Nernst-Einstein relationship

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... 

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. 

The Nernst-Einstein relationship plotted in Fig. 36.1 motivated the development of DiagnoSwiss technology, namely reducing the analysis time and reaction volumes by replacing the commonly used microtitre plate wells with typical dimensions of 100 pL volume and 1 cm2 surface area by a microchannel with a volume of 60 nL and a surface area of 0.03 cm2. The microchannel is an isotropically etched microstructure (see Section 36.2) with a minimal cross-section dimension of 40 pm,... 

Figure 36.1. Nernst-Einstein relationship representing the time for a protein of 14 kD to diffuse through different distances (diffusion coefficient 10-lom2/s).

Generally, ions can be transported by three mechanisms diffusion, conduction and convection. To / z F these transport components contribute by -DVc, -D z Fc Vyf /RT and c v, respectively, where according to the Nernst-Einstein relationship, see 14.3.55), the ion mobility is written as I z l FD / FT. In Volume I we derived the Nemst-Planck equation for the corresponding fluxes, see 11.6,7.1 and 2] and 13.13.12). In the present context we write the Nemst-Planck equation as... 

Using the Nernst-Einstein relationship and converting all units to SI units, one obtains the following expression for the conductivity ... 

Substituting for cr ef, using the Nernst-Einstein relationship [Eq. (7.34)], and integrating gives... 

Experimental results have been used to obtain averaged activity coefficients.60 Another approach toward characterization of degenerate semiconductors has been to include the nonidealities associated with degeneracy within a modified Nernst-Einstein relationship.61"64 The modified Nernst-Einstein relationship is given by65... 

D and u are related by the Nernst-Einstein relationship. The conductivity a is given by Equation 19.25... 

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