Einstein relation ionic mobility

According to (1.3b), the nonconvectional electro-diffusion flux component j. is a superposition of the following two terms. The first is the diffusional Fick s component proportional to the concentration gradient VC. The second is the migrational component, proportional to the product of the ionic concentration Cj and the electric force —ZiFV

proportionality factor. Einstein s equality (1.3c) relates ionic mobility to diffusivity >. ... 

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. 

Ionic mobility — Quantity defined by the velocity of an ion moving in a unit electric field (SI unit m2 V-1 s-1). The ionic mobility of ion i (uf) is related to its molar ionic conductivity (A ) by A = zfFui, where Z is the charge number of the ion. The ionic mobility is also related to the -> diffusion coefficient (A) by the Nernst-Einstein... 

Einstein presented an equation relating the diffusion coefficient of species i (Df to the frictional coefficient in a diffusion process [ii], this can be transformed into a relationship between -> ionic mobility iq and -> diffusion coefficient D ... 

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 concepts of ionic mobility ui, m /s-V) and effective ionic mobility ( f, m /s-V) are introduced as representative parameters of electromigration (ionic migration). The effective ionic mobility defines the velocity of the ionic species under the effect of a unit electric field, which can be theoretically estimated using the Nemst-Townsend-Einstein relation (Holmes, 1962). Ionic mobility is related to the ionic valence (z,) and molecular diffusion coefficient (Z) m /s) of species as follows ... 

The movement of ions toward the oppositely charged electrode is called electromigration, which is quantified by the effective ionic mobility. The effective ionic mobility (f/f) is defined as the velocity of ion within the pore space under the influence of a unit electrical potential gradient. The Nernst-Einstein equation is used to relate the ionic mobility to the diffusion coefficient of the ion in a dilute solution (Koryta, 1982) as follows ... 

In diluted solutions the Nemst-Einstein equation holds for the relation between the diffusion coefficient and ionic mobility ... 

As is well known, the mobility of charge carriers and, in particular, the ionic mobility /x is related to the diffusion coefficient D by Einstein s relationship... 

This general relation was obtained by Einstein and is sometimes called the Einstein relation. For ionic systems, as we have seen in section 10.1 (10.1.5), Tfc = Fzk = eNxZk and F = ezkhk- Since R — k Nwhere is the Boltzmann constant (= 1.381 x 10 JK ) and Np, the Avogadro nnmber, equation (10.3.18) for the ionic mobility Ffc becomes... 

In addition, the intrinsic diffusivity of a species i is related to the ionic mobility. This relationship is given in Chapter 4 as the Nemst-Einstein equation defined by... 

In polar liquids, ionic mobilities have been extensively studied, for instance by conductometry. They are known for many solvents (Robinson and Stokes, 1959 Janz and Tomkins, 1972). In low polar liquids, K has been deduced most often from transit time measurements, after irradiation of a thin layer of liquid or photoexcitation of the cathode, or after applying a voltage step. K values are now available in different liquids (Tables 1 and 2) but the nature of the corresponding ion is often unknown. So, to estimate the value of the mobility of a given ion in a liquid, the relation, Kt) = e/6nr, between K and the liquid viscosity Vi (Stokes-Einstein relation) is often used, where r is the hydrodynamic radius of the ion. In fact, this relation is only a crude approximation even for rather large ions, since the viscous forces are not the only retarding force acting on the ion, and the solvent is... 

This result is called the Nernst-Einstein relation it relates the diffusivity of the ion to the ionic mobility and is valid only at infinite dilution. It is clear from relations (3.1.108h) and (3.1.108i) and the definition (3.1.86) of diffusion coefficient D at infinite dilution that... 

Since the ionic conductivity and diffusivities are related by the Nemst-Einstein relationship, what governs one governs the other. Fast ionic conductors are a class of solids in which the ionic conductivity is much larger than the electronic conductivity. For a solid to exhibit fast ion conduction, the concentration and mobility of ionic defects must be quite large. The band gap of the material must also be quite high to minimize the electronic contribution to the overall conductivity. The electronic conductivity depends critically on the concentration of free electrons and holes. There are essentially three mechanisms by which mobile electronic carriers can be generated in a solid ... 

The mobility fi, and hence the ionic conductivity, are directly related to the diffusion through the Nernst-Einstein equation, iikT = ZeD, giving equation (4) ... 

This dilute-solution approach does not account for interaction between the solute molecules which could be dominant for many concentrated ionic solutions. Also, this approach will either use too many or too few transport coefficients depending on if the Nemst-Einstein relationship is used to relate mobility and diffusivity. 

Equations (20.1.2-2) and (20.2.1-7) relate the diffusion potential to mass-transport within the electrolyte and to transference numbers, respectively. If both the transference numbers are equal to one-half in the latter equation, then the diffnsion potentials in solution arising from contact of solutions with differing concentrations will be eliminated. The limiting values of the ionic diffusion coefficient are related to their mobility by the Einstein law, equation (20.2.1-10), which provides justification for the argument that balancing the mobilities of the electrolyte s constituent ions wiU nullify any diffusion potential. This approach to eliminating diffusion potentials clearly suffers from the severe restriction that... 

---