Theoretical relative permittivity

FIGURE 5.1 A plot of the theoretical relative permittivity e of a suspension of spherical particles lOOnm in radius in a 1 mM KCl solution at 25°C, as a function of the frequency V of the applied field. The zeta potential of the particles is -lOOmV. The left and right axes correspond, respectively, to the real (e = Re e ) and imaginary (e" = Im e ) components of the relative permittivity. Volume fraction of solids ( )= 1%. The vertical arrows indicate the approximate positions of the a- (left) and MWO- (right) relaxations. 

A further problem is that ion association, that is, the tendency of oppositely charged ions to form pairs or larger aggregates in solution, becomes increasingly important as the temperature rises unless the density is kept constant this is because ion association is inversely related to the dielectric constant (relative permittivity) of the medium, which is correlated with density for a given solvent. Helgeson and co-workers have attacked these problems theoretically for aqueous solutions up to 1000 °C.28 For our purposes, it is enough to note that quantitative treatment of ionic reactions in sub- and supercritical aqueous solutions is extremely difficult at present, and likely to remain so for some time. 

The problem of relative permittivity of mixed solvents (denoted by e for simplicity) was theoretically studied by Debye and Onsager [34], Onsager derived the following equation ... 

The pressure dependence of the relative permittivity has been determined for relatively few solvents only, although it is an important quantity in theoretical... 

Because of the complicated interactions between solvents and solutes, the prediction of solvent effects on reaction rates, and the correlation of these effects with intrinsic solvent properties, is very difficult. Nevertheless, many authors have tried to establish -empirieally or theoretically - correlations between rate constants or Gibbs energies of aetivation and characteristic solvent parameters such as relative permittivity, r, dipole moment, fi, refractive index, n, solubility parameter, 5, empirical solvent polarity parameters, etc., as schematically shown by Eq. (5-9). 

Although the solvent effects are small, the alkene formation diminishes as predicted with increasing water content (corresponding to increased solvent polarity). The Sn2/E2 reaction of 2-phenylpropyl tosylate with sodium cyanide (in hexamethyl-phosphoric triamide and in A,A-dimethylformamide as solvents at 100 °C) gives a-methylstyrene (elimination product) and l-cyano-2-phenylpropane (substitution product) [75]. It has been found, in accordance with the predictions of the Hughes-Ingold rules, that the elimination/substitution ratio decreases as the polarity of the solvents (measured by the relative permittivity) increases [75]. Theoretical investigations of the... 

The first theoretical treatment of infrared solvent shifts was given in 1937 by Kirkwood [166] and by Bauer and Magat [167], Eq. (6-8) - known as the Kirkwood-Bauer-Magat (KBM) relationship - has been derived on the basis of Onsager s reaction field theory [80] using the simple model of a diatomic oscillator within a spherical cavity in an isotropic medium of macroscopic relative permittivity r. 

Experimental measurements confirm the linear dependence of relative permittivity on density. Furthermore, the agreement on the value of K (mean experimental value 0.326 compared with 0.327 theoretical) is very satisfactory. 

Although Maxwell (1892) considered a rather special case when he first examined the effect of discontinuities in dielectrics theoretically, he obtained a very important result. He examined the effect of a field applied across a specimen consisting of layers of two different materials with relative permittivities e i 2 and conductivities crj, a2 respectively. His results showed that charges will accumulate in time at the interfaces between the layers whenever <7i ia. ... 

Acetonitrile is a polar solvent with a relative permittivity of 35.9. It may be represented as a hard sphere with a diameter of 427 pm. Estimate the Gibbs energy of solvation of Na in acetonitrile according to the Born and MSA models. Compare the theoretical estimates with the experimental estimate given that the Gibbs energy of transfer for Na" " from water to acetonitrile is 15.1 kJmoP ... 

Question Can it possibly be valid to use the macroscopic relative permittivity in the theoretical treatments ... 

The theoretical basis of his calculation is less secure than Bjerrum s, but his work had the merit of inspiring high precision work on the behaviour of electrolyte solutions over a range of relative permittivities. Bjenum s and Fuoss theories predicted different dependencies of association (sce Section 12.16). 

The relative permittivity measures the alignment of the solvent dipoles and production of induced dipoles by an electric field. An ion produces an intense field on bound solvent molecules, and will cause partial, if not complete, alignment of the dipoles of the solvent molecules affected by the ion. This results in a drop in the observed relative permittivity of the solution relative to the pure solvent. This drop is related to the number of bound solvent molecules. Controversy exists as to whether the effect is restricted to bound molecules only, or whether other solvent molecules are involved. Both theoretical and experimental studies have been carried out. The dependence of the relative permittivity on the distance from a given ion is of fundamental importance in theories of electrolyte solutions where generally the bulk relative permittivity is used in the theoretical expressions. But it is more likely that a varying relative permittivity should be used. 

Here /u, is the dipole moment of the molecules of the liquid, Sr is the relative permittivity of the liquid, and is the square of its refractive index at the frequency of the sodium D-line. The theoretical expression requires the square of the infinite frequency refractive index to represent the polarizability of the molecules of the liquid, but 1.1 d is a good empirical approximation of it. The KDOCP of a non-structured liquids is ideally = 1, but in practice = 1,0 0.3 holds for such liquids, whereas for ordered liquids g > 1.7 (Marcus 1992). Water has the value g = 2.90 at 25 °C, comparable with other hydrogen-bonded liquids, see Table 1.5, but by no means outstandingly large (compare, e.g., N-methylformamide with g = 3.97) (Marcus 1992). 

In considering the theoretical modeling of the electrical double layer at the cell surface, it may be useful to consider the way in which the value of the relative permittivity of the electrolyte varies with distance from the membrane surface. Usually the permittivity is assumed to be constant up to the surface and to have a value equivalent to that of normal bulk water. By comparing the value obtained for the zeta potential from electrophoresis measurements with the surface potential obtained from free radical quenching studies, more detailed information concerning the structure of the electrical double layer may be forthcoming. 

Even though the slow dynamics of water in PEMs has been experimentally studied [190], no theoretical or numerical works have been reported yet. Some recent simulation studies [68,75,191] have provided an important insight about the dielectric constant of water in concentrated electrolyte solutions where electrostatic interactions can be almost twice as strong as in pure water. In the work of Qiao et al. [160] the slow dynamics of water inside PSS/PDADMA systems was investigated via the static dielectric constant (i.e., static relative permittivity) and the water diffusion coefficient (see Table 1). Error estimates are determined by block... 

A UV-visible spectroscopic study of 3 and related substances revealed a strong solvatochromic effect, which served as the basis of the establishment of a solvent polarity scale (Buncel and Rajagopal, 1989, 1990,1991). The theoretical study of Rauhut et al. (1993) was based on AMI methodology (Dewar and Storch, 1985,1989) but used a double electrostatic reaction field in a cavity, dependent on both the relative permittivity and the refractive index. Nuclear motions interact with the medium through the relative permittivity, but electronic motions are too fast only the extreme high-frequency part of the dielectric constant is relevant. These authors were able to evaluate solvent-specific dispersion contributions to the solvation energy. The calculations reproduced satisfactorily the experimental solvatochromic results for 3 in 29 different solvents. The method has also been successfully applied to other solvatochromic dyes, including Reichardt s .j,(30) betaine. 

It has been mentioned that microwave intensities are determined by the size of the electric dipole moment, so one might suppose that accurate measurements of intensities might provide values of /. This turns out not to be practical for various reasons. However, another very accurate procedure can be used. If an electric field is applied to a rotating molecule, a well-understood phenomenon known as the Stark effect splits the rotational transitions into a number of components. Precise measurements of these small splittings (typically several MHz) lead to very precise values of the electric dipole moment. Values of jj, determined by this method refer to particular quantum states and are thus much more meaningful theoretically than those determined by classical bulk-gas relative permittivity (dielectric constant) measurements. 

It should be noticed that, in many theoretical works, the term solvent polarity is defined by the values of the relative electric permittivity, also called dielectric constant. However, such a definition is by no means precise. The existence of hydrogen bonds (H-bonds) between solute and solvent molecules is one of the important limitations of the use of the continuum models based on the theory of dielectrics. In modern physical chemistry of solutions in order to quantitatively describe the solvatochromism phenomenon various empirical scales of the polarity are used. The exhaustive reviews on this topic have been presented by Reichardt [1, 2],... 

Practically, DMTA is limited to low frequencies (up to tens of hertz) and, consequently, provides information about relatively slow processes. Dielectric spectroscopy is a related approach in which an alternating electric field is applied to a sample and the complex permittivity is then obtained from phase and amplitude measurements of current and voltage again, it is possible to consider data in the frequency domain, the temperature domain, or even as frequency/temperature contour maps.2 ° 2 See Refs. 230 and 232 for a theoretical account of the underlying physics. The approach can provide information in the frequency range W -io" coupling the applied electric field... 

The presence on the furan ring of several substituents, each of which can give rise to rotational isomerism, increases the number of possible conformations. Furan-2,5-dicarbaldehyde represents a classic example. In principle four conformations are possible for this compound (59). Analysis of the electric dipole moments of furan-2,5-dicarbaldehyde in solvents of low dielectric permittivity enabled the relative populations of the conformers to be determined <89JST(196)227>. Experimental data suggest that the compound exists as a mixture of mixed ( ,Z)/(Z, )- and ( , )-conformers with ( ,Z) + (Z, ) = 0.68 and 0.96, ( , ) = 0.32 and 0.04, according to the solvent. Molecular orbital AMI-calculated enthalpies of all the conformers confirm the presence of the mixtures with populations of ( ,Z) -H (Z,E) = 0.38 and ( , ) = 0.60. Another conformational analysis of the same derivative, obtained by an experimental dipole moment determination combined with theoretical MNDO calculations <89H(29)657>, indicates the presence of a conformational equilibrium in which the ( ,Z)-conformation has the biggest contribution, and that the two aldehyde groups are not equivalent. Experimental confirmation has been performed for two model reactions monoprotection with MeOH/TsOH and aldolic condensation. 

The MO measurements provide information about the angular distribution of molecules in the x, y, and z film coordinates. To extract MO data from IR spectra, the general selection rule equation (1.27) is invoked, which states that the absorption of linearly polarized radiation depends upon the orientation of the TDM of the given mode relative to the local electric field vector. If the TDM vector is distributed anisotropically in the sample, the macroscopic result is selective absorption of linearly polarized radiation propagating in different directions, as described by an anisotropic permittivity tensor e. Thus, it is the anisotropic optical constants of the ultrathin film (or their ratios) that are measured and then correlated with the MO parameters. Unlike for thick samples, this problem is complicated by optical effects in the IR spectra of ultrathin films, so that optical theory (Sections 1.5-1.7) must be considered, in addition to the statistical formulas that establish the connection between the principal values of the permittivity tensor s and the MO parameters. In fact, a thorough study of the MO in ultrathin films requires judicious selection not only of the theoretical model for extracting MO data from the IR spectra (this section) but also of the optimum experimental technique and conditions [angle(s) of incidence] for these measurements (Section 3.11.5). 

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