Nernst-Einstein relation

The oxide solid elecU olytes have elecuical conductivities ranging from lO Q cm to 10 cm at 1000°C and these can be converted into diffusion coefficient data, D, for die oxygen ions by the use of the Nernst-Einstein relation... 

The flux by diffusion is described by the diffusivity Di and the migration by the conductivity cr-. The conductivity is proportional to the product of the mobility and the concentration of the mobile species. The diffusivity and mobility are related by the Nernst-Einstein relation [3J. The flux is in general given by... 

The mobility of eh was determined by measuring the equivalent conductance following pulse irradiation (Schmidt and Buck, 1966 Schmidt and Anbar, 1969). After correcting for the contribution of H30 and OH ions, they found the equivalent conductance of eh = 190 10 mho cm2. From this, these authors obtained the mobility p(eh) = 1.98 x 10"3 cm2/v.s. and the diffusion coefficient D(eh) = 4.9 x 10-5 cm2/s using the Nernst-Einstein relation, with about 5% uncertainty. The equivalent conductance of eh is the same as that for the OH - ion within experimental uncertainty. It is greater than that of the halide ion and smaller than that of eam... 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

Breaking and reforming of such cross-links relate to the renewal time Tr of the DDH model [321] of the conductivity. Here the mean squared displacement without renewal events saturates after a short time to a value of r (°°)) until a restructuring establishes the start of a new diffusion step. Then D(0)=(r (oo))/(6TR) and via the Nernst-Einstein relation ... 

Equation (6.41) is known as the Nernst-Einstein relation, originally deduced for the mobility of colloid particles in a liquid, but also valid for ionic solids. 

Nernst-Einstein relation D show subtle differences because ionic motion in tracer experiments is correlated. The Haven ratio, is proportional to the correlation... 

In order to evaluate this expression, we need to know the force v / that is responsible for producing the molecular flux. It could be an external force such as an electric field acting on ions. Then evaluation of Eq. 18-48 would lead to the relationship between electric conductivity, viscosity, and diflusivity known as the Nernst-Einstein relation. 

Equation (5.56) relates the correlation factor fA with the cross coefficient LAA . From the Nernst-Einstein relation we know that LAA = bA-cA = DAcA/R T. For a tracer experiment with a negligible fraction of A, the jump conservation requires that Da = Dv-Nv, so that instead of Eqn. (5.56) we have... 

The Nernst-Einstein relation shows the dependence between the self-diffusion coefficient Dt and the equivalent conductivity A of molten salts ... 

There are numerous data in the literature [59] which demonstrate that for molten halides the equivalent conductivity calculated by means of the Nernst-Einstein relation is significantly higher than the directly measured conductivity value. This is due to the fact that the structural entities of molten salts make unequal contributions to diffusion and electrical conductivity. 

The Einstein relation also permits experiments on diffusion to be linked up with other phenomena involving the mobility of ions, i.e., phenomena in which there are forces that produce drift velocities. Two such forces are the force experienced by an ion when it overcomes the viscous drag of a solution and the force arising from an applied electric field. Thus, the diffusion coefficient may be linked up to the viscosity (the Stokes-Einstein relation) and to the equivalent conductivity (the Nernst-Einstein relation). 

There were several aspeets of the Stokes-Einstein relation that reduced it to being only an approximate relation between the diffusion coefficient of an ionic species and the viscosity of the medium. In addition, there were fundamental questions regarding the extrapolation of a law derived for macroscopie spheres moving in an incompressible medium to asituation involving the movement ofions in an environment of solvent molecules and other ions. In the case of the Nernst-Einstein relation, the factors that limit its validity are more subtle. 

Test of the Nernst-Einstein Relation for Equivalent Conductivity of Molten... 

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. 

Mass transport in ceria as well as in other fluorite-related materials is several orders of magnitude faster for anions than for the metal . Measurements of bulk oxygen diffusion are relatively scarce in the literature and they arc summarised in Table 2.6. Alternatively, values can be estimated from ionic conductivity data using the Nernst-Einstein relation ... 

The expression a = nDI2/3 for the ionic conductivity a follows directly from the comparison with the phenomenological theories [2, 8] and indeed gives the well-known Nernst-Einstein relation connecting the conductivity with the mutual diffusion coefficient. Note that D = lirri, , 0 D(k, z) is the hy-... 

The ionic mobility ji and diffusion constant D satisfy the Nernst-Einstein relation kT = qD, where T is the temperature, q the ionic charge, and k the Boltzmann constant. Since conductivity a is equal to Nqii, with N the density of current carriers, it follows that... 

The validity of the Nernst-Einstein relation rests on the fact that the driving force for both migration and diffusion is the gradient... 

Diffusivity, Mobility and Conductivity The Nernst-Einstein Relation... 

If the electrical conductivity of a substance is the result of field-induced diffusion of a single ionic species, the electrical conductivity, diffusion coefficient, D, of that species via the Nernst-Einstein relation ... 

The mechanism leading to this behavior has been the subject of many studies, with at least 10 suggested explanations. Examination of the diffusion coefficients of sodium and potassium in such a series of glasses reveals that each coefficient decreases monotonically as the concentration of the other alkali ion increases. The cross-over in the diffusivities occurs near the minimum in conductivity, and the calculated conductivity using the Nernst-Einstein relation is near the experimental value. The cause of the trends in the diffusivity remains uncertain at this time. 

---