Effective Ionic Diffusivities

An expression for the effective ionic diffusivity may be obtained by setting equal the right-hand sides of Eqs. 2.4.31 and 2.4.32. 

DATA The components in the mixture will be numbered as follows  

The infinite dilution Maxwell-Stefan diffusion coefficients for the ions in water are (from Newman, 1991). 

SOLUTION The effective diffusivities may be calculated from Eq. 2.4.33 we approximate the ratio of concentration gradients Vcy/Vc, by the ratio of concentration differences Acy/Ac thus  

As a basis for calculating the effective diffusivities we make Cnaci 1 kmol/m. We further define the concentration ratio r as 

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Effective ionic diffusivities at a rotating-disk electrode are calculated from the Levich equation as derived for constant physical properties, used here in inverted form ... 

Table VII should be 1.939 for the ratio k = 0.5. Part of the 17% discrepancy between the results of Lin et al. (L9) and Eq. (27) may be ascribed to the use of incorrect diffusivities. An estimate of the errors is possible for part of their experiments. The value of the product nD/T of K3Fe(CN)6 based on the electric mobility at infinite dilution as used by Lin et al. is 11% too high, according to more recent measurements of the effective ionic diffusivity of Fe(CN)(% by Gordon et al. (G5). Similarly, the mobility product of K4Fe(CN)6 is 16% too high, and that of 02 no less than 26% too high, compared with data of Davis et al. (D7) (see Table III). According to Eq. (27) the value of D would have to be 27% too high to account fully for a coefficient that is 17% too high consequently, the discrepancy cannot be attributed entirely to incorrect diffusivities.

Due to the lack of a reliable description, the diffusion of an ionic species in a molecular species is usually represented by the effective ionic diffusivity in the liquid phase [52]. The calculation of the diffusion coefficient for an ionic component in another ionic species is reduced to the arithmetical mean of both effective ionic diffusivities [52]. 

Figure 2.10. Comparison of calculated experimental values of the effective ionic diffusivity. Data from Vinograd and McBain (1941).

Additional calculations of the effective ionic diffusivities are shown in Figure 2.10 as a function of the square root of the concentration ratio r. The experimentally determined effective diffusivities are shown in the same figure for comparison. The agreement between theory and experiment is very good, especially for the Cl and Ba ions. The theory overestimates the effective diffusivity of the H ions but the decrease in the effective diffusivity of the H ions as the concentration ratio increases is predicted correctly. 

It follows from Eqs. (2.6.6), (2.6.8) and (2.6.10) that the presence of the solvent has two effects on the ionic mobility the effect of changing viscosity and that of changing the ionic radius as a result of various degrees of solvation of the diffusing particles. If the effective ionic radius does not change in a number of solutions with various viscosities and if ion association does not occur, then the Walden rule is valid for these solutions ... 

It 1s well known that water absorption bears a direct relation with the number of polar groups in the polymer and that Ionic diffusion occur via "hopping" along hydrophilic sites. Therefore, the hydrophilic/ hydrophobic characteristics of the Inhibitor exert a profound effect on dissolution rate. 

In summary, from the above theoretical and experimental results, it is concluded that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent selfsimilar fractal dimension rather than the self-affine fractal dimension. In addition, the triangulation method is one of the most effective methods to characterize the self-similar scaling property of the self-affine fractal electrode. 

For inter diffusion between same-valence ions (ionic exchange) in an aqueous solution, or a melt, or a solid solution such as olivine (Fe +, Mg +)2Si04, an equation similar to Equation 3-135c has been derived from the Nemst-Planck equations first by Helfferich and Plesset (1958) and then with refinement by Barter et al. (1963) with the assumption that (i) the matrix (or solvent) concentration does not vary and (ii) cross-coefficient Lab (phenomenological coefficient in Equation 3-96a) is negligible, which is similar to the activity-based effective binary diffusion treatment. The equation takes the following form ... 

Meade (1966) shows that claystones have a porosity decreasing to 0% at 1 Km depths and sandstones, 20% porosity at the same depth. Manheim (1970) shows that ionic diffusion rates in sediments are 1/2 to 1/20 that of free solutions when the sediments have porosities between 100 - 20%. It is evident that the burial of sediments creates a very different physical environment than that of sedimentation. As a result of reduced ionic mobility in the solutions, a different set of silicate-solution equilibria will most certainly come into effect with the onset of burial. The activity of ions in solution will become more dependent upon the chemistry of the silicates as porosity decreases and the system will change from one of perfectly mobile components in the open sea to one approaching a "closed" type where ionic activity in solution is entirely dictated by the mass of the material present in the sediment-fluid system. Although this description is probably not entirely valid even in rocks with measured zero porosity, for practical purposes, the pelitic or clayey sediments must certainly rapidly approach the situation of a closed system upon burial. 

The treatment of ionic diffusion properties is a more difficult task. The obstruction effect of the polymer matrix can be represented by a number of semi quantitative equations, the most popular of which is 111 ... 

Figure 13 shows the potential and concentration distributions for different values of dimensionless potential under conditions when internal pore diffusion (s = 0.1) and local mass transport (y = 10) are a factor. As expected the concentration and relative overpotential decrease further away from the free electrolyte (or membrane) due to the combined effect of diffusion mass transport and the poor penetration of current into the electrode due to ionic conductivity limitations. The major difference in the data is with respect to the variation in reactant concentrations. In the case when an internal mass transport resistance occurs (y = 10) the fall in concentration, at a fixed value of electrode overpotential, is not as great as the case when no internal mass transport resistance occurs. This is due to the resistance causing a reduction in the consumption of reactant locally, and thereby increasing available reactant concentration the effect of which is more significant at higher electrode overpotentials. 

These equations are called the phenomenological equations, which are capable of describing multiflow systems and the induced effects of the nonconjugate forces on a flow. Generally, any force Xt can produce any flow./, when the cross coefficients are nonzero. Equation (3.175) assumes that the induced flows are also a linear function of non-conjugated forces. For example, ionic diffusion in an aqueous solution may be related to concentration, temperature, and the imposed electromotive force. 

Another strategy to improve the membrane permeant properties of a drug is based on the effect of non-ionic diffusion . An example is provided by the two ganglion-blocking agents hexamethonium and mecamylamine, which act as antagonists at certain receptors of the transmitter acetyl-... 

Non-ionic diffusion can also produce unwanted effects, as in the case of aspirin (acetylsalicylic acid figure 2.8b). In the acidic milieu of the stomach, this molecule will be protonated and thus uncharged, which promotes its diffusion into the cells of the stomach mucous membrane. Inside the cell, the pH is very close to neutral, which will lead to deprotonation of aspirin. Diffusion of the deprotonated (charged) form out of the cell will be much slower than entry, so that aspirin will accumulate inside the cells to con-... 

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