A mechanism for electron transfer reactions is described, in which there is very little spatial overlap of the electronic orbitals of the two reacting molecules in the activated complex. Assuming such a mechanism, a quantitative theory of the rates of oxidation‐reduction reactions involving electron transfer in solution is presented. The assumption of ``slight‐overlap'' is shown to lead to a reaction path which involves an intermediate state X* in which the electrical polarization of the solvent does not have the usual value appropriate for the given ionic charges (i.e., it does not have an equilibrium value). Using an equation developed elsewhere for the electrostatic free energy of nonequilibrium states, the free energy of all possible intermediate states is calculated. The characteristics of the most probable state are then determined with the aid of the calculus of variations by minimizing its free energy subject to certain restraints. A simple expression for the electrostatic contribution to the free energy of formation of the intermediate state from the reactants, ΔF*, is thereby obtained in terms of known quantities, such as ionic radii, charges, and the standard free energy of reaction.

This intermediate state X* can either disappear to reform the reactants, or by an electronic jump mechanism to form a state X in which the ions are characteristic of the products. When the latter process is more probable than the former, the over‐all reaction rate is shown to be simply the rate of formation of the intermediate state, namely the collision number in solution multiplied by exp(—ΔF*/kT). Evidence in favor of this is cited. In a detailed quantitative comparison, given elsewhere, with the kinetic data, no arbitrary parameters are needed to obtain reasonable agreement of calculated and experimental results.

Topics
Electrostatics, Free energy, Calculus of variations, Transfer reactions, Redox reactions, Chemical reaction dynamics
1.
See review articles:
Zwolinski
,
Marcus
 (Rudolph J.), and
Eyring
,
Chem. Revs.
55
,
157
(
1955
);
Google Scholar
Crossref
Search ADS
 
C. B.
Amphlett
,
Quart. Revs.
8
,
219
(
1954
);
Google Scholar
Crossref
Search ADS
 
O. E. Myers and R. J. Prestwood, Radioactivity Applied to Chemistry, edited by Wahl and Bonner (John Wiley and Sons, Inc., New York, 1951), Chap. 1;
Betts
,
Collinson
,
Dainton
, and
Ivin
,
Ann. Repts. on Progr. Chem. (Chem. Soc. London)
49
,
42
(
1952
);
Google Scholar
 
R. R.
Edwards
,
Ann. Revs. Nuclear Sci.
1
,
301
(
1952
);
Google Scholar
Crossref
Search ADS
 
M.
Haissinsky
,
J. Chim. Phys.
47
,
957
(
1950
);
Google Scholar
Crossref
Search ADS
 
and recent reviews in Ann. Rev. Phys. Chem.
2.
W. F.
Libby
,
J. Phys. Chem.
56
,
863
(
1952
).
Google Scholar
Crossref
Search ADS
 
3.
Marcus
 (Rudolph J.),
Zwolinski
, and
Eyring
,
J. Phys. Chem.
58
,
432
(
1954
). These authors summarize some of these data in their Table I. In Table II, reactions are given having apparent positive entropies of activation. However, in at least all but one of the reactions in Table II the mechanism is complex and the concentrations of the actual reactants are unknown. Accordingly, the so‐called entropies of activation of such reactions have no immediate theoretical significance. The lone possible exception, incidentally, does not involve reacting ions of like sign.
Google Scholar
Crossref
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4.
J.
Weiss
,
Proc. Roy. Soc. (London)
A222
,
128
(
1954
), has also discussed the electronic jump process. Unlike reference 3 the necessity for the reorganization of the solvent occurring prior to the electronic transition was not considered there.
Google Scholar
 
5.
The mechanism used there was incomplete in that only one fate of the intermediate state in the reaction was considered. It was tacitly assumed that this state involving the reorganized solvent could only produce products, but not reform the reactants. (The former would occur by an electron jump process, the latter by a disorganizing motion of the solvent.) It is shown later that this omission can significantly affect the role played by the electronic jump process. The number of times per second that the electron in one of the reactants struck the barrier was not included in the over‐all calculation. Effectively, this made electron tunnelling appear about one thousand‐fold less frequent than would otherwise have been estimated.
6.
The Schrödinger equation can be written as Hφ = Eφ; E is the energy of an atomic configuration. The Hamiltonian operator H includes terms expressing the interaction of the electrons and nuclei of the reacting particles with each other and with the solvent molecules. In the case of no overlap, φx and φx* were shown to be solutions to this wave equation. Let their corresponding energies be Ex and Ex*, respectively, so that we have: Hφx = Exφx and Hφx* = Ex*φx*. If c is any constant, a linear combination of φx and φx* is (φx+cφx*). When introduced into the wave equation this yields: H(φx+cφx*) = Exφx+Ex*cφx*. Only when Ex equals Ex* is the right‐hand side equal to Ex(φx+cφx*). That is, only under these conditions does (φx+cφx*) satisfy the equation Hφ = Eφ. It is also seen that for such a linear combination, the total energy E equals Ex and therefore Ex*.
(a) L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1935).
7.
Glasstone, Laidler, and Eyring, The Theory of Rate Processes (McGraw‐Hill Book Company, Inc., New York, 1941).
8.
R. A.
Marcus
,
J. Chem. Phys.
24
,
979
(
1956
).
Google Scholar
Crossref
Search ADS
 
9.
Numerous theoretical treatments of the free energy of solvation which have assumed this model include: (a)
J. D.
Bernal
 and
R. H.
Fowler
,
J. Chem. Phys.
1
,
515
(
1933
);
Google Scholar
Crossref
Search ADS
 
(b)
D. D.
Eley
 and
M. G.
Evans
,
Trans. Faraday Soc.
34
,
1093
(
1938
);
Google Scholar
Crossref
Search ADS
 
(c)
E. J. W.
Verwey
,
Rec. Trav. Chim.
61
,
127
(
1942
);
Google Scholar
Crossref
Search ADS
 
(d)
R. W.
Attree
,
Dissertation Abstr.
13
,
481
(
1953
).
Google Scholar
 
10.
This is especially true when the valence of the ion before and after the reaction differs by only one unit. This will be shown to be the case of greatest interest, in later applications of this paper.
11.
See
R.
Platzman
 and
J.
Franck
,
Z. Physik
138
,
411
(
1954
).
Google Scholar
Crossref
Search ADS
 
12.
E.g., if during some point of the charging process the ion has a charge q, then the potential at any point in the dielectric medium distant r from the center of the ion is q/Dsr. The potential at the surface of the sphere is .q/Dsa. The work required to add an infinitesimal charge dq to the ion is therefore (q/Dsa)dq. Upon integrating this from q = 0 to q = e, the total work required to charge up the ion is seen to be e2/2aDs. In passing it is observed that when one subtracts from this the work, e2/2a, required to charge up the sphere in a vacuum (Ds = 1), one obtains the usual expression for the contribution to free energy of solvation of an ion, −(e2/2a)(1−1/Ds), arising from the dielectric outside of the sphere. See reference 9.
13.
E.g., A. A. Frost and R. G. Pearson, Kinetics and Mechanism (John Wiley and Sons, Inc., New York, 1953), Chap. 7.
14.
C. E. Moore, Atomic Energy Levels (National Bureau of Standards, 1952), circular 467, Vol. II.
15.
Hasted
,
Ritson
, and
Collie
,
J. Chem. Phys.
16
,
1
(
1948
).
Google Scholar
Crossref
Search ADS
 
16.
See
D. H.
Everett
 and
C. A.
Coulson
,
Trans. Faraday Soc.
36
,
633
(
1940
).
Google Scholar
 
17.
Equation (48) of the present paper may be obtained from Eq. (53) of reference 8 by observing that in that equation, (a) E = −∇ψ, (b) D = 1+4πα, (c) Ec is the value of E when D = 1 (vacuum).
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1956
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