Nernst-Einstein

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... 

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... 

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... 

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)... 

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. 

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Among them are Haber, Planck, Nernst, Einstein, Perutz, Bernal, Staudinger, Pauling, and Flory. A man molded by the largeness of his actions, accomplishments, and contacts, he has very little regard for parochial interests. These interests in his mind include national boundries. As a result he has shared his time and knowledge, as well as, the time and experience of those under his leadership, so freely that there are polymer science establishments in nearly all the industrialized nations which in some way or another owe a part of their existence to him. 

Proton conductivity, c7h+/ can be related through the Nernst-Einstein relationship to the activity of protons ( h+) in the membrane as well as to the mobility (Uh+) of those protons ... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

It may be shown (Ratner, 1987) that assuming the Nernst-Einstein relationship (Eqn (6.4)), a free volume expression for the conductivity in the form of Eqn (6.7) may be derived, provided the electrolyte is fully dissociated and... 

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... 

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... 

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. 

Baird et al. [350]). In the following analysis, the functional forms, p(E), which have been proposed (see below) to represent the field-dependence of the drift mobility are used for electric fields up to 1010Vm 1. The diffusion coefficient of ions is related to the drift mobility. Mozumder [349] suggested that the escape probability of an ion-pair should be influenced by the electric field-dependence of both the drift mobility and diffusion coefficient. Baird et al. [350] pointed out that the Nernst— Einstein relationship is not strictly appropriate when the mobility is field-dependent instead, the diffusion coefficient is a tensor D [351]. Choosing one orthogonal coordinate to lie in the direction of the electric field forces the tensor to be diagonal, with two components perpendicular and one parallel to the electric field. 

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. 

As our final task in this subsection, let us determine what the potential difference is between the layers of solution adjacent to the anode and the cathode (or, more exactly, between the outer regions of the double layers at the two electrodes, see Sect. 1.2). For this purpose, we turn to eqn. (63), set x — L, and replace D2/u2 by the Nernst—Einstein term RT/ z2 F. Thence... 

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... 

Equation 3.43, which expresses the link between the mobility and the diffusivity in this case, is known as the Nernst-Einstein equation. 

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