The statistical thermodynamics of symmetrical “primitive-model” electrolytes is formulated in such a way that all ions are uniquely paired. The behavior of the resulting fluid of “polar molecules” may conveniently be described by a wavelength-dependent dielectric constant ε(k). A rigorous formula of the Kirkwood type for ε(k) is derived. Since ion-atmosphere mean charge densities may be obtained from ε(k), this dielectric function is utilized in construction of an electrolyte free-energy expression [Eq. (50)], as well as to establish an exact second-moment condition on the ion atmospheres [Eq. (73)]. From the latter it is demonstrated that for rigid spherical ions of diameter a, the ion atmospheres necessarily each have nonuniform charge sign when κa > 61 / 2 (κ− 1 = Debye length).
The rigorous second-moment condition previously derived for “primitive-model” electrolyte ion atmospheres in equilibrium is generalized to arbitrary mixtures of electrolytes of unrestricted charge species. No special assumptions regarding the nature of solvent dielectric behavior are required, and the condition remains valid even in the presence of specific chemical interactions that lead to complex ion formation.
A formal derivation is presented for the equilibrium relation between the singlet density in a fluid and the direct correlation function and for the equivalent relation involving the pair number density. It is shown that this relation is equivalent to the macroscopic condition for hydrostatic equilibrium when the singlet density varies sufficiently slowly to permit the introduction of local thermodynamics. Some aspects of the usage of this in the determination of the singlet density in the liquid–vapor transition region are discussed.
On the basis of model calculations, we argue heuristically that the dilatation-transformation method is applicable to all potentials, even those that are not dilatation analytic. In particular, we show that resonances are converted into bound (localized) states on the nonphysical sheet of the complex energy plane under the action of a dilatation transformation.
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