Mobility Equations

One equation for the mobility coefficient of thermaUzed ions drifting through a gas atmosphere in an electric field was given by Mason and cowoikers, as shown in Equation 10.9  

Mason and coworkers described the collision cross section using Equation 10.10  

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If we postulate diat die chemical potentials of all species are equal in two phases in contact at any interface, dieii Einstein s mobility equation may be simply applied, in Pick s modified form, to describe die rate of a reaction occun ing dirough a solid product which separates die two... 

A number of metals, such as copper, cobalt and h on, form a number of oxide layers during oxidation in air. Providing that interfacial thermodynamic equilibrium exists at the boundaries between the various oxide layers, the relative thicknesses of the oxides will depend on die relative diffusion coefficients of the mobile species as well as the oxygen potential gradients across each oxide layer. The flux of ions and electrons is given by Einstein s mobility equation for each diffusing species in each layer... 

The analysis of oxidation processes to which diffusion control and interfacial equilibrium applied has been analysed by Wagner (1933) who used the Einstein mobility equation as a starting point. To describe the oxidation for example of nickel to the monoxide NiO, consideration must be given to tire respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality tliroughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the elecuic potential gradient... 

Apart from fundamental constants and the liquid temperature, the variable parameters in the effective mobility equation are the quasi-free mobility, the trap density, and the binding energy in the trap. Figure 10.2, shows the variation of prff with e0 at T = 300 K for /tqf = 100 cm3v 1s 1 and nt = 1019cm-3. It is clear that the importance of the ballistic mobility (jl)l increases with the binding... 

For small molecules (metabolites, monomers, and small oligomers) the mobility equation may be empirically approached with the Offord model (linear relation to charge-to-size ratio, the charge being obtained directly from the ionization constants and the size being approached with the molecular mass exponent a factor a) [see Offord (1966)]. 

Since ionic conductivity is the function of the number of carrier ions and their mobility (equation 22.1), the dissociation constant of the salt has a great influence on the ionic conductivity ... 

This model predicts that the sum of the exponents of the current decay before and after tT will be 2. As a tends to zero, the temporal distribution j/(2) broadens and the sharp knee seen in Fig. 8.25(b) becomes much less prominent, since the rate of decay of the current is similar both before and after tT. tT will depend on the ratio of the sample thickness to the average displacement of the carrier in the field direction during its random walk through the sample. Use cjf the formal definition of mobility, Equation (4.2), leads to the result that the mobility has a dependence on electric field and sample thickness, L, of the form ... 

Here and t] are, respectively, the relative permittivity and the viscosity of the electrolyte solution. This formula, however, is the correct limiting mobility equation for very large particles and is valid irrespective of the shape of the particle provided that the dimension of the particle is much larger than the Debye length 1/k (where k is the Debye-Htickel parameter, defined by Eq. (1.8)) and thus the particle surface can be considered to be locally planar. For a sphere with radius a, this condition is expressed by Ka l. In the opposite limiting case of very small spheres (Ka 3> 1), the mobility-zeta potential relationship is given by Hiickel s equation [2],... 

Henry [3] derived the mobility equations for spheres of radius a and an infinitely long cylinder of radius a, which are applicable for low ( and any value of Ka. Henry s equation for the electrophoretic mobility p of a spherical colloidal particle of radius a with a zeta potential C is expressed as ... 

Banks [34BAN] made careful measurements of the conductivity of zinc selenate solutions at 298.15 K. The concentration range used was approximately 2 x 10 to 1 X 10 M. On the assumption that only ZnSe04(aq) was formed the data were evaluated by an iterative procedure in which the inter-ionic attraction was corrected for using the Debye-Huckel (activity coefficient) and Onsager (ionic mobility) equations. The result for ... 

With CZE as described in section 3.3.2, it is often impossible to separate different nucleic acids from each other because they have a similar charge to size ratio and, thus, similar electrophoretic mobilities (equation 3.5). The same is true for SDS denatured proteins. Introducing a gel into the capillary, leads to an additional molecular sieving effect. Large analytes are retained more than smaller ones, enabling separation of analytes with similar mobilities. 

Corrections to Mobility Equations in the High-Field Regime... 

Equation (2.16) is applicable for all for 10 < ka < oo. To obtain an approximate mobility expression applicable for ka < 10, it is convenient to express the mobility in powers of and make corrections to higher powers of in Henry s mobility equation (2.6), which is correct to the first power of Ohshima [25] derived a mobility formula for a spherical particle of radius a in a symmetrical electrolyte solution of valence z and bulk (number) concentration n under an applied electric field. The drag coefficient of cations, A+, and that of anions, A, may be different. The result is... 

An approximate analytic mobility equation apphcable for arbitrary values of was daived by Levich [28], and Ohshima et al. [30] derived a more accurate mobility expression correct to order 1 /ka for the case of symmetrical electrolytes of valence z. The leading term of their expression is given by [30]... 

Assuming (as it is reasonable) that for conditions in which the approximation ko 5> 1 is valid, the dynamic mobility also contains the (1 — Cq) dependence displayed by the static mobility (Equation (3.37)), one can expect a qualitative dependence of the dynamic mobility on the frequency of the field as shown in Figure 3.14. The first relaxation (the one at lowest frequency) in the modulus of u can be expected at the a-relaxation frequency (Equation (3.55)) as the dipole coefficient increases at such frequency, the mobility should decrease. If the frequency is increased, one finds the Maxwell-Wagner relaxation (Equation (3.54)), where the situation is reversed Re(Cg) decreases and the mobility increases. In addition, it can be shown [19,82] that at frequencies of the order of (rj/o Pp) the inertia of the particle hinders its motion, and the mobility decreases in a monotonic fashion. Depending on the particle size and the conductivity of the medium, the two latter relaxations might superimpose on each other and be impossible to distinguish. 

The ionic conductivity, cr, of an electrolyte is given by the product of the concentration of ionic charge carriers and their mobility (equation... 

The movement of charged analytes is now considered as a consequence of the combination of their own individual electrophoretic mobilities (Equation [3.61]) and their participation in the bulk electro-osmotic flow (EOF). The net speed of motion of an analyte ion in the field direction (along the length of the capillary) is the vector sum of its electrophoretic velocity (Equation [3.61])... 

The conductivity (or resistivity) of a semiconducting material, in addition to being dependent on electron and/or hole concentrations, is also a function of the charge carriers mobilities (Equation 18.13)—that is, the ease with which electrons and holes are transported through the crystal. Furthermore, magnitudes of electron and hole mobilities are influenced by the presence of those same crystalline defects that are responsible for the scattering of electrons in metals—thermal vibrations (i.e., temperature) and impurity... 

For a general case where the particle diameter-to-Debye length ratio is large (a the mobility equation becomes ... 

These materials are oxides of transition metals, with conduction taking place through hopping (see section 11.1.4). The concentration of earriers is fixed by the doping element concentration. The temperature dependency of the resistanee is therefore that of mobility (equation [11.10]) ... 

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