Onsager equation validity

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. 

Up to concentration of 2 X 10-3 gram-equivalents per litre there is a satisfactory agreement between the results calculated from the Debye-Hiickel-Onsager equation and the actual values of conductance of uni-univalent electrolytes. The validity of this equation has been verified even for uni-bivalent electrolytes, while for bi-bivalent electrolytes there are greater deviations to be observed. 

Equations of type (2.17) for the interrelation of the rates of conjugate stepwise reactions are valid for any intermediate linear transformation pathways (including catalytic reactions). The value of A may be expressed by relations that are much more complicated than (2.15) and depends not only on parameters Sy but also on thermodynamic rushes of some external reactants of the stepwise reactions (see Section 2.3.5 for exam pies). At the same time. A > 0 always. However, the relationship between the cross coefficients Ay and Aj may be more intricate than that in the traditional Onsager equations. 

W course, the validity of the calculation depends upon whether the theoretical expression for the equivalent conductivity (e.g., the Debye-Hiickel-Onsager equation) is valid in the given concentration range. 

Before proceeding with a description of the experiments that have been made to test the validity of the Onsager equation, attention may be called to the concentration term c (or c) which appears in the equations (33) to (36). This quantity arises from the expression for k [equation (12)], and in the latter it represents strictly the actual ionic concentration. As long as dissociation is complete, as has been assumed above, this is equal to the stoichiometric concentration, but when cases... 

Validity of the Debye-Huckel-Onsager Equation.—For a uni-univalent electrolyte, the Onsager equation (36), assuming complete dissociation, may be written in the form... 

Less accurate measurements of the conductances of aqueous solutions of various electrolytes have been made, and in general the results bear out the validity of the Onsager equation. A number of values of the experimental slopes are compared in Table XXIV with those calculated... 

The first conclusion regarding figures 4.26 and 27 is that for xa /(xa) approaches 1.5 (fig. 4.26) whereas for xa -4 0 /(xa) -4 1.0 (fig. 4.27). In other words, in the appropriate limits the Helmholtz-Smoluchowski [4.3.4) and Huckel-Onsager equations (4.3.5) are retrieved. So it is concluded that these two limiting laws also remain valid when the double layer is polarized. This is an extension of Morrison s result, quoted in connection with 14.3.20). 

II) For very low xa the Huckel-Onsager equation (4.3.5] is valid. Example particles in media of low polarity. The applicability limit can be deduced from fig. 4.27. 

The most recent and accurate test of the equation is, however, due to Shedlovslcy.0,10 Using the methods for determining conductances outlined in Chapter 3, he was able to make measurements, with precision, at salt concentrations as low as 0.00003 normal. Some of his results for aqueous solutions of potassium chloride at 25° are given in Table I. A sensitive method for testing the validity of the Onsager equation is to compute values of the limiting conductance, A0, with its aid, for which purpose the equation may be conveniently rearranged in the form... 

The evidence just given, which is typical of that obtained from all recent measurements, shows that the Onsager equation is valid for very dilute aqueous solutions of strong electrolytes. This fact is important as it lends additional and strong support to the correctness and utility of the interionic attraction theory. As has already been emphasized Onsager s equation is a limiting equation and deviations from it, even for completely dissociated electrolytes, are to be expected as the concentration is increased. 

Evidence from Transference Numbers for the Correctness of the Onsager Equation. An important and quite sensitive test of the validity of the Onsager equation, and of the ideas underlying it, is afforded by transference number measurements. It will be recalled from the discussion in Chapter 3, that the Arrhenius theory of ionization assumes that ions have the same mobilities at all concentrations. The transference number, f+, of the cation constituent of a binary electrolyte is... 

This equation shows that even for solutions dilute enough for the Onsager equation to hold, the transference numbers should in general change with the concentration, if the interionic attraction theory is valid. 

It must be emphasized that the transference number measurements are at concentrations at which the Onsager equation, on which expressions (28) and (28a) are based, is only approximately valid. In general the transference data lend strong support to the interionic attraction theory of electrolytic conductance. 

Within the range of concentrations for which the Fuoss-Onsager equation is expected to be valid, this equation accounts well for the effects of non-ideality in solutions of symmetrical electrolytes in which there is no ion association. It can thus be taken as a base-line for non-associated electrolytes and any deviations from this predicted behaviour can be taken as evidence of ion association (see Section 12.12). 

The large ionic interaction often renders the Onsager equation useless (it is still presumably correct) for the extrapolation to obtain A . The solutions for which the Onsager relation is valid are so dilute that it is not possible to obtain reliable measurements of their conductivity. In these cases, special methods of obtaining A are used. If the electrolyte is weakly dissociated, then the A can be obtained by application of the Ostwald dilution law, modifying it in precise work to correct for the interionic forces. 

The diffusion coefficients Du and D22 are the principal or "self diffusion coefficients and the off-diagonal quantities D12 and D21 are mutual diffusion coefficients. Even when Onsager s reciprocal relations (31) are valid for the appropriate flow equations so that D12 = D21, there are still three diffusion coefficients generally required to describe the diffusion process. It is noted that even if dC Jbx = 0, the flow of Component 1 is linked to that of Component 2 through the term — Di2dC2/dx, and is not zero. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Onsager s reciprocal relations state that, provided a proper choice is made for the flows and forces, the matrix of phenomenological coefficients is symmetrical. These relations are proved to be an implication of the property of microscopic reversibility , which is the symmetry of all mechanical equations of motion of individual particles with respect to time t. The Onsager reciprocal relations are the results of the global gauge symmetries of the Lagrangian, which is related to the entropy of the system considered. This means that the results in general are valid for an arbitrary process. 

Booth73 used Frolich s108 modification of the Onsager expressions101 for the cavity field in non-associated polar liquids, and corresponding modifications of Kirkwood s equation for associated polar media. Booth s assumptions in deriving Eq. (28) for the Kirkwood case are important to determine the validity of his final expressions. He used the Onsager-Frolich cavity field ratio... 

This set of equations obeying the Onsager reciprocity conditions obtains only near equilibrium. It is generally the case that the linear approximation for the reactions is valid over a far more limited range than is the linear approximation for the types of processes discussed in Sections 6.7-6.11. 

It is essential that the relations that are similar to the phenomenological Onsager reciprocal equations are also valid for many types of chemically reactive systems that are far from thermodynamic equilibrium (see Section 2.3.4). 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

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