Long-range self diffusion

Fig. 3.1.5 Temperature dependence of the coefficient of long-range self-diffusion of ethane measured by PFG NMR in a bed of crystallites of zeolite NaX (points) and comparison with the theoretical estimate (line). The theoretical estimate is based on the sketched models of the prevailing Knudsen diffusion...

In general, for zeolitic self-diffusion at sufficiently high temperatures, the mean molecular displacements outside the crystal are much larger than those inside the zeolites that is to say, long-range self-diffusion, Di.r., is much faster than intracrystalline self-diffusion, Dintra- For observation times comparable with the mean lifetimes of the adsorbed molecules in the individual crystallites, the spin-echo attenuation can be approximated by the superposition of two exponentials of the type of Eq. (6)... 

Three quantities that have a key function for the understanding of mass transfer in granules are illustrated in Pig. 2 (i) the coefficients of intracrystalline self-diffusion, Dmm, and of (ii) long-range self-diffusion, Du., as well as (iii) the molecular mean lifetime, Tma- The coefficient of long-range self-difffusion is approximated by... 

Translational self-diffusion and rotation of the water molecules. Since the above two models fail to explain the data, one may think of a model which combines both, and which is certainly more realistic. We h ve2tr e< such a possibility with p = 0.95 and D 1.6 x 10 cm /s. This last value is the long range self-diffusion coefficient of water in this membrane, measured by radioactive tracers. We found that no fit is possible with these values whatever D is chosen. As for the preceeding section, we find that the fiE improves considerably if we take either p or Dt as parameters. With Dt fixed, we should increase p to 3 A, as above, and with p fixed, we should increase D to 10 cm /s. These results suggest that one should think of a model which contains these two features. The simplest one is a model where the water molecules, more precisely the protons, are restricted to diffuse (diffusion goejjficient D) in a sphere of radius a, where we expect D 10 cm and a 3 S. 

For root mean square displacements 1 much less than the mean crystallite diameters,the thus determined self-diffusion coefficient exclusively refers to migration in the intracrystalline space. In the opposite limiting case of large molecular displacements one obtains the coefficient of long-range self-diffusion (Di.v.). This quantity is related to the relative... 

Figure 23 Coefficients of (a) intracrystalline and (b) long-range self-diffusion, and (c) intracrystalline mean lifetimes Tj , and (d) Timra " for methane in granulated zeolite NaCaA at 293 K. Also, comparison with (e) the breakthrough capacities for a petroleum raffinate and (0 the specific retention volume for /i-pentane all plotted against the temperature of hydrothermal pretreatment applied over a time interval of 7 h (0) and 14 h (O), respectively. (From Ref. 175.)...

From the measurement of two-component long-range self-diffusivities, direct information about adsorption selectivity may be deduced. With xj and denoting the mole fractions of the mixture component in the adsorbed and gaseous phases, the sorption separation factor is defined by the relation 203]... 

Coefficient of long-range self-diffusion (=pinterDinter)... 

These experiments suggest that as the long time self-diffusion coefficient approaches zero the relaxation time becomes infinite, suggesting an elastic structure. In an important study of the diffusion coefficients for a wide range of concentrations, Ottewill and Williams14 showed that it does indeed reduce toward zero as the hard sphere transition is approached. This is shown in Figure 5.6, where the ratio of the long time diffusion coefficient to the diffusion coefficient in the dilute limit is plotted as a function of concentration. 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

The reduction of the long-range diffusivity, Di by a factor of four with respect to bulk water can be attributed to the random morphology of the nanoporous network (i.e., effects of connectivity and tortuosity of nanopores). For comparison, the water self-diffusion coefficient in Nafion measured by PFG-NMR is = 0.58 x 10 cm s at T = 15. Notice that PFG-NMR probes mobilities over length scales > 0.1 /rm. Comparison of QENS and PFG-NMR studies thus reveals that the local mobility of water in Nafion is almost bulk-like within the confined domains at the nanometer scale and that the effective water diffusivity decreases due to the channeling of water molecules through the network of randomly interconnected and tortuous water-filled domains. ... 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

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