The Lippmann equation for the ideally polarizable interface is normally derived by thermodynamics, using the Gibbs dividing surface. Therefore, the quantities appearing in the Lippmann equation can have no reference to the actual charge distribution in the interfacial region. For example, the quantity referred to as surface charge is actually a sum of surface excesses, rather than the integral of a true charge density. In this article we derive, by statistical mechanical methods, the Lippmann equation for a model at the molecular level, thus giving a precise physical definition to all quantities which appear. First, we derive the conditions for mechanical equilibrium for a system (the interface between metal and solution) in which an electric field is present, and whose properties are inhomogeneous and anisotropic. From the balance of forces, we obtain equations for the surface tension. in terms of the pressure, electric field, electric charge density, and electric polarization at each point within the system. Considering a spherically symmetric system (mercury drop), we then proceed to a direct calculation of the change in the surface tension produced by a change in the potential drop across the interface, maintaining thermal equilibrium, constant temperature, and the pressure and chemical composition in homogeneous regions (on the boundaries of the interfacial region). Since an ideally polarizable interface does not permit charge transport across it, we introduce a surface within the interface on which the charge density is always zero. This surface serves to divide the interfacial region into two parts, thus allowing the surface charge to be defined as the integral of the charge density over the metal side of the interface. Only the solution side is treated by statistical mechanics. Boltzmann distributions for charged and polarizable species (solute and solvent) are used to guarantee thermal equilibrium. The Lippmann equation is obtained (a) considering only ions and supposing a dielectric constant equal to that of vacuum and (b) considering ions and molecules in thermal equilibrium, and a dielectric constant varying from point to point and changing with field. Finally, the response of our system to an imposed alternating potential is considered. A direct calculation of the impedance shows that it behaves, in the low-frequency limit, as a pure capacitance, and that the value of this capacitance is the derivative of the previously defined surface charge density with respect to the potential drop across the interface.
Badiali, Jean- Pierre and Goodisman, Jerry, "The Lippmann Equation and the Ideally Polarizable Electrode" (1975). Chemistry - Faculty Scholarship. 73.
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