Onsager relationships

In discussing the Onsager relationship (Chap. 7, Sect. 2.3), use was made of the reciprocity of the Green s function. This has been discussed... 

In accordance with the Onsager relationships, each of the fluxes under consideration is conjugate with both thermodynamic forces Xj = Ap and X2 = An. Then... 

This process will increase the dielectric constant of the solution. Superimposed on this effect is the tendency of the solute to aggregate. This aspect has been recognized by many workers (13, 14, 15, 17). Dielectric measurements give information concerning the contribution of both processes, as shown in Table III, using the Onsager relationship (12). 

Some electrolytes containing large ions, particularly soaps, dyes and many synthetic detergents, behave as normal electrolytes only in very dilute solution. At higher concentrations they show unusually low osmotic pressures and their conductances show large deviations from the Onsager relationship. 

The second aspect is more fundamental. It is related to the very nature of chemistry (quantum chemistry is physics). Chemistry deals with fuzzy objects, like solvent or substituent effects, that are of paramount importance in tautomerism. These effects can be modeled using LFER (Linear Free Energy Relationships), like the famous Hammett and Taft equations, with considerable success. Quantum calculations apply to individual molecules and perturbations remain relatively difficult to consider (an exception is general solvation using an Onsager-type approach). However, preliminary attempts have been made to treat families of compounds in a variational way [81AQ(C)105]. 

As already mentioned, the criterion of complete ionization is the fulfilment of the Kohlrausch and Onsager equations (2.4.15) and (2.4.26) stating that the molar conductivity of the solution has to decrease linearly with the square root of its concentration. However, these relationships are valid at moderate concentrations only. At high concentrations, distinct deviations are observed which can partly be ascribed to non-bonding electrostatic and other interaction of more complicated nature (cf. p. 38) and partly to ionic bond formation between ions of opposite charge, i.e. to ion association (ion-pair formation). The separation of these two effects is indeed rather difficult. 

The first approximate calculation was carried out by Debye and Hiickel and later by Onsager, who obtained the following relationship for the relative strength of the relaxation field AE/E in a very dilute solution of a single uni-univalent electrolyte... 

Onsager s theorem consists of proving that a reciprocal relationship of the type Lap = Lpa between the affinities and fluxes of coupled irreversible processes is universally valid in the absence of magnetic fields. 

Onsager found a proof of the general reciprocal relationship by consideration of natural fluctuations that occur in equilibrium systems. It is argued that an imbalance like that which initially exists when two reservoirs are connected as in figure 2, could also arise because of natural fluctuations. When the occasional large fluctuation occurs in an equilibrium system, the subsequent decay of the imbalance would be indistinguishable from the decay that follows deliberate connection of the reservoirs. 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

Studies on multicomponent systems have been mainly restricted to relatively simple ternary systems containing a solvent as component 1 and-solutes as components 2 and 3. For such a system, under zero-volume flow conditions (Eq. (3)), exact expressions for the fluxes of the components 2 and 3 may be written as independent quantities of the forces involved so that the linear laws to which the Onsager reciprocal relationship applies may be written as follows341 ... 

The Onsager reciprocity relationships (Dl, 01, 02) become of importance in situations where there are combined material and thermal transport and under these circumstances may be expressed as... 

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

These constitute a set of linear relationships between the potential differences pi — p q, which drive the Y) toward equilibrium and their corresponding rates, dYi/dt. In terms of the Onsager coefficients, they have the form... 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

The physical significance of these variables is apparent when they are evaluated in the Onsager cavity description of solvation, which treats the solute as a sphere (which we will assume here is unpolarizable) of radius a. The solvent is modeled as a uniform dielectric medium with a static dielectric constant s and an optical dielectric constant op. The following relationships apply in the Onsager cavity description... 

A simple model for C(t). In this subsection we explore the relationship of C(r) to dynamic properties of the solvent, in terms of the Onsager cavity description, following the work in the literature on this subject [12-14, 53-57]. Theories that go beyond the Onsager model are described in Sections II.E and II.D. 

The physical meaning of the relationship described in the previous subsection becomes apparent when we consider the popular special case of the Onsager cavity model that arises if we assume that the solvent s dielectric properties are well described by a Debye form. 

A theoretical approach for explaining the relationship between S and the characteristics of the electrolyte was provided by Onsager on the basis of the model of ions plus ionic cloud developed in the Debye-Hiickel theory, obtaining [4]... 

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