Dielectric constant of the electrolyte solutions

Debye-Huckel parameter. (V2 Laplace operator, o -> permittivity of vacuum, er -> dielectric constant of the electrolyte solution, cf bulk concentrations of all ions i, zp charges of the ions i, f electric potential, k - Boltzmann constant, and T the absolute temperature). See also -5- Debye-Huckel theory. 

Here v is the velocity of the particle, E is the electric field strength, eo is the - permittivity of vacuum, eT is the dielectric constant of the electrolyte solution, ( is the equilibrium potential at the plane of shear (- zeta potential), and tj is the -> viscosity. See also -> Smolu-chowski equation (for the case of xr 1). 

The dielectric constant of the electrolyte solution is equal to that of the pure solvent. 

This scattered layer plus the electrode positive charge array is called the diffuse double-layer or diffuse layer. In the literature, this diffuse layer is also called the Gouy point charge layer or model or the Gouy-Chapman model. We will use the diffuse layer term in this chapter. The thickness of the diffuse layer is dependent on the temperature, the concentration of the electrolyte, the charge number carried by the ion, and the dielectric constant of the electrolyte solution. 

In general, the Helmholtz layer can be treated as a linear capacitor. In a theoretical model of the electric double-layer, the compact Helmholtz layer is generally treated as an ideal capacitor with a fixed thickness (d), and its capacitance is considered unchanging with the potential drop across it. Therefore, fhe capacifance of fhe Helmholtz layer can be treated as a constant if fhe femperafure, fhe dielectric constant of the electrolyte solution inside the compact layer, and its thickness are fixed. However, if the specific ion adsorpfion happened on the electrode surface, the dielectric constant of the electrolyte solution inside the compact layer may be affected, leading to non-linear behavior of the Helmholtz layer. This will be discussed more in a later section. 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

Strong electrolytes are dissociated into ions that are also paired to some extent when tlie charges are high or the dielectric constant of the medium is low. We discuss their properties assuming that the ionized gas or solution is electrically neutral, i.e. 

In the relationship shown above, A and B are constants depending on temperature, viscosity of the solvent, and dielectric constant of the solvent, C is the concentration expressed in gram equivalents per liter, and Ac represents the equivalent conductance of the solution. A0 is the equivalent conductance at infinite dilution - that is, at C = 0, when the ions are infinitely apart from one another and there exists no interionic attraction, a represents the degree of dissociation of the electrolyte. For example, with the compound MN... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

This expression assumes that the dielectric constant and viscosity of the electrolyte solution are the same in the electric double layer as in the bulk solution. 

The activity of ions in a solution is governed by the dielectric constant of the medium they are dissolved in and by the total concentration of ions in solution. For solutions of electrolytes in water with concentrations < 0.5 M the activity of the ions present in solution is usually approximated to their individual concentrations. The mean activity coefficient for an ion in solution is defined as ... 

For molten salts one sets so = 1. For electrolyte solutions solvent-averaged potential [37]. Then, in real fluids, eo in Eq. (11) depends on the ion density [167]. Usually, one sets so = s, where e is the dielectric constant of the solvent. A further assumption inherent in all primitive models is in = , where is the dielectric constant inside the ionic spheres. This deficit can be compensated by a cavity term that, for electrolyte solutions with e > in, is repulsive. At zero ion density this cavity term decays as r-4 [17, 168]. At... 

At a quantitative level, near criticality the FL theory overestimates dissociation largely, and WS theory deviates even more. The same is true for all versions of the PMSA. In WS theory the high ionicity is a consequence of the increase of the dielectric constant induced by dipolar pairs. The direct DD contribution of the free energy favors pair formation [221]. One can expect that an account for neutral (2,2) quadruples, as predicted by the MC studies, will improve the performance of DH-based theories, because the coupled mass action equilibria reduce dissociation. Moreover, quadrupolar ionic clusters yield no direct contribution to the dielectric constant, so that the increase of and the diminution of the association constant becomes less pronounced than estimated from the WS approach. Such an effect is suggested from dielectric constant data for electrolyte solutions at low T [138, 139], but these arguments may be subject to debate [215]. We note that according to all evidence from theory and MC simulations, charged triple ions [260], often assumed to explain conductance minima, do not seem to play a major role in the ion distribution. 

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