Nemst-Einstein

Equation 11 gives the conductivity for a particular ion having a transference number, in a crystal, which is the Nemst-Einstein relationship. 

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]... 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

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... 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

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... 

Several famous equations (Einstein, Stokes-Einstein, Nemst-Einstein, Nernst-Planck) are presented in this chapter. They derive from the heyday of phenomenological physical chemistry, when physical chemists were moving from the predominantly thermodynamic approach current at the end of the nineteenth century to the molecular approach that has characterized electrochemistry in this century. The equations were originated by Stokes and Nernst but the names of Einstein and Planck have been added, presumably because these scientists had examined and discussed the equations first suggested by the other men. 

The observed conductivity is always found to be less than that calculated from the sum of the diffusion coefficients (Table 5.27), i.e., from the Nemst-Einstein relation [Eq. (5.61)]. Conductive transport depends only on the charged species because it is only charged particles that respond to an external field. If therefore two species of opposite charge unite, either permanently or temporarily, to give an uncharged entity, they will not contribute to the conduction flux (Fig. 5.34). They will, however, contribute to the diffusion flux. There will therefore be a certain amount of currentless diffusion, and the conductivity calculated from the sum of the diffusion coefficients will exceed the observed value. Currentless diffusion will lead to a deviationfrom the Nernst-Einstein relation. 

Possible Molecular Mechanisms for Nemst-Einstein Deviations.. . ... 

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. 

Equation (10.19) may be simplified for predominant electronic conduction, assuming that the classical Nemst-Einstein relationship can be represented as... 

It is interesting to ask the question, how high a conductivity can be achieved in FIC glasses. From the Nemst-Einstein relation, a is given by ne DIkT, where n is the carrier concentration. The maximum value of D, which can be realized in glasses is about 2-5 x 10" cm sec, (this value is deliberately assumed to be smaller than the D of alkali ions in silicate melts). The maximum value of n can be similarly estimated as 2-5 x 10 cc. The maximum value of cr thus works out to be 2.4-15 Scm (at... 

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)... 

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