (m1+m2) respectively. The rotational partition function is [27, p. 32]:
where μ is the reduced mass:
and R is the distance between the centers of gravity of the reactants. The vibrational partition function for motion within the cage is equal to one, within a factor of three.
Introducing the earlier results into a statistical mechanical expression for the equilibrium constant of reaction (2.1) [27, 7, p. 93], one then gets:
It now so happens that k−1 has the value of 1013 sec−1, which is also the value of kT/h. Substituting then k−1 with kT/h one gets, after cancellations,
This expression is Eq. (2.36) with Z the collision number in solution taken as approximately equal to that in the gas phase [7, p. 130]. ΔF∗, the free energy of formation of the intermediate state X∗ from the reactants
is the free energy for the reaction A + B → products.
(c) Estimation of k2 and k−2
The free energy difference between states X∗ and X is −T ΔSe where ΔSe is given by Eq. (2.23).
The equilibrium constant for the interconversion reaction of X∗ and X, Eq. (2.2), is then:
This ratio will in general be approximately or exactly equal to 1.
In order to estimate k2 and k−2, some model is to be assumed for the electronic jump process. Such process was treated as an electronic tunneling process in Ref. [29].(11) The probability κe of an electron tunneling through a barrier from one reactant to the other was estimated to depend exponentially on the tunneling distance rab between A and B in X∗, that is:
For the ferrous-ferric