Nernst-Einstein diffusion equation

The relationship between the ion diffusion and ionic conductivity has been studied by the Nernst-Einstein (ME) equation (5) for molar conductivity 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... 

The results attaching to the steady state may be deduced as special cases of the equations developed in Sect. 4.4, but it is instructive to develop these results ab initio for this so-called unsupported case. To begin, we shall not even assume the Nernst—Einstein relationship (Sect. 2.5) between the diffusion coefficient and mobility. 

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. 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

Nernst-Einstein equation - Nernst derived the relationship between the - diffusion coefficient Dr a of 1 1 electrolyte and the mobilities of the individual ions (nK, uA) [i] ... 

Solid electrolyte — is a class of solid materials, where the predominant charge carriers are -> ions. This term is commonly used for -> conducting solids with ion -> transport number equal to or higher than 0.99 (see also -> electrolytic domain). Such a requirement can only be satisfied if the -> concentration and -> mobility of ionic -> charge carriers (usually -> vacancies or interstitials) both are relatively high, whilst the content of -> electronic defects is low. See also -> superionics, -> defects in solids, - diffusion, and -> Nernst-Einstein equation. 

The interpretation of the pre-factor as a conductivity as well as the correlation between defect diffusion coefficient Dk and mobility uk known as Nernst-Einstein equation follow directly from Table 3. 

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 ... 

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... 

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. 

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. 

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