Electrical mobility equation

In this equation, Zp is the electrical mobility of the particle. In the case of fine particles, the slip correction must be taken into account, and the mobility is given by... 

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 the respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality throughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the electric potential gradient... 

The movement of a charged particle in an electric field is often defined in terms of mobility, /r, the velocity per unit of electric field (Equation 4.2). 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

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 ... 

The smaller aerosol particles can be captured from the air for subsequent counting and size measurement by means of so-called thermal precipitators. In these instruments, metal wires are heated to produce a temperature gradient. Aerosol particles move away from the wire in the direction of a cold surface, since the impact of more energetic gas molecules from the heated side gives them a net motion in that direction. The particles captured are studied with an electron microscope. Another possible way to measure Aitken particles is by charging them electrically under well-defined conditions. The charged particles are passed through an electric field and are captured as a result of their electrical mobility (see equation [4.6]). Since size and electrical mobility are related, the size distribution of particles can be deduced. These devices are called electrical mobility analyzers. 

Figure 8. Variation of the electrical mobility of ChSO -Na (1.5 g/L) with added NaCl concentration (A) experimented values ( ) values derived from Equation 9.

The mobile ions begin to migrate perpendicularly to the electrodes in the constant electric field causing, as stated above, a charge flow or electric current. Equation (21.11) gives us the migration velocity under these conditions ... 

Remember that the electric mobilities u of anions are negative.) Equation (21.33) and the relation c,- = v,c can be used to find a relationship between the transport number of an ion and the ion conductivity ... 

Here, D+ and D, the respective diffusion coefficients of H" and Cl", as well as u+ and u their electric mobility values (see definition in section 4.2.1.4), are considered as being independent of the concentration and therefore of x. As a consequence, they are considered as constants in the equations, if we consider the volume mass balance of these species , then two different equations emerge to define the concentration... 

There are also expressions using the electric mobility, m, which is the steady-state relative velocity of a charge species submitted to a unit electric field, namely 1 V m . Its SI unit is therefore m s W . Simple equations can be found linking the electrochemical and electrical mobility.and the molarconductivity of a given species ... 

This latter is known as the Nernst-Einstein equation K It can also be written involving the electric mobility ... 

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... 

Here F is the Faraday constant, Zi is the valence, c is the concentration, and is the electrical mobility of the ions. From this equation it can be seen that two important quan-... 

In fact, this equation is nothing other than Ohm s law (cf. case study Gl) by remarking that the conductivity is the product of the electrical mobility Uq by the charge concentration ... 

However, in cementitious material, research related to the diffusion coefficient, the permeation coefficient, and quantification of the electric mobility has just made a start, and models corresponding to various materials and environmental conditions have not yet been constructed as of this stage. In the past, apparent diffusion coefficients were used including aggregates as shown on the left side of Fig. 4, but in practice, substances such as pore solution and ions did not move within the aggregates or the hydrates themselves, but they can move in capillary porosity and transition zones. Thus, an effective diffusion coefficient D, calculation equation is proposed as... 

For given electron and hole densities Ce and Ch as determined by the dopant content, the temperature dependence temperature dependence of the electric mobilities i/ei,e and j/ei,h- For a hopping motion with varying hopping distances theories predict a temperature dependence of the mobility and, thus, of the electric conductivity as given by the equation... 

Equation (21) shows that the electrical mobility of a carbon nanotube is a function of diameter, length, and number of charges. This implies that different size nanotubes can have the same mobiUty, which leads to ambiguity as to the dimensions of the nanotubes. Therefore, it is necessary to obtain an additional expression relating the number of charges carried by a nanotube to the nanotube dimensions. If such an expression can be derived, the electrical mobiUty can be calculated knowing the diameter and length of the nanotube. 

Equation (21) can now be used to describe the electrical mobility of a carbon nanotube as a function of the dimensions of the nanotube and the DMA settings. Equating Eqs. (21) and (29) leads to an expression that includes only nanotube diameter and length as variables ... 

Figure 6 shows the field dependence of hole mobiUty for TAPC-doped bisphenol A polycarbonate at various temperatures (37). The mobilities decrease with increasing field at low fields. At high fields, a log oc relationship is observed. The experimental results can be reproduced by Monte Carlo simulation, shown by soHd lines in Figure 6. The model predicts that the high field mobiUty follows the following equation (37) where d = a/kT (p is the width of the Gaussian distribution density of states), Z is a parameter that characterizes the degree of positional disorder, E is the electric field, is a prefactor mobihty, and Cis an empirical constant given as 2.9 X lO " (cm/V). ...

More precise coefficients are available (33). At room temperature, cii 1.12 eV and cii 1.4 x 10 ° /cm. Both hole and electron mobilities decrease as the number of carriers increase, but near room temperature and for concentrations less than about 10 there is Htde change, and the values are ca 1400cm /(V-s) for electrons and ca 475cm /(V-s) for holes. These numbers give a calculated electrical resistivity, the reciprocal of conductivity, for pure sihcon of ca 230, 000 Hem. As can be seen from equation 6, the carrier concentration increases exponentially with temperature, and at 700°C the resistivity has dropped to ca 0.1 Hem. 

The relatively high mobilities of conducting electrons and electron holes contribute appreciably to electrical conductivity. In some cases, metallic levels of conductivity result ia others, the electronic contribution is extremely small. In all cases the electrical conductivity can be iaterpreted ia terms of carrier concentration and carrier mobiUties. Including all modes of conduction, the electronic and ionic conductivity is given by the general equation ... 

Particle Mobility By equating the electrical force acting on a... 

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