Molecular Imaging. Markus Rudin

href="#fb3_img_img_8eb98438-3ba2-5fe1-96e3-b176ac368bcb.png"/> at equilibrium [1, p. 215]. Fluctuations δP_u and δc_i carry the system away from equilibrium and moves to corresponding nonequilibrium values In order to get the particular expression for F_e at equilibrium, M. minimizes F_e setting δF_e = 0 and considering two equations of restraint. For the variation δF_e, we have from Eq. (4.6b):

      In δF the term χ ∫_M dσdS will appear.

      In the minimization process, one should consider two equations of constraint of immediate physical interpretation:

      (i)∫ δc_i(r)dV = 0 for each i

      [2.215] because the total number n_i of ions of species i is fixed in solution ⁽²⁾ and:

      (ii)

      [2.220] which means that if the charge density on the electrode varies by σ, the charge in the solution must vary by an equal and opposite amount because of the electroneutrality of the whole electrode and solution system.

      Solving the variational problem for δF_e subject to the earlier constraints, M. shows that at equilibrium P_u(r) is determined by E(r)(as it was intuitively to be expected):

      and that c_i(r) depends on ϕ^eq(r) around a central ion:

      [1, p. 215], that is, the concentration c_i(r) of volume charges of ions of species i around a central ion depends on the potential ϕ^eq(r) and is Boltzmann weighted by the ratio:

      Notice, moreover, that the integral in the denominator has the dimension of a volume and so there is a concentration in both members of Eq. (4.15).

      The equilibrium electrostatic free energy (4.16a) can be deduced directly from Eqs. (4.6b) and (4.10) [2, p. 215]

      An equivalent expression is:

      [2.222] where The formula is of transparent physical meaning: the equilibrium free energy of the electrodic system is the energy necessary to charge up all of the surfaces present in it. If the system is supposed to be made up of

      (i)A central ion

      (ii)Mobile ions

      (iii)An electrode M

      (iv)A medium of dielectric constant D_s the potential ϕ^eq is built up from the contributions of (i), (ii), and (iii) [2, pp. 196, 197]:

      “where r and r_i denote the distances from the field point to the center of the ion and to the center of its electrical image, respectively. is the contribution from the mobile ions and from the electrode charges they induce. On the electrode surface, ϕ^eq is a constant, β = χ. On the surface of the central ion of radius a, see Fig. 2, and surface 4πa², the surface density dσ equals dq/4πa².” It can be shown that (see Ref. [4, p. 974] with note on the Born charging formula):

      where R is the distance from the center of the ion to its electrical image as in Fig. 2.

      We have then:

       is, on its turn, the sum of three independent contributions, when the ion is outside the double layer region:

      (1) arising from a spherically symmetric ionic atmosphere about the central ion,” see Fig. 1.

      (2)“A contribution due to the electrode charge density induced by this atmosphere. It is symbolized by a dashed spherical shell in Fig. 1. Since the atmosphere is concentric with the central ion, spherically symmetric, and has a total charge of −q, it can be shown that the same electrode charge density would be introduced by a point charge −q situated at the center of the central ion. The image of this charge is q and its contribution to the potential is therefore q/D_s R.” This contribution to the potential will give a contribution q²/2D_s R to that will cancel the contribution −q²/2D_s R in Eq. (4.16b). This means that the contributions of the image of the central ion and of the image of its ionic atmosphere cancel each other.

      (3)“φ_S arising from the ions of the electrical double layer together with the electrode charges they induce,” see Fig. 1.

      “Remembering that both and φ_S are constant over the surface of this ion, we obtain:

      Since where γ_q is the electrostatic contribution to the activity coefficient of the ion of charge q in the body of the solution, Eq. (4.17) becomes:

       4.7.Theory of Overvoltage for Electrode Processes Possessing ET Mechanism

       4.7.1. Introduction

Among