Diffusion coefficient and mobility

As already stated, the diffusion coefficient and mobility of a solvated electron, though much smaller than that of a free electron in a conduction band, are some 4-5 times larger than for the solvated cation. There is some doubt whether the electron carries along a solvated shell with it, and there has been discussion in the literature of mechanisms by which molecules drift across the cavity, changing their orientation as they do so. 

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. 

We remember that minority point defect concentrations in compounds depend on the activity of their components. This may be illustrated by the solubility of hydrogen in olivine since it depends on the oxygen potential in a way explained by the association of the dissolved protons with O" and O- as minority defects [Q. Bai, D. L. Kohlstedt (1993)]. Similarly, tracer diffusion coefficients and mobilities of Si and O are expected to depend on the activity of Si02. The value (0 lnDf/0 In aSio2)> = Si and O, should give information on the disorder type as discussed in Section 2.3. 

A comparison of the results of the exact calculation and the approximate one is shown in Figure 1 for the values c = 5 X 10 3, r = 5. The error is fairly small, and the relative error is generally less than 5%. There is no great advantage to using the perturbation theory for the example just discussed because an exact solution is available. However, a gel for which the diffusion coefficients and mobilities are... 

This is the Einstein relation, and shows the direct proportionality between diffusion coefficient and mobility. 

Einstein pointed out that the same random forces that produce Brownian motion are also in operation when the particle is dragged through the medium. Whereas diffusion is characterized by the diffusion coefficient, drag is usually quantified in terms of a mobility ju, a proportionality constant between the applied force F and the terminal velocity v, where v = fiF. In fact, the diffusion coefficient and mobility are related by... 

The ion diffusion coefficient and mobility are related by the Nemst-Einstein equation ... 

If diffusion is ambipolar, which is usually the case at higher ionization degrees, relation (3-90) can be applied to find Da in a magnetic field. Obviously, free diffusion coefficients and mobilities for electrons and ions should be replaced by those in the magnetic field (see (3-267) and (3-77)) ... 

The Einstein relation between diffusion coefficient and mobility is assumed to apply for the motion of orientational defects but may not apply for the quantum tunnelling motion of ion states. (It should be noted in passing that some workers define with a sign opposite to that used here.)... 

The absolute magnitude of the calculated value of Qa depends on the way in which the relation between diffusion coefficient and mobility for ion states differs from the classical Einstein form. If we introduce parameters 0+ and 6 and write... 

The diffusion coefficient and mobility of a particle are linked together. As has been shown in Section 4.4, an estimation D Uik holds for extremely diluted... 

The aim here is not to go into thorough detail about mass transport in porous materials. We will merely describe these phenomena in macroscopic terms using the effective mass transport parameters (diffusion coefficients and mobilities) denoted by the index m and adapted to the material s macroscopic geometry. 

In the absence of any specific properties (which is wholly opposite to the case of selective membranes) these effective parameters can sometimes be related to microscopic parameters, such as the porosity and tortuosity in the materials used. Assuming that the membrane is not ion-selective with respect to the different ions, one can consider the transport numbers to be constant in all the electrolytic media. In the examples outlined below (see sections 4.4.2.2 and 4.4.2.3), it is assumed that the effective diffusion coefficients and mobilities are significantly lower than in the free electrolyte. This is due for exampie to a low porosity in the material used. Note that if these particular experimental conditions are chosen, then it implies that there is a significant ohmic drop in the eiectroiysis cell, which requires high electrolysis voltages (typically over 10 V). These conditions are not suitable for industrial applications. 

The diffusion coefficients and mobilities of OH anions are less than that of... 

J is the electronic transfer integral which describes the overlap of the electron trap state with the empty neighboring state. Making use of the relation between diffusion coefficient and mobility, the mobility Pg, of the small polaron is then given as... 

The energy distribution functions in liquid argon and liquid xenon are shown in Figure 16. These results should serve only as illustrative examples. The precision and accuracy of the calculations depend critically on the input data for Aq and A (see Gushchin et al, 1982). Aq can be taken as constant, while Aj is extracted from mobility measurements. The mobility data available in the literature show considerable scatter (see Chapter 3). A direct estimation of the electron mean energy comes from the measurement of the transversal diffusion coefficient. The ratio of diffusion coefficient and mobility is a measure of the mean energy. 

Table III. Diffusion coefficients and mobility selectivities of polyimide films ... 

Here tb is a relaxation time for the Brownian velocities, and rg is far smaller than t, even in the nominal t 0 limit of Eq. 4.25. Diffusion coefficients and mobility tensors are related by a generalized Einstein relation... 

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