Some Theoretical Aspects of Electron-Transfer Processes in Aqueous Solution

Abstract

A theoretical treatment has been developed for the rates of electron-transfer reactions in aqueous solution, with particular reference to the ferric-ferrous system. T h e reactions are considered to be diffusion-co~ltrolled processes, the approach of the ions being hindered by the electrostatic repulsion between them. Calculations have been ~u a d e of the free energy of the diffusion process and for the repulsion, account being taken of the variation in dielectric constant with the electric field. The form of the potential-energy barrier between the ions is calculated for various separations, and the transmission coefficient calculated using the quantum-mechanical expression corresponding to a rectangular barrier. The total free energy of activation for the reaction, which is the sum cf the contributions due to diffusion, repulsion, and tunnelling, is found to pass through a minimum a t a separation of about 4 A.The calc~~lated free energy of activation for the reaction is 15.4 kcal, in good agreement with the experimental value of 16.8 lccal. The energy and entropy of activation for the reaction arc also briefly discussed. INTRODUCTIONDuring the past few years a considerable number of kinetic studies have been concerned with electron-transfer processes between ions in different valency states. A well-known process of this type is that between ferric and ferrous ions in aqueous solution,Although many complexities exist, the evidence indicates that in some cases these reactioils occur by a simple transfer of an electron from one ion to another. This is possibly true of the ferric-ferrous reaction, and Silverman and Dodson (1) have determined values of 9.9 kcal and -25 e.u. for the energy and entropy of activation of the process.This paper is concerned with the theory of simple electron-transfer reactions, and for the most part deals quantitatively with the factors influencing the rate of the Fe++-Fe++f reaction. Several previous theoretical discussioils of these reactions have been presented, notably by Libby (2), Weiss (3), Eyring and co-workers (4, 5 ) , and by Marcus (Ge). In the case of the latter two treatments, an attempt was made to calculate the free energies of activation (AF;" of a number of reactions, including the Fe++-Fe+++ one. The quantitative agreement with experiment was not, however, completely satisfactory. T h e experimental AF" for the Fe++-Fe+++ system is 16.8 kcal a t 25' C: Eyring et al. obtain agreement only by arbitrarily adding a term of 8.1 kcal, while Marcus (Ga) calculated a value of 9.8 ltcal.* I t has appeared to the present writer that this lack of good agreement could be attributed to failure to take into consideration certain important factors which must influence the rate. In particular it would seem that neither Eyring nor Marcus treated in sufficient detail the collisions (or, more correctly, encounters) between the reacting ions.I n the present work an attempt has been made to bring about some improvement, particularly with respect to the two points noted above. If ...

Solvent Influence on Rates and Mechanisms

“…The respective y values indicate that the transition state is about 2 kcal/mole more solvated in the dipolar aprotic solvents D, DMAC, and N than in the protic solvent E. For the back reaction which is one between two polar molecules with the same transition state as the forward reaction, the product of the activity coefficients D y E CH3Br · D y E CH = 36 which is in agreement with the finding that the back reaction has about the same rate in ethanol as in dimethylformamide [99]. The S N 2 mechanism for the forward and back reactions were written [99]:…”

“…A second approach used in accounting for the microscopic dielectric constant in the vicinity of an ion has been to treat the dielectric constant as a function of the distance from an ion quantatively in terms of the variation of the dielectric constant with field strength [53,99]. These calculations were made for simple ions ranging in valence from 1 to 4.…”

Electrical and electronic conductivity of CaO-SiO2-FeOx slags at various oxygen potentials: Part II. Mechanism and a model of electronic conduction

“…The optimized value of r* coincides very well with the typical optimum distance for charge hopping in other systems. For example, Laidler [34] reported an optimum distance of 4 Å for charge transition between Fe 2ϩ and Fe 3ϩ in aqueous solutions. The value of r* Ӎ 4 Å was also found by Nasu and Soga [35] to be the maximum distance at which charge hopping between ferric and ferrous ions can take place in Fe 2 O 3 -Na 2 O-P 2 O 5 glasses.…”

“…Therefore, the net probability of an electron transfer from Fe 2ϩ to Fe 3ϩ is given by Boltzman statistics as [31] or in the case EF RT, [32] c. The diffusion-assisted charge transfer model Transfer of n charges per unit volume in unit time creates a current density of [33] Combining Eqs. [25], [29], and [32] with e ϭ i e /E, and presents the expression for electronic conductivity via the diffusion-assisted charge transfer (DACT) mechanism: [34] where [Fe] in this equation is the total concentration of iron ions in (mol и cm Ϫ3 ).…”

“…From Eq. [34], electronic conductivity follows a relationship with temperature, similar to that of ionic conductivity, i.e., [35] where E D is the activation energy for diffusion of involved species in the electronic conduction process, i.e., ferrous ions. From Eq.…”