Nemst-Einstein relation

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... 

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... 

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. 

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... 

When this is introduced into Eq. (32) we obtain by analogy with the Nemst-Einstein relation O = RTJD... 

We have above introduced concentrations and relations between concentrations of species of interest The area-specific flux (flux density) of a species, i, resulting from a driving force, F, is proportional to its concentration and to its mechanical mobility (ease of movement) j j = Cj Bj Fj as we come back to in the next section. First we briefly recall from textbooks that for species with an activated diffusion, the self-diffusion coefficient, D, mechanical mobility, B, charge mobility, u, and conductivity, a, are linked through the Nemst-Einstein relation (1.14) ... 

The quantity RTUi is called the diffusion coefficient (Nemst-Einstein relation)... 

Here u is the mobility, ix the chemical potential, and y the activity coefficient of the mobile species. The denotes that the relations are written for a neutral species. Now A is the component diffusion coefficient which, as pointed out in Section 2.1.2.4, obeys the Nemst-Einstein relation... 

OxXdf holds, the ion density is obtained by using the Nemst-Einstein relation (D = T/e, where kT is the thermal energy and e the unit charge) as... 

The flux by diffusion is described by the diffusivity and the migration by the conductivity mobile species. The diffusivity and mobility are related by the Nemst-Einstein relation [3]. The flux is in general given by... 

As long as the electrons remain thermal, D j is related to the electron mobility via the Nemst-Einstein relation (see Chapter 1, Equation 77). 

This relation is called the Nemst-Einstein relation. This relation and also the effect of an apphed electric field on migration of charged species in a homogeneous crystal may also be derived from the following model, in which we will understand also when and why conduction is termed a hnear process. 

From the Nemst-Einstein relation it is also seen that the temperature dependence of the product aiT is the same as that of Df. Thus in evaluating the activation energy associated with the diffusion coefficient from conductivity measurements, it is necessary to plot (aiT) vs 1/T. 

However, in interstitialcy diffusion the charge displacement is larger than the atom jump distance, and a displacement factor S must be included in the Nemst-Einstein relation. In colhnear interstitialcy diffusion (Fig. 5.9) the effective charge is, for instance, moved a distance twice that of the tracer atom and Dt/Dr is given by... 

The multitude of transport coefficients collected can thus be divided into self-diffusion types (total or partial conductivities and mobilities obtained from equilibrium electrical measurements, ambipolar or self-diffusion data from steady state flux measurements through membranes), tracer-diffusivities, and chemical diffusivities from transient measurements. All but the last are fairly easily interrelated through definitions, the Nemst-Einstein relation, and the correlation factor. However, we need to look more closely at the chemical diffusion coefficient. We will do this next by a specific example, namely within the framework of oxygen ion and electron transport that we have restricted ourselves to at this stage. 

Eq. 7.4 is written with conductivity as the transport coefficient. Flux of a neutral species is not affected by the electrical potential gradient and can not be expressed by conductivity. It is therefore meaningfiil to express the equation using the random diffusion coefficient for the chemical potential gradient and the conductivity for the electrical potential gradient. Do this splitting/substitution. (This is mainly a simple exercise in the Nemst-Einstein relation). Check that both parts of the resulting equation will have units that confer with flux density. 

Two diffusion coefficients are of interest in MIECs the component diffusion coefficient, Dk, and the chemical diffusion coefficient, D. The component diffusion coefficient reflects the random walk of a chemical component. It is therefore equal to the tracer diffusion coefficient, except for a correlation factor which is of the order of unity. It is also proportional to the component mobility as given by the Nemst-Einstein relations. The chemical diffusion coefficient, I), reflects the transport of neutral mass under chemical potential gradients. In MIECs mass is carried by ions, and transport of neutral mass occurs via ambipolar motion of ions and electrons or holes so that the total electric current vanishes. b can be determined from steady-state permeation measurements, as mentioned in Section IV.H. However, D is usually determined from the time dependence of a response to a step change in a parameter, e.g., the applied current. Alternatively, D is determined from the response to an ac signal applied to the MIEC. 

This is known as the Nemst-Einstein relation. For charged particles i we should also take into consideration the migration due to the diffusion potential ( ). In other words, we are now interested in the change in Equations (15.4) and (15.5) if the moving particles (i) are charged. The electrochemical potential r of the particles i is given by ... 

Substituting the Nemst-Einstein relation Equation (15.6) into Eqnation (15.12) gives ... 

The temperature dependence of conductivity (see Sect. 2.5) depends on temperature in the same way as the diffusion coefficient of ions. Often it is convenient to estimate actual values of D by means of its relation to conductivity. Two fundamental equations describe this relation. It is Eq. (2.19), the Einstein relation (with u the molar ion mobility and F the Faraday constant), and Eq. (2.20), the Nemst-Einstein relation (with 2 the molar ion conductivity). 

The conductivity of molten salts has been related to the existence of free volume in the melt [268] and it was argued that the Arrhenius activation energy Ba should be lower than the corresponding one for ion diffusion in the melt (see below), Bd as was in fact found. This would explain why the conductivity does not adhere to the Nemst-Einstein relation A = F D+ + D-)/RT for the diffusion or to Stokes law as mentioned above. The significant structure theory in this case [160] specifies that only the solid-like particles contribute to the conductivity. Their number per unit volume is where Fsd is the molar volume of the solid... 

The mobility Uj of ion j in the electrolyte is an important parameter related to the diffusion coefficient Dj through the Nemst-Einstein relation [1] ... 

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