Dielectric medium, thermodynamics

Figure 1. Thermodynamic cycle to achieve the insertion of an arbitrary ion into a continuum dielectric medium.

We choose the total system to be the condenser and the entire dielectric medium. The condenser is immersed in the medium which, for purposes of this discussion, is taken to be a single-phase, multicomponent system. The pressure on the system is the pressure exerted by the surroundings on a surface of the dielectric. In setting up the thermodynamic equations we omit the properties of the metal plates, because these remain constant except for a change of temperature. The differential change of energy of the system is expressed as a function of the entropy, volume, and mole numbers, but with the addition of the new work term. Thus,... 

It is of interest to obtain thermodynamic relations that pertain to the dielectric medium alone. The system is identical to that described in Section 14.11. However, in developing the equations we exclude the electric moment of the condenser in empty space. We are concerned, then, with the work done on the system in polarizing the medium. Instead of D we use (D — e0E), which is equal to the polarization per unit volume of the medium, p. Finally, we define P, the total polarization, to be equal to Fcp. Now the equation for the differential of the energy is... 

Not to be forgotten is the assumption that neither the presence of the electrolyte nor the interface itself changes the dielectric medium properties of the aqueous phase. It is assumed to behave as a dielectric continuum with a constant relative dielectric permittivity equal to the value of the bulk phase. The electrolyte is presumed to be made up of point charges, i.e. ions with no size, and responds to the presence of the charged interface in a competitive way described by statistical mechanics. Counterions are drawn to the surface by electrostatic attraction while thermal fluctuations tend to disperse them into solution, surface co-ions are repelled electrostatically and also tend to be dispersed by thermal motion, but are attracted to the accumulated cluster of counterions found near the surface. The end result of this electrical-thermodynamic conflict is an ion distribution which can be represented (approximately) by a Boltzmann distribution dependent on the average electrostatic potential at an arbitrary point multiplied by the valency of individual species, v/. 

The topic of interactions between Lewis acids and bases could benefit from systematic ab initio quantum chemical calculations of gas phase (two molecule) studies, for which there is a substantial body of experimental data available for comparison. Similar computations could be carried out in the presence of a dielectric medium. In addition, assemblages of molecules, for example a test acid in the presence of many solvent molecules, could be carried out with semiempirical quantum mechanics using, for example, a commercial package. This type of neutral molecule interaction study could then be enlarged in scope to determine the effects of ion-molecule interactions by way of quantum mechanical computations in a dielectric medium in solutions of low ionic strength. This approach could bring considerable order and a more convincing picture of Lewis acid base theory than the mixed spectroscopic (molecular) parameters in interactive media and the purely macroscopic (thermodynamic and kinetic) parameters in different and varied media or perturbation theory applied to the semiempirical molecular orbital or valence bond approach [11 and references therein]. 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Plotting ixbase VS. pH gives a sigmoidal curve, whose inflection point reflects the apparent base-pAi, which may be corrected for ionic strength, I, using Equation 6.11 in order to obtain the thermodynamic pATa value in the respective solvent composition. Parameters A and B are Debye-Hiickel parameters, which are functions of temperature (T) and dielectric constant (e) of the solvent medium. For the buffers used, z = 1 for all ions ao expresses the distance of closest approach of the ions, that is, the sum of their effective radii in solution (solvated radii). Examples of the plots are shown in Figure 6.12. 

In a medium with a relatively low dielectric constant like a glass, the dissociation constant is expected to be small, and the thermodynamic ionic activities proportional to their concentrations. An approximate expression for (4.12) is then... 

A number of attempts have been made to predict thermodynamic functions for ionizations on the basis of electrostatic theory (Benson, 1960d Frost and Pearson, 1961a). The simple Born treatment, which considers the solvent as a continuous dielectric, gives for the free energy of separation of a pair of spherical charges, ZK e and ZB e, in a medium of dielectric constant D,... 

The micelle formation process and structure can be described by thermodynamic functions (AG°mjc, AH°mjc, AS°mic), physical parameters (surface tension, conductivity, refractive index) or by using techniques such NMR spectroscopy, fluorescence spectroscopy, small-angle neutron scattering and positron annihilation. Experimental data show that the dependence of the aggregate nature, whether normal or reverse micelle is formed, depends on the dielectric constant of the medium (Das et al., 1992 Gon and Kumar, 1996 Kertes and Gutman, 1976 Ward and du Reau, 1993). The thermodynamic functions for micellization of some surfactants are presented in Table 1.1. 

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