Diffusion coefficient, average

If average diffusion coefficients are used, then the finite difference equation is as follows. 

Water molecules diffusing past each other cause further intermolecular magnetic dipole-dipole interactions. These are a function of the concentration of spins, N, the average diffusion coefficient, D, and the distance of closest approach between the spins, b. 

Obviously, in the case of isotopic exchange, a = 1. Furthermore, in isotopic exchange, the diffusion coefficients are equal to the self-diffusion coefficient of each ion. For different ions (not isotopes), the self-diffusion coefficient is substituted by an empirical constant diffusion coefficient. In the general case of ion exchange, the diffusion coefficient is not constant and for practical reasons, an appropriate average value is used. For the case of complete conversion into A-form (infinite solution volume), this average diffusion coefficient is (Helfferich, 1962)... 

The use of the Stokes-Einstein relation with the above value of the average diffusion coefficient leads to a hydrodynamic radius of roughly 30 nm, which is consistent with the specification of the manufacturer. ... 

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

Mixture-Averaged Diffusion Coefficient The ordinary diffusion coefficient of a species k into a mixture may be evaluated as... 

Species fluxes calculated by either the multicomponent (Section 12.7.2) or the mixture-averaged (discussed subsequently in Section 12.7.4) formulations are obtained from the diffusion velocities V, which in turn depend explicitly on the concentration gradients of the species (as well as temperature and pressure gradients). Solving for the fluxes requires calculating either all j-k pairs of multicomponent diffusion coefficients Dy, or for the mixture-averaged diffusion coefficient D m for every species k. 

See also Problem 12.12.) The mixture-averaged diffusion coefficient of Eq. 12.176 was derived for use in calculating the molar diffusion velocity J with respect to the molar average velocity V, as in Eq. 12.172. 

To calculate the mixture-averaged diffusion coefficient relating the mass flux with respect to the mass-average velocity V in terms of the mass fraction gradient, write an expression analogous to Eq. 12.158 ... 

Using the assumptions mentioned above, the mixture-averaged diffusion coefficient Dkm appropriate for use in Eq. 12.177 can be calculated from (see Problem 12.13)... 

By the same approach as outlined above, the derived mixture-averaged diffusion coefficient D km is [211]... 

Solve this two-point boundary-value problem by any of the methods mentioned in Section 12.8 or Problem 12.16. Calculate the mixture-averaged diffusion coefficients Dkm using Eq. 12.176. Report the molar fluxes Ni, N2 (mol/m2-s) for SO2 and H2O. 

Evaluate the four mixture-average diffusion coefficients D m, based on Eq. 12.178. 

Calculate the four species diffusion velocities V using these mixture-averaged diffusion coefficients and Eq. 12.177. 

Our concentration dependent diffusion coefficient model equation 3.21 gives an average diffusion coefficient, Davg, which is constant over the concentration range of the diffusion process. In a similar manner we derive the following equation for aavg ... 

Figure 15. Average enhanced- and retarded-diffusion coefficients ofP9 As, and Sb following direct nitridation experiments. Data are from Fahey (43). Abbreviations , time-averaged diffusion coefficient ) a, intrinsic diffusion coefficient. (Reproduced with permission from reference 118. Copyright 1984...

For the purpose of this calculation it was necessary to know the diffusion coefficient for ligand and whole molecules of hemocyanin (Table I) and the value of the diffusion coefficient for half molecules of hemocyanin. If the level of calcium ion is reduced in pH 9.6 glycine buffer to the point where no whole molecules remain, the half molecules always are contaminated with one-twelfth size (5S) subunits, so that it is impossible to measure directly the diffusion coefficient of half molecules. For a self-associating system, the Gouy method leads to an average diffusion coefficient which is the Z-average, defined by... 

From Eq. 33, it can be shown that the weight average diffusion coefficient... 

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