In a redox process, the oxidizing and reducing centers can react with or without a change in their coordination spheres. In some reactions, the electron transfer can only be accomplished by the transfer of a ligand from reducing agent to the oxidizing agent.

There are two stoichiometric mechanisms: the inner sphere mechanism involves a ligand transfer, and a transient shared ligand, while the outer sphere mechanism includes the simple electron transfers, without the presence of a shared ligand.

Inner-sphere Mechanism

The reduction of the non-labile Co complex by the aqueous Cr complex produces a reduced Co complex and an oxidized CrCl complex. The chloride ligand has been transferred between the metal centers, as proven by the fact that addition of ³⁶Cl^- to the solution results in no incorporation of ³⁶Cl^- into the Cr complex.

The reaction is faster than reactions which remove Cl^- from Co^III or introduce Cl^- to Cr³⁺_(aq), and hence the Cl^- ion must have moved directly from the coordination sphere of one complex to the other during the reaction.

The intermediate has a bridging Cl ligand.

The Cl^- ion is a good bridging ligand as it has more than one pair of electrons, and so can form bonds to each of the metal centers simultaneously. Other good bridging ligands include SCN^-, N₂, N₃^-, and CN^-.

The Outer Sphere Mechanism

When both the species in the redox reaction have non-labile coordination spheres, no ligand substitution can take place on the very short time scale of the redox reaction. The electron transfer must proceed by a mechanism involving transfer between the two complex ions in outer-sphere contact.

If the redox reaction is faster than the ligand substitution, then the reaction has an outer-sphere mechanism.

When the reaction involves ligand transfer from an initially non-labile reactant to a non-labile product, there is no difficulty in assigning the inner-sphere mechanism.

When the products and reactants are labile, it is difficult to make an unambiguous assignment of either an inner- or an outer-sphere mechanism.

The Marcus Equation

When an electron transfer occurs in a redox equation, it does so after the atoms have adopted the geometries such that the electron transfer becomes facile, when the electron has the same energy in each of the two nuclear geometries.

According to the Franck-Condon principle, the motion of the electrons is much faster than the motion of the atoms, such that the nuclei can be considered as stationary during the electronic rearrangement. Therefore, the rate of electron transfer, and the activation energy for the process, is determined by the ability of the involved nuclei to adopt arrangements which are equal in energy.

This can be investigated by considering the self exchange reaction of hydrated iron.

This is a redox process where an electron is transferred from the unlabeled iron atom (Fe) to the labeled iron atom (Fe^*). The Fe-O bond lengths are longer for Fe²⁺ than Fe³⁺ due to the decreased electrostatic interaction, and so for the transfer to occur the two complexes must adopt configurations in which the Fe-O bond lengths are the same. This costs energy, and this is known as the inner-sphere rearrangement energy, ΔG^*_IS.

When the complexes rearrange, there is a resultant reorganization of the solvation shell around the complex. This also costs energy, known as the outer-sphere reorganization energy, ΔG^*_OS.

We must also consider the change in electrostatic interaction energy, ΔG^*_ES, between the two complexes at the point of electron transfer, and hence the total activation energy, ΔG^*, can be defined

The rearrangement of the Fe-O systems to give the same geometry in the two complexes can be examined in terms of their potential energy surfaces. If the Fe-O stretching vibrations are assumed to be harmonic, then the potential energy curves are as shown. The activated complex is at the position of intersection of the two curves.

Potential energy diagram for the self transfer in Iron
The initial reaction coordinates on the left are those for FeII and Fe*III, and the product reaction coordinates on the right are those for FeIII and Fe*II. The activated complex, where electron transfer occurs, is in the middle.

However, the non-crossing rule says that molecular energy curves of two states of the same symmetry do not cross, but instead split into an upper and a lower curve. This means that the reactants in their ground states slowly distort to reach the activated complex point, and then slowly distort, following their minimum energy pathways, into the reactants in their ground states after the electron transfer.

This mechanism can be extended to cases where the transfer occurs between non-identical species, when the energies of the reactants and products are not the same. When the energy of the products is higher than that of the reactants, ie. the reaction is endothermic, the activation energy increases as the energy of the curve crossing point is higher than in the symmetrical case. Similarly, if the energy of the products is lower than that of the reactants, ie. the reaction is exothermic, the activation energy decreases as the energy of the curve crossing point is lower.

The rate of transfer therefore depends on two factors: the shape of the potential energy curves and the Gibbs reaction energy.

The shape of the potential energy curves: If they rise steeply, their crossing points will be high and so there will be a large activation energy. This is the case where there are stiff bonds, ie. a large increase in energy with increasing bond length. If the curves are shallow, then the activation energy is low.

The standard reaction free energy: The more negative the reaction free energy, then the lower the activation energy.

The rate constant, k, for an outer-sphere reaction can be expressed, in the Marcus equation, in terms of the exchange rate constants for each of the redox couples, k_i, and the equilibrium constant for the overall reaction, K. These reflect, respectively, the shape of the potential energy surface and the standard reaction free energy.

The Marcus Equation:

f is a complex parameter which depends on the rate constants and the complex encounter rate (which depends on their diffusion), but can be taken as one for approximate calculations. The overall reaction rate, k, is therefore expressed in term of a weighted average of the two self-exchange rates, and so the Marcus equation is also known as the Marcus cross relation.