Nernst-Einstein equations

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. 

The conductivity of ionic liquids can be modeled in the same manner as the viscosity, i.e. despite the high ionic strength of the liquid, ionic migration is limited by the availability of suitably sized voids [130]. Since the fraction of suitably sized holes in ambient temperature ionic liquids is effectively at infinite dilution, migration should be described by a combination of the Stokes-Einstein and Nernst-Einstein equations. This is explained in greater detail in Chapter 11.3 on process scale-up but it is sufficient to say that an expression can be derived for the conductivity, k... 

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. 

The type and concentration of defects in solids determine or, at least, affect the transport properties. For instance, the -> ion conductivity in a crystal bulk is usually proportional to the -> concentration of -> ionic charge carriers, namely vacancies or interstitials (see also -> Nernst-Einstein equation). Clustering of the point defects may impede transport. The concentration and -> mobility of ionic charge carriers in the vicinity of extended defects may differ from ideal due to space-charge effects (see also - space charge region). 

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] ... 

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

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. 

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 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 ... 

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

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. 

---