Equation Nemst-Einstein

The mobility ratio equal to the diffusion ratio in this equation would naturally follow from application of the Nemst-Einstein equation, Eq. (88), to transport gels. Since the Nemst-Einstein equation is valid for low-concentration solutes in unbounded solution, one would expect that this equation may hold for dilute gels however, it is necessary to establish the validity of this equation using a more fundamental approach [215,219]. (See a later discussion.) Morris used a linear expression to fit the experimental data for mobility [251]... 

On the other hand, the diffusivity of an ion, for example, Cu2+, is only known in the limit of infinite dilution where the Nemst-Einstein equation is... 

Substituting for the mobility using the Nemst-Einstein equation and the definition of the transport number... 

Making use of the Nemst—Einstein equation, expression (33) can be rewritten for the combined migration—diffusion flux as... 

The mobility, m, and diffusion coefficient, D, are related by the Nemst-Einstein equation ... 

The ionic resistance of a polymer electrolyte membrane is an important parameter in determining the mobility of protons through the membrane and the corresponding voltage loss across the membrane. Currently, the most commonly used membranes in PEM fuel cells are Nafion membranes produced by DuPont. However, these membranes are limited to low-temperature uses (usually below 80°C) because membrane dehydration at high temperatures can lead to reduced water content and then a lower proton transfer rate, resulting in a significant decrease in conductivity. The relationship between conductivity and the diffusion coefficient of protons can be expressed by the Nemst-Einstein equation ... 

Gottlieb MH and Sollner K (1968) Failure of the Nemst-Einstein equation to correlate electrical resistances and rates of ionic self-exchange across certain fixed charge membranes. Biophys J 8 515-35... 

The Nemst-Einstein equation connecting diffusion and partial equivalent ionic conductivity... 

The mobility /r, and hence the ioiuc conductivity, are directly related to the diffusion through the Nemst Einstein equation, iikT = ZeD, giving equation (4) ... 

Transient solutions for the Nemst-Einstein equation have been derived assuming a constant source concentration [69]. An approximate solution may be obtained when the decline in source concentration is small over the time period of the study by assuming (a) a constant iontophoretic permeability coefficient Fionp (b) the donor iontophoretic compartment is a well-stirred solution (c) sink conditions exist and (d) the solute concentration in the donor solution Q of volume changes at a rate equal to the total flux JjA out of the compartment into the outer layer of the epidermis ... 

The limits of integration are the oxygen partial pressures maintained at the gas phase boundaries. Equation (10.10) has general validity for mixed conductors. To carry the derivation further, one needs to consider the defect chemistry of a specific material system. When electronic conductivity prevails, Eqs. (10.9) and (10.10) can be recast through the use of the Nemst-Einstein equation in a form that includes the oxygen self-diffusion coefficient Dg, which is accessible from ionic conductivity measurements. This is further exemplified for perovskite-type oxides in Section 10.6.4, assuming a vacancy diffusion mechcinism to hold in these materials. 

A fully microscopic theory of chemical diffusion can be constructed, however, it requires a careful distinction between the motions of the observed species and the underlying host, and is made complicated by the fact that, as defined, the diffusion coefficient relates flux to the concentration gradient while the actual force that drives diffusion is gradient of the chemical potential. An alternative useful observable is the so-called conductivity diffusion coefficient, which is defined for the motion of charged particles by the Nemst-Einstein equation (11.69)... 

By combining Equations (3.67), (3.71), (3.77) and (3.78), it is possible to obtain a very important expression which is often referred to as the Nemst-Einstein equation ... 

The theory presented here resolves itself into a generalization of the well-known Nemst-Einstein equation D = 22T/C>) to several components and optional concentration characteristics. The cases of two- and three-component mixtures are treated in detail. The latter case is also shown to be of interest in treating self diffusion in a binary mixture, a system which results from letting two components become diffusionally identical although still distinguishable. 

According to Nemst-Einstein equation (e.g., Daniels and Alberty, 1955), the diffusivity is proportional to temperature for diffusion in dilute solution. The diffusivity at T = 60°C are about 4 times of the values at T = 15°C. The decrease of the effect aperture by 30% implies that the proportionality k in the linear relation is increased by 30%. We then have P = 3900r for 60°C. Four times increase of D implies only two times of increase of /rsince k -Jd... 

According to Nemst-Einstein equation, matter flux J caused due to the grainboundary diffusion is driven by stress and electromigration, which can be described... 

Following Nemst-Einstein equation, one may relate DC equivalent conductivity to ion diffusion coefficient ... 

Diffusion in liquid-filled pores occurs by essentially the same mechanism-as in gaseous systems. However, methods of correlation and prediction are less accurate since the fundamental theory of diffusion in the liquid phase is less well developed than the theory of molecular diffusion in the vapor phase. Correlations based on the Stokes-Einstein and Nemst-Einstein equations must be treated with caution. A wide range of empirical and semiempirical correlations is available but it is generally necessary to select the appropriate correlation with care, taking due account of the nature of the components. Predictive methods are at their best for mixtures of two nonpolar species and at their worst for mixtures of a polar and nonpolar species. 

Here D is known as the component diffusion coefQcient. The importance of this dehnition lies in the fact that Nemst-Einstein proportionality between a diffusion coefficient and a mobility has been retained, even though the condition of ideality has been relaxed. This is important since the apparent violation of the Nemst-Einstein equation in nonideal solutions is not a failure of the proportionality between mobility and mean displacement it is a weakness in the method of formulating the driving force for diffusion in terms of a concentration gradient (Pick s law) rather than in terms of an activity or chemical potential gradient. 

Self-diffusion coefficients of distinct mobile species measured using PFG-NMR are based on spectral selectivity. In the context of IL selfdiffusion, PFG-NMR measures the time-averaged (miUisecond timescale) diffusion coefficients. Since ion-pair (cation-anion) interactions take place on a timescale faster than that, the measured diffusion coefficients are a weighted average over charged and neutral species. This is the reason for the variation between conductivity calculated from diffiasion coefficients determined by the Nemst-Einstein equation [15] (Eq. 1) and conductivity measured using impedance analysis ... 

The mobility of electrons far exceeds that of the interstitial cations ( 2 Mi). The equation (9.43) can therefore be simplified to yield (9.44), where we used the Nemst-Einstein equation (4.127) to replace the mobilities by the diffusion coefficients. 

In diluted solutions the Nemst-Einstein equation holds for the relation between the diffusion coefficient and ionic mobility ... 

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