Free energy, electric part

The third exponential term in eqn. (187) is identical to the exponential term in the Butler—Volmer equation, eqn. (80), in the absence of specific adsorption. The first two exponential factors in eqn. (187) corresponding to the variation in the electrical part of the free energy of adsorption of R and O with and without specific adsorption A(AGr) and A(AG0), respectively. The explicit form of as, the activity of the adsorption site, and potential dependence of as, A(AGr) and A(AG0) is necessary for a complete description of the electrode kinetics. 

The Helmholtz free energy F consists of an electric part Fel and an osmotic part F°sm. In Appendix A of reference [13], the electric part Fel is represented generally as a functional of the mean electric potential (x) as follows ... 

Substitution of the (x) in Equation 6.17 into this formula yields the electric part of the Helmholtz free energy of the region R as... 

Using the formulae in Equation 6.29 and Equation 6.30, the Helmholtz free energy is obtained for each region under the imposed boundary conditions. For the region R-, the electric part of the Helmholtz free energy is... 

Here kt, kl and kr0 are the moduli, and Ni , N0 l and V0 r are the numbers of small ions in the regions R R, and R). Summing up these electric and nonelectric parts of the Helmholtz free energies of all regions, we obtain the adiabatic potential of the system in the form... 

We can divide this Gibbs free energy G into an electric part Gel and an osmotic part Gosm, namely... 

For the outer region RJ u Rj outside the plates, the electric part and the osmotic part of the Helmholtz free energy are given by Equation 6.34 and Equation 6.35, respectively. Similarly, from Equation 6.45 and Equation 6.46, the electric part and the osmotic part of the Gibbs free energy in the outer region RJ u R, are determined as... 

The stability of inverse micelles has been treated by Eicke (8,9) and by Muller (10) for nonaqueous systems, while Adamson (1) and later Levine (11) calculated the electric field gradient in an inverse micelle for a solution in equilibrium with an aqueous solution. Ruckenstein (5) later gave a more complete treatment of the stability of such systems taking both enthalpic (Van der Waals (VdW) interparticle potential, the first component of the interfacial free energy and the interparticle contribution of the repulsion energy from the compression of the diffuse part of the electric double layer) and entropic contributions into consideration. His calculations also were performed for the equilibrium between two liquid solutions—one aqueous, the other hydrocarbon. 

If the dissociation of ionizable groups on the particle surface can be regarded as complete, then N = and Sc can be dropped so that the surface free energy increase Ft is just equal to the electrical part of the double-layer free energy F (Eq. (5.4)), namely. 

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