Molecular Imaging. Markus Rudin

that:

      where r_im is the distance from the field point to the center of the electrical image of this ion” [1, p. 218].

       4.5.Mixing and Electrostatic Entropy and Free Energy

I report here the Note added in proof in p. 983 of Ref. [10]: “The charge distribution at the end of Stage I of the charging process, denoted by ρ⁰ and σ⁰, is a fictitious distribution used to produce the specified U-type polarization P_u(r). By contrast, the charge distribution at the end of Stage II is the actual distribution of charges on the ions of the system. The complete charging process is performed at fixed configurations of these ions, though they may be uncharged in Stage I and charged up in Stage II. Thus the reversible work given by Eq. (4.6)

      was also performed at fixed configuration of these ions. There is therefore an additional free energy term which should be considered, namely the entropy term associated with the preliminary formation of this given uncharged configuration of the ions from a random configuration. Let e_i, c_i(r), and denote the charge, the concentration, and the average concentration of ions of the ith species which make up the continuous volume charge distribution ρ(r); Then, this additional entropy term associated with the formation of the given configuration from a random one is the well-known excess entropy of mixing and is given by Eq. (4.6a):

      (typo in the original). The total electrostatic contribution to the free energy includes this term and F given by Eq. (4.6). It is the sum of these two terms, rather than F alone, which is the electrostatic free energy, F_e, say

      where F is given by Eq. (4.6). In a very dilute solution c_i equals and F_e equals F.

      Similarly, the total electrostatic contribution to the entropy is the sum of the term in Eq. (4.6a) and of S given by Eq. (4.7) in the following section

      Altogether the total electrostatic entropy will be:

      F is computed by the two stages charging process. The work done during each stage can be calculated from the equation:

      where λ is a charging parameter which is increased from 0 to 1 during each charging stage and where φ^λ denotes the value of φ at any λ (Notice that dρdV is a charge and that φdρdV is then a product potential × charge, that is, electrical work. The same for φdσdS, compare Eq. (4.9) in Ref. [10]). Introducing Eqs. (4.2) and (4.3) in this expression for W, that is, going from treating ET in solution to the parallel treatment of ET at the electrode, Eq. (4.9) becomes:

      Eq. (4.10) differs from the corresponding expression used for the redox case [10] only in the last term. For this reason, the formula deduced for F by the two stages charging process in Ref. [10], see Chapter 2, will differ from the one for the electrode system only in last term:

      where E_c(r) is the electric field which the given ionic and electrode charge distribution would exert in a vacuum:

      On the electrode, this σ(r) arises from all the induced charges, including those induced by the polarized dipoles. M. defines then a function E_v(r) which “depends only on the ionic charges in the solution and on the surface charge density σ_v(r) which they would induce in a vacuum.” The potential ψ_v(r′) in that system must be a constant on the electrode.

      E_v(r′) and ψ_v(r′) are given by:

      where ρ_v(r) = ρ(r) and where σ_v(r) and σ(r) are equal on the surface of each central ion but differ on the electrode. They differ there by an amount equal to the surface charge density induced in the electrode by the polarized dielectric, that is, by P(r).

      M. shows that Eq. (4.12) can be rewritten in terms of E_v:

      For electrode systems, Eq. (4.8) will be:

      where stands for and the internal energy will be U = F_e + TS_e.

      4.6.Evaluation of the Fe of an Equilibrium System

      Equations (4.11) and (4.12) give the general expressions for F_e valid for equilibrium and nonequilibrium dielectric polarization systems. F_e will have its minimum value