Onsager’s equation

Diffusion of the electroactive species within the electrode toward or away from the interface with the electrolyte is an irreversible process. The sum of the products of the forces and fluxes corresponds to the entropy production. In order to avoid space charge accumulation, the motion of at least two types of charged species has to be considered for charge compensation. Onsager s equations read in the isothermal case (neglecting energy fluxes)... 

L length of separation path column or tube length Lkj coefficient in Onsager s equation of reciprocity L mean external length of molecule or colloid m mass... 

The value of the fraction representing the ratio of the conductances of two differently concentrated but fully dissociated solutions can be calculated from Onsager s equation (see III-14) which enables us to determine the effect of electrostatic forces of attraction in strong, i. e. fully dissociated electrolytes. In the case of weak electrolytes, however, it is necessary to substitute ct in Onsager s equation by the real concentration of ions, i. e. by equivalent conductance of a hypothetical, fully dissociated solution is considered. In this way we obtain the following equation ... 

In the case of strong electrolytes with a = 1 the last, mentioned equation changes to the form of Onsager s equation, while for very weak electrolytes or substantially diluted solutions (o, si 0, /A 1) it changes to the classical formula of Arrhenius. If we consider the problem of electrolytes from the point of view of the equation (III-31) we can see that there is no sharp dividing line between strong and weak electrolytes. It can equally be seen that the Debye-Hiickel-Onsager theory does not replace the theory of Arrhenius but merely corrects and suitably supplements it. 

The fact that the type of deviation from Onsager s equation under discussion is not observed, at least up to relatively high concentrations, with many simple electrolytes, e.g., the alkali halides in both aqueous... 

Constant in Onsager s equation (Equation 17, Chapter 18), degree of association... 

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. 

Although this equation is empirical it reduces to Onsager s equation for very small values of c, and the terms Be and Dc log c are of the form required by consideration of factors neglected in the derivation of the simple equation.7 17 The utility of an equation of the form of (32) is shown in Table II in which the conductance data of Shed-... 

Another method, also based on the assumption that Onsager s equation is the true limiting relation, depends upon Shedlovsky s equation, which for uni-univalent electrolytes takes the form... 

The Conductance of Strong Electrolytes in Methyl and Ethyl Alcohol. Careful studies of the conductances of electrolytes in methyl and ethyl alcohol have been carried out by Hartley and associates.3 A plot of the equivalent conductance, A, values for a series of sulplio-cyanates in methyl alcohol as functions of the square root of the concentration are given in Pig. 1 It will be seen that the plots are all straight lines as required by Onsager s equation for uni-univalent electrolytes, equation (18), Chapter 18. Since methyl alcohol at 25° has a dielectric constant of 31.5 4 and a viscosity of 0.00545 poise, that equation takes the form ... 

However, although the results of the conductance measurements follow an equation of the form A = A0 — KyJc the value of the constant K does not in every case correspond to that predicted by Onsager s equation. A comparison of the observed values of AT, (the slopes of the lilies in plots similar to Fig. 1) and the computed values from equation (1) are given in Table I, Although the data for a number of the salts agree with the predictions of the theory almost within the experimental error, pronounced deviations are observed particularly for cesium salts and some of the nitrates. However, as mentioned in Chapter 18, nitrates... 

Table I. Test of Onsager s Equation for Some Uni-Univalent Salts in Methyl Alcohol at 25°...

The value of K is, in most cases, considerably greater than that predicted by Onsager s equation which for ethyl alcohol at 25° takes the form8... 

More recently, Harris and Alder,24 keeping the general principles of Kirkwood s theory, have tried to calculate the polarization effects more rigorously. Unfortunately their final equation does not coincide as it should, with Onsager s equation when it is assumed that there are no short-range interactions cos y) — 0). This is because some of Kirkwood s equations are only valid when the assumptions of the author are justified, and cannot be used as was done by Harris and Alder, when a deformation polarization is superimposed on the orientation polarization. For instance, in presence of deformation effects boundary conditions cannot be introduced in the same manner as in Kirkwood s model (cf. Frdhlich 21). 

Onsager s equation has been used for slightly polar solvents such as toluene. With strongly polar solvents, chloroform for instance, Kirkwood s or Frohlich s theories must be resorted to, and no value of the dipole moment can be obtained unless the correlation factor g is known by independent data concerning the structure of the solution. 

Incomplete dissociation of electrol3rtes of high valence tj pe has been widely postulated to account for negative de dations from Onsager s equation and semi-empirical extensions of it (see Davies, Endeavour, 1945,4,114). In many cases independent values of are not available, so that both and K have to be determined from the same data. 

If the polarization vector P is first calculated with these constants and the result substituted in Eq. (11.20) the well-known Onsager s equation for the permittivity is obtained ... 

It is well established that the second Wien effect is particularly large in linear polyelectrolytes/ Compared to simple electrolytes the linear range of the conductivity increase with increasing electric fields starts already at relatively low field strengths (5-10 kV cm" ). Because of the rather extended linear region Onsager s equation for the dissociation field effect has been used as the basis for a qualitative discussion of the second Wien effect in polyelectrolytes. 

All principles, cited above, are equivalent to one another and to Onsager s equations, coimecting forces and fluxes by a symmetrical matrix of kinetic coefficients. They differ in the parameters chosen as variable and fixed. In this sense, extremum principles do not present any peculiarly new results. However, there exists at least one extremum principle, which, though not proven strictly so far, long ago became a powerful heuristic means of forecasting the system s evolution in material... 

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