Electric parts, Helmholtz free energy

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... 

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... 

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