Relative complex permittivity

Ion pairing is due to electrostatic forces between ions of opposite charges in a medium of moderate to low relative permittivities. It should be distinguished from complex formation between metal cations and anionic ligands, in which coordinative bonds (donation of an electron pair) takes place. One distingnishing feature is that, contrary to complex formation, the association is nondirectional in space. The association of a cation and an anion to form an ion pair can, however, be represented as an equilibrium reaction by analogy to complex formation with an equilibrium constant A)ass [3,5]. If a is the fraction of the electrolyte that is dissociating into ions and therefore (1 - a) is the fraction that is associated, then... 

There are two sets of quantities that are often used to describe optical properties the real and imaginary parts of the complex refractive index N = n + ik and the real and imaginary parts of the complex dielectric function (or relative permittivity) e = c + ie". These two sets of quantities are not independent either may be thought of as describing the intrinsic optical properties of matter. The relations between the two are, from (2.47) and (2.48),... 

Probably the most familiar parameter used by the coordination chemist investigating the effect of solvent on rates of reaction of coordination complexes is the relative permittivity R (dielectric constant) of the medium. If the solvent can be regarded as an inert medium then the effect of the solvent can be evaluated, semiquantitatively at least, if only electrostatic forces are considered. 

It is much more difficult to predict the effect of change of solvent on A 5 than it is to predict the effect on the ideal rate constant. This is because the analysis for the dependence of A S on solvent includes the temperature dependence of the relative permittivity. This complexity in the analysis occurs for ion-ion and ion-molecule reactions, and, because of the difficulty in disentangling the effects, this will not be pursued further. 

Since the reactants are molecules, the extent of solvation will be small and approximately the same for all solvents. But the extent of charge separation in the activated complex is very dependent on the solvent, by virtue of the magnitude of the relative permittivity, and a large variation in the extent of solvation of the activated complex is expected. The entropy of activation and the A factor are expected to vary with change of solvent. 

Cole-Cole plot in the complex plane of the loss factor (the imaginary part) e" against the real part s in Eq. (3.30), for a solvent with the high frequency relative permittivity... 

If the capacitor is now filled with a lossy dielectric, its effect can be taken into account by introducing a complex relative permittivity e = e r — je", where s r and s" are respectively the real and imaginary parts of the relative permittivity. It follows that... 

Ceramic dielectrics and insulators cover a wide range of properties, from steatite with a relative permittivity of 6 to complex ferroelectric compositions with relative permittivities exceeding 20000. For the purposes of this discussion insulators will be classed with low permittivity dielectrics, although their dielectric loss may be too high for use in capacitors. Reference should be made to Table 5.10 and Fig. 5.40. 

The methodology for the calculation of the complex relative permittivity for the dipolar relaxation mechanism is founded on the calculation of the dielectric response function, f(t), for a depolarization produced by the discharge of a previously charged capacitor. In Figure 1.29a, a circuit is shown where a capacitor is inserted in which a dipolar dielectric material is enclosed in the parallel plate capacitor of area, A, and thickness, d, with empty capacitance C0 = Q0/U0 = 0(A/d), and E0 = U0ld. In Figure 1.29b, the corresponding depolarization process is shown. 

In Figure 1.30, the plots of e (co) are shown, that is, the real part of the complex relative permittivity, and e" (co), that is, the imaginary part of the complex relative permittivity. 

The methodology for the calculation of the complex relative permittivity for the cation-hopping relaxation mechanism is very similar to those applied in the previous (Section 1.7.5). It is also based on... 

The study of materials by dielectric methods is carried out in alternating electric fields and the dielectric behavior is described, in a formal way, by the complex relative permittivity (or complex dielectric constant) (see Section 1.7.2) [15,103,104]... 

D) is the real part of the complex relative permittivity (or real component of the complex dielectric constant)... 

The solvent molecules form an oriented parallel, producing an electric potential that is added to the surface potential. This layer of solvent molecules can be protruded by the specifically adsorbed ions, or inner-sphere complexed ions. In this model, the solvent molecules together with the specifically adsorbed, inner-sphere complexed ions form the inner Helmholtz layer. Some authors divide the inner Helmholtz layer into two additional layers. For example, Grahame (1950) and Conway et al. (1951) assume that the relative permittivity of water varies along the double layer. In addition, the Stern variable surface charge-variable surface potential model (Bowden et al. 1977, 1980 Barrow et al. 1980, 1981) states that hydrogen and hydroxide ions, specifically adsorbed and inner-sphere... 

A second limitation of the Hughes-Ingold theory concerns the fact that the solvent is treated as dielectric continuum, characterized by one of the following its relative permittivity, e, the dipole moment, fi, or by its electrostatic factor, EF, defined as the product of and [27]. The term solvent polarity refers then to the ability of a solvent to interact electrostatically with solute molecules. It should be remembered, however, that solvents can also interact with solute molecules through specific inter-molecular forces like hydrogen bonding or EPD/EPA complexation cf. Section 2.2). For example, specific solvation of anionic solutes by pro tic solvents may reduce their nucleophilic reactivity, whereas in dipolar aprotic solvents solvation of anions is less,... 

As the data for the Menschutkin reactions indicate, the character of the solute-solvent interactions is more complex than described by Eq. (5-87). It is evident that functions of relative permittivity alone, as given in Eq. (5-87), are not useful for describing the solvent effect on reactions between dipolar reactants, except in certain special cases, such as when a mixture of two solvents is used. In addition to electrostatic forces, non-electrostatic interactions, such as dispersion forces and hydrogen-bonding, must also be involved in Menschutkin reactions. 

It is obvious that such a definition of solvent polarity cannot be measured by an individual physical quantity such as the relative permittivity. Indeed, very often it has been found that there is no correlation between the relative permittivity (or its different functions such as l/sr, (sr — l)/(2er + 1), etc.) and the logarithms of rate or equilibrium constants of solvent-dependent chemical reactions. No single macroscopic physical parameter could possibly account for the multitude of solute/solvent interactions on the molecular-microscopic level. Until now the complexity of solute/solvent interactions has also prevented the derivation of generally applicable mathematical expressions that would allow the calculation of reaction rates or equilibrium constants of reactions carried out in solvents of different polarity. 

Wagner (1914) gave an approximate treatment of the important practical case where a very highly insulating dielectric suffers from inclusions of conductive impurities. Taking the model where the impurity (relative permittivity e2, conductivity a2) exists as a sparse distribution of small spheres (volume fraction f) in the dielectric matrix (relative permittivity e, negligible conductivity), he derived equations for the components of the complex relative permittivity of the composite ... 

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