The Nernst-Einstein Equation

Now the Einstein relation (4.172) will be used to connect the transport processes of diffusion and conduction. The starting point is the basic equation relating the equivalent conductivity of a z z-valent electrolyte to the conventional mobilities of the ions, i.e., to the drift velocities under a potential gradient of 1 V cm,  

By using the relation between the conventional and absolute mobilities, Eq. (4.163) can be written 

This is one form of the Nernst-Einstein equation from a knowledge of the diffusion coefficients of the individual ions, it permits a ealeulation of the equivalent eondue-tivity. A more usual form of the Nernst-Einstein equation is obtained by multiplying numerator and denominator by the Avogadro number, in which case it is obvious that 

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Substituting for the mobility using the Nernst-Einstein equation and die deh-nition of die naiisport number... 

Furdiertiiore, using the Nernst-Einstein equation to substimte in the general equation above yields... 

Note The values in the parentheses are experimental results.A j is the deviation from the Nernst-Einstein equation expressed by K= (F2/VmRTXz+D+ 1 - A ). 

The ratio of the diffusion coefficient and the electrolytic mobility is given by the Nernst-Einstein equation (valid for dilute solutions)... 

According to the Nernst-Einstein equation (Nl, El), the diffusion of a single particle or solute molecule A through a medium B may be described by the relation... 

Equation 3.43, which expresses the link between the mobility and the diffusivity in this case, is known as the Nernst-Einstein equation. 

In spite of the high ionic conductivity, there is no guarantee that the IL can transport the desired ions such as metal ions or protons. It is therefore important to analyze the ion transport properties in ILs. The ion conduction mechanism in ILs is different from that in molecular solvents. The ionic conductivity is generally coupled to carrier ion migration and ionic conductivity (a) correlates to diffusion coefficient (D) according to the Nernst-Einstein equation (see Eq. (3.10)) where n and q imply the number of carrier ions and electric charge, respectively. R, T, and F stand for the gas constant, the temperature in K, and the Faraday constant, respectively. 

Diffusion coefficients (Dimp) obtained from measurement are calculated via the Nernst-Einstein equation. Furthermore, electrochemical diffusion coefficient measurements are possible which directly measure the diffusion coefficient. The degree of dissociation of a component ion in the IL can be estimated from the relation (DNMR/Amp) between Dimp and the diffusion coefficient measure by PFG-NMR (Dnmr) [132], This parameter is called the Haven ratio and should be unity... 

This fundamental expression for D is the Nernst-Einstein equation. It shows still another dimension to the importance of friction coefficient /, which is now seen to control diffusion as well as nearly all other transport. 

Further development results in the Nernst-Einstein equation... 

Bz particles is constant, this equation transforms into Ohm s law. When the electrical potential is constant, this equation may be transformed into the first -> Fick s law using the Nernst-Einstein equation. 

The microscopic mechanisms for ionic conduction are the same as those for atomic diffusion, namely, the vacancy and interstitial models discussed in the previous section. In fact, the diffusivity can be related to the conductivity via the Nernst-Einstein equation ... 

The Nernst-Einstein equation is applied to calculate molar conductivity (A mr) from the PGSE-NMR diffiision coefficients ... 

An implicit but principal requirement for the Nernst-Einstein equation to hold is that the species involved in diffusion must also be the species responsible for conduction. Suppose now that the species M exists not only as ions but also as ion pairs of the type described in Section 3.8.1. 

In systems where ion-pair formation is possible, the mobility calculated from the diffusion coefficient =D/kT is not equal to the mobility calculated from the equivalent conductivity u yZieo = (A/ZjeQ)F and therefore the Nernst-Einstein equation, which is based on equating these two mobilities, may not be completely valid. In practice, one finds a degree of nonapplicability of up to 25%. 

Another important limitation on the Nernst-Einstein equation in electrolytic solutions may be approached through the following considerations. The diffusion coefficient is in general not a constant. This has been pointed out in Section 4.2.3, where the following expression was derived. 

The above argument brings out an important point about the limitations of the Nernst-Einstein equation. It does not matter whether the diffusion coefficient and the equivalent conductivity vary with concentration to introduce deviations into the Nernst-Einstein equation, D and A must have different concentration dependencies. The concentration dependence of the diffusion coefficient has been shown to be due to nonideality (f 1), i.e., due to ion-ion interactions, and it will be shown later that the concentration dependence of the equivalent conductivity is also due to ion-ion interactions. It is not the existence of interactions perse that underlies deviations from the Nernst-Einstein equation otherwise, molten salts and ionic crystals, in which there are strong interionic forces, would show far more than the observed few percent deviation of experimental data from values calculated by the Nernst-Einstein equation. The essential point is that the interactions must affect the diffusion coefficient and the equivalent conductivity by different mechanism and thus to different extents. How this comes about for diffusion and conduction in solution will be seen later. 

In solutions of electrolytes, the terms in the expressions for D and A tend to zero as the concentration of the electrolytic solution deaeases, and the differences in the concentration variation of D and A become more and more negligible in other words, the Nernst-Einstein equation becomes increasingly valid for electrolytic solutions with increasing dilution. 

This explanation is all very well for the liquid sodium chloride type of case, but deviations from the predictions of the Nernst-Einstein equation occur in dilute aqueous solutions also, and here the + and - ions are separated by stretches of water, and ion pairs do not form significantly until about 0.1 M. 

Because deviations from the Nernst-Einstein equation are so widespread, and because the reasoning that gives rise to the equation is phenomenological, it is better to work out a general kind of noncommittal response—one that is free of a specific model such as that suggested in the molten salt case (see Section 5.2). The response... 

In Section 5.6.6, one will see the details of the modeling treatment of the deviations for the Nernst-Einstein equation outlined here. Of course, a modeling explanation is more enlightening than a general-explanation type of approach. However, the difficulty is that the model of paired ions jumping together applies primarily to a pure liquid electrolyte, or alloy, where the existence of paired vacancies is a fact. Other models would have to be devised for other kinds of systems where deviations do occur (Fig. 4.66). 

The self-diffusion coefficients of CF and Na" in molten sodium chloride are, respectively, 33 x 10 exp(-8500// 7) and 8x10 exp(-4000// 7) cm s". (a) Use the Nernst-Einstein equation to calculate the equivalent conductivity of the molten liquid at 935°C. (b) Compare the value obtained with the value actually measured, 40% less. Insofar as the two values are significantly different, explain this by some kind of structural hypothesis. 

The Nernst-Einstein reiation can be tested by using the experimentally determined tracer-diffusion coefficients D,. to calcuiate the equivalent conductivity A and then comparing this theoreticai vaiue with the experimentally observed A. It is found that the vaiues of A caicuiated by Eq. (5.61) are distinctly greater (by 10 to 50%) than the measured values (see Table 5.27 and Fig. 5.33). Thus there are deviations from the Nernst-Einstein equation and this is strange because its deduction is phenomenological. ... 

The Na" and Cr ions that make coordinated jumps into paired vacancies, i.e., the NaCl species, contribute to diffusion but not to conduction since such a coordinated pair is effectively neutral. Hence, the Nernst-Einstein equation is only applicable to the ions that jump independently, i.e.. 

Radiotracer Method of Calculating Transport Numbers in Molten Salts. In the discussion of the appiicabiiity of the Nernst-Einstein equation to fused salts, it was pointed out that the deviations could be ascribed to the pairedjump of ions resuiting in a currentiess diffusion. With fused NaCI as an example, it has been shown that there is a simpie reiation between the experimentally determined equivaient... 

The determination of transport numbers in aqueous electrolytes is relatively easy (Chapter 3), but in molten salts it poses difficulties of concept, which in turn demand specialized apparatus. Explain why direct determination is difficult. Would it not be better to abandon the direct approach and use the approximate applicability of the Nernst-Einstein equation, relying on self-diffusion determinations Any counter considerations ... 

In deviations from the Nernst-Einstein equation in a molten salt, one hypothesis involved paired-vacancy diffusion. Such a model implies that holes of about twice the average size are available at about one-fifth the frequency of averagesized holes. Use the equation in the text for the distribution of hole size to test this model. 

The conductance calculated from the Nernst-Einstein equation is several tens of percentages more than that measured. An interpretation is that the diffusion coefficient includes contributions fromjumps into paired vacancies and these (having no net charge) would contribute nothing to the conductance while counting fully for the diffusion. 

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