Relative permittivity of the solution

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. 

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Walden [Wa 24] introduced the following empirical correlation between the molar fraction (N) of the solute and the relative permittivity ( ) of the solution ... 

Liszi [Li 75] studied the phenomenon of dipole association in nitrobenzene-carbon tetrachloride, nitrobenzene-n-heptane and nitrobenzene-benzene mixtures, i.e., mixtures in which the second component is an apolar solvent. With a knowledge of the densities, he used the dielectrometrically (at a constant frequency of 3 MHz) determined relative permittivities of the solutions to calculate the molar... 

The parameters d, Sx,Q, and k are the diameter of the droplets emitted at the jet (the primary droplet size), the relative permittivity of the solution, the feed rate or flow rate of the solution, and the electrical conductivity of the solution, respectively. All of these processes influence the morphology of the deposited layer. A detailed description has been reported by Chen et al. [116]. 

A further complication that sets in when organic or mixed aqueous-organic solvents are used, which is aggravated when the relative permittivity of the medium, e, falls below 40, is ion pairing. This phenomenon does occur in purely aqueous solutions, mainly with higher-valence-type electrolytes 2 2 and higher, and with 2 1 or 1 2 electrolytes only at high concentrations. Ion pairs may also form in aqueous solutions of some 1 1 electrolytes, provided the ions are poorly hydrated and can approach each other to within <0.35 nm. Such ion pairs are of major importance in solvents that are relatively poor in water or that are nonaqueous. 

When the relative permittivity of the organic solvent or solvent mixture is e < 10, then ionic dissociation can generally be entirely neglected, and potential electrolytes behave as if they were nonelectrolytes. This is most clearly demonstrated experimentally by the negligible electrical conductivity of the solution, which is about as small as that of the pure organic solvent. The interactions between solute and solvent in such solutions have been discussed in section 2.3, and the concern here is with solute-solute interactions only. These take place mainly by dipole-dipole interactions, hydrogen bonding, or adduct formation. 

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. 

Most solvents consist of molecules that are intrinsic dipoles and have permanent dipole moments (pi). If such molecules are placed between the two plates of a capacitor as a vapor (or as a dilute solution in a nonpolar liquid), they are oriented by the electric field. Then, the orientational polarization and the induced polarization occur simultaneously, as described above. If er is the relative permittivity of the vapor, there is a relationship ... 

For reactions in solution, the exponential term appropriate to the gas phase has to be modified to include the contribution accounting for the charges and the solvent. Calculations show that AG has to be modified by a term involving er, the relative permittivity of the solvent, and the charges on the ions. This term turns out to be the same as that appearing in the collision formula, Equation (7.3), i.e. z.KZ.ne1 / 47T o r ry, so that... 

In the analysis of Eq. (1) we assume that the relative permittivity of the liquid phase is the same as that of the membrane phase, for simplicity. If this is not satisfied, Eq. (1) with i — 1 needs to be modified [32-34]. Although the solution procedure similar to those introduced above can be used, the analysis becomes much more complicated. 

Figure 4(a) shows a metal-insulator-metal (MIM) gap. The solution to the gap is equivalent to that for a TM dielectric waveguide, except that the relative permittivity of the cladding layers is negative and the field is a hyperbolic cosine inside the gap. The equation to find the solution for the effective index of the MIM waveguide mode is as follows ... 

First, immediately after ionization, contact ion pairs are formed, in which no solvent molecules intervene between the two ions that are in close contact. The contact ion pair constitutes an electric dipole having only one common primary solvation shell. The ion pair separated by the thickness of only one solvent molecule is called a solvent-shared ion pair In solvent-shared ion pairs, the two ions already have their own primary solvation shells. These, however, interpenetrate each other. Contact and solvent-shared ion pairs are separated by an energy barrier which corresponds to the necessity of creating a void between the ions that grows to molecular size before a solvent molecule can occupy it. Further dissociation leads to solvent-separated ion pairs Here, the primary solvation shells of the two ions are in contact, so that some overlap of secondary and further solvation shells takes place. Increase in ion-solvating power and relative permittivity of the solvent favours solvent-shared and solvent-separated ion pairs. However, a clear experimental distinction between solvent-shared and solvent-separated ion pairs is not easily obtainable. Therefore, the designations solvent-shared and solvent-separated ion pairs are sometimes interchangeable. Eventually, further dissociation of the two ions leads to free, i.e. unpaired solvated ions with independent primary and secondary solvation shells. The circumstances under which contact, solvent-shared, and solvent-separated ion pairs can exist as thermodynamically distinct species in solution have been reviewed by Swarcz [138] and by Marcus [241],... 

Eq. (4-10) can be used only for solvents of equal acid and base strength, because only the effect of the solvent relative permittivity on the degree of ionization is considered. Under these conditions, Eq. (4-10) predicts that the logarithm of the ionization constant of HA should be inversely proportional to the relative permittivity of the solvent in which HA is dissolved. However, one has to take into account the fact that the relative permittivities near solute ions can differ considerably because of the effect of dielectric saturation, which hinders the precise calculation of electrostatic interactions. Because of these restrictions, Eq. (4-10) can be expected to yield only semiquantitative results. Nevertheless, it allows us to predict qualitatively how the charge type of an acid affects the ionization constant in solvents of different relative permittivities. 

In conclusion, it can be said that the electrostatic theory of solvent effects is a most useful tool for explaining and predicting many reaction patterns in solution. However, in spite of some improvements, it still does not take into account a whole series of other solute/solvent interactions such as the mutual polarization of ions or dipoles, the specific solvation etc., and the fact that the microscopic relative permittivity around the reactants may be different to the macroscopic relative permittivity of the bulk solvent. The deviations between observations and theory, and the fact that the relative permittivity cannot be considered as the only parameter responsible for the changes in reaction rates in solution, has led to the creation of different semiempirical correlation equations, which correlate the kinetic parameters to empirical parameters of solvent polarity (see Chapter 7). 

We calculate the self-atmosphere potential of an ion near the planar surface. Imagine a planar uncharged plate in contact with a solution of general electrolytes composed of N ionic mobile species of valence z, and bulk concentration (number density) nf (i=l,2,. . . , N) (Fig. 3.10). Let and p, respectively, be the relative permittivities of the electrolyte solution and the plate. Consider the potential distribution around an ion. By symmetry, we use a cylindrical coordinate system r(s, x) with its origin 0 at the plate surface and the x-axis perpendicular to the surface. 

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