The problem is now that of determining the free energy of a system whose “solvent configuration” is not in equilibrium with the ionic charges of the TS and so cannot be determined by standard electrostatics, just knowing the ionic charge distribution, the dielectric constant and the overall radius of the TS considered as a sphere whose charge is the sum of the charges of the reactants, as is usually done when the polarization is in equilibrium with the charges [5]. Ref. [2] was devoted to finding an expression for this free energy in terms of equilibrium and nonequilibrium electrostatic macroscopic functions for a two spheres system.
For the description of systems with nonequilibrium electrical polarization, M. uses three vectors [2], the electric field strength E, the polarization P, and the electric field strength Ec which the charge distribution would exert “if it were in a vacuum rather than in a polarized medium” [2]. The electric field strength in the absence of polarized medium is the negative gradient of the potential due to the polarized spheres and so
Eq. (2.11a) can be expressed in the form:
which is valid for any system, equilibrium or not. Only in an equilibrium system can P be expressed in terms of E.
where α is the total polarizability of the medium and considering that E = −∇ψ. The values of Ec, Pe, Pu, P, and E which obtain in the intermediate state X∗ will be designated by an asterisk, while those characteristic of state X will bear no asterisk. Since the U-type polarization is the same in both states,
The electrostatic free energy of any state is generally defined as the reversible work to charge up that state [1, 13, 14]. The electrostatic free energy of the nonequilibrium systems X∗ and X will be derived in the next starred paragraph.
We give here the final results of the derivation:
Where the dot denotes the dot product of two vectors and where αu is the polarizability of the U-type polarization. αu can be expressed [2] in terms of the static dielectric constant Ds and the optical constant Dop. Dop is the square of the refractive index in the visible region of the spectrum [1, 12]:
If the solvent is water, Ds = 78.5 and Dop ~ 1.8 at 25∘C.
“The electrostatic contribution ΔF∗ to the free energy of formation of the intermediate state X∗ from the reactants in the dielectric medium” [1] is found by subtracting from F∗ the reversible work
Similarly for X:
The reversible work required to charge up a conducting sphere of radius a in vacuum with charge e is e2/2a and e2/2aDs in the dielectric medium [15, p. 204 ff.].
In Fig. 1∗, the uncharged system, the charged one with reactants at infinite distance from each other and the charged system X∗ with reactants at distance R, are represented using schematic orientational polarization diagrams.
Fig. 1∗.
2.4.∗The Electrostatic Free Energy for the Nonequilibrium Systems X∗ and X
Derivation of the Formulas
In [2], there are two derivations of Eq. (2.14). For Marcus’ famous expression of the nonequilibrium polarization expression by a reversible two steps charging process see Chapter 3; a second one, more intuitive, is reported here.
We have just seen how much free energy is stored in a charged sphere. Likewise, the free energy stored in an induced dipole p is given by:
where α0 is the electronic polarizability. It is the energy necessary to create the induced dipole in a field E. The free energy of interaction of the dipole with the field is [12, p. 285]:
If we now imagine turning off the field E but of holding fixed the dipole moment, the energy of interaction (2.19) would become zero, but the energy (2.18) stored in the dipole would remain there.
We are interested in a system made up of a distribution of charges in a polarized medium with polarizabilities Pu(r) and Pe(r). The free energy of the whole system is the sum of the free energies of each volume element, considered isolated, and the free energy of interaction among different volume elements. It follows from Eq. (2.18) that the free energy stored in each volume element is