Self-diffusivities of mixture components

Fig. 11 Self-diffusivities of mixture components in silicalite as a function of the 2-methyl-pentane fraction in the gas phase (left) and as a function of the 2-methylpentane loading (total hydrocarbon pressure 6.6 kPa, 433 K)...

Fig. 14 Self-diffusivities of mixture components in both MFI-type zeolites as a function of 2-methylpentane loading, 433 K...

General three-component diffusion equations may be reduced in two ways to concern only two chemically different components. One of these ways leads to the ordinary two-component equation presented above. The other leads to equations for self diffusion of a component in a mixture with a second component (Lamm > > ). The former component is split in two parts, (ideally) labelled by the isotope tracer procedure, which form a diffusion gradient. The latter component is assumed to have a constant concentration during the self-diffusion experiment (the more general case is of minor interest). We will mainly reproduce here the result which has a bearing upon the (relative) constancy of the resistivities. Let the chemically different components be a and b. The former is composed of two isotopically different, but with respect to diffusion properties identical, substances (a) 1 and (a)2 c = -(- c. In view of what has been stated... 

These two equations, (50) and (51), illustrate how the macrodynamic theory can give a kinematic framework, which may be of interest in connection with work on the molecular dynamics of diffusion. It is seen that on a fundamental basis it is not possible to express the mutual coefficient in terms of the two selfdiffusion coefficients (of component a) and D (of component b). As a matter of fact, self diffusion of a component in a mixture is a more complicated process than mutual diffusion, as far as frictions are concerned, because according to (50) it depends on two kinds of friction, viz., internal friction within this component... 

Figure 11 shows the self-diffusion coefficients obtained from the TEX-PEP experiments for both alkanes as a function of the gas-phase mixture composition. Evidently, we find that the self-diffusivity of n-hexane is an order of magnitude higher than that of the 2-methylpentane. Indeed, the kinetic diameter of n-hexane (4.3 A) is smaller than that of isohexane (5.0 A) [51]. Moreover, we observe a decrease in mobility with increasing fraction of the branched alkane in the gas phase. Analogous behavior was found for CH4/CF4 mixtures, where the self-diffusivity of both components decreased as the loading of the slower diffusing tetrafiuoromethane increased [52]. 

A comparison between sihcalite-1 and H-ZSM-5 teaches that acid sites have a profound influence on the self-diffusivity of alkanes. The self-diffusivities of both components decrease strongly, and we observe a significant preferential adsorption of the linear over the branched hexane. This is caused by the relatively stronger interaction of the linear hexane with the acid sites. On the contrary, 2-methylpentane loadings in mixtures in sihcahte-1 and H-ZSM-5 are very close. In H-ZSM-5, the diffusivity of the hnear alkane in mixtures with the branched alkane is influenced by two factors... 

Catalytic systems are inherently multicomponent. An attractive method for determining the difFusivities in multicomponent systems is Fourier transform (FT) PFG NMR spectroscopy, which allows the simultaneous determination of the self-difFusivities of the individual components in a. mixture. If the chemical shifts of the individual species are sufficiently different, the Fourier transform of the spin echo yields separate peaks for the various adsorbates, and then similar to normal PFG NMR the attenuation of the separate peaks with increasing applied field gradient intensity yields the self-diffusivities. Because the technique can also be applied at elevated temperatures, it provides the opportunity for in-situ diffusion measurements under reaction conditions. The experiment also yields the time dependence of the relative concentrations of the reactant and product molecules and thus the intrinsic reaction rate. 

Figure 32 Values for the coefficients of (a) intracrystalline self-diffusion of the two components in /i-heptane-benzene mixtures in NaX at 400 K and (b) long-range selfdiffusion at 361 K, plus the separation factors calculated from them. (From Refs. 145, 163, and 165.)...

The FTIR technique has proven to be a powerful method for investigating adsorption, desorption, and diffusion of single components or binary mixtures in microporous solids such as zeolites. In the latter case of mixtures, the phenomena of codiffusion and counter-diffusion became accessible to measurement, which was not possible with methods of investigation based on changes of weight, volume, or pressure. Even with the powerful and most important NMR techniques (see Chap. 3 of the present volume), the study of multicomponent (e.g., H2-D2) self-diffusion rather than co- and counterdiffusion experiments is possible (see Sect. 1 and [6]). The only prerequisite for the IR method is that the IR spectra, which are contributed by the components of the mixture, can be sufficiently decomposed. This, however, was easily achieved for all systems studied so far, owing to appropriate computer programs nowadays available. Certainly, the computational methods... 

Figure 20 shows the self-diffusivities of n-pentane and n-hexane as a function of gas mixture composition. The loadings of both components depend on the gas phase composition fraction. Note that in these experiments the total hydrocarbon pressure is kept constant (6.6 kPa). The loading of a feed of pure n-hexane under these conditions (433 K, 6.6 kPa) is 3.6 molecules per unit cell. Unexpectedly, we observe a lower diffusion coefficient for pure n-pentane than for n-pentane in a mixture with n-hexane. Tentatively, we ascribe this to the more drastic increase in n-hexane dif-fusivity with loading than for n-pentane. As discussed earlier, this can be ascribed to stronger repulsive interactions for the longer hexane hydrocarbons. It is clear from Fig. 20 that at low n-hexane concentrations, its diffusion is slower than of n-pentane. The diffusivity of hexane increases with n-hexane loading, while the diffusivity of pure n-pentane was found to be independent on the concentration at this temperature. Repulsion between n-pentane... 

Self-diffusion coefficients for the two components have been calculated separately by using Equations 2.10 (MSD route) and 2.11 (VACF route), and finally, the mean values have been plotted in Figure 2.3. The upper and middle panels of Figure 2.3 are the plots for the self-diffusion coefficients of Cl (D, ) and C2 (Dl), respectively, as a function of Xj. D decreases almost steadily with Xj, although the extent of decrease is somewhat small (-12%). D is nearly equal to ut Xj = 0.9, which is expected because self-diffusion of any species in a binary mixture dominated by that... 

However, self-diffusion is not limited to one-component systems. As illustrated in Figure 4.4-1, the random walk of particles of each component in any composition of a multicomponent mixture can be observed. 

Solid-phase reactions are usually activated by high-temperature treatment [1-4]. However, the practical efficiency of this process is rather low since the diffusion rate through a product layer is small, no tight contacts between the particles of components are provided, a particle size distribution is non-uniform, etc. Therefore, the search for new methods of performance of solid-phase reactions is carried out. Some new methods are successfully used for these purposes. These methods include, for example, self-propagating high-temperature synthesis [5], shock waves [6,7], mechanical activation of mixtures in grinding apparatus [8-15]. The latter method becomes more widely used at present due to its relative simplicity and availability. 

Electrolyte Solutions. - The self-association of relatively small hydro-phobic organic components in aqueous mixtures has been investigated by Sacco. He determined the association aparameter A(22), which is essentially obtained from experimental intermolecular dipole-dipole relaxation data of nuclei and from NMR measurements of the self-diffusion coefficients of the organic solvents. The effect of electrolytes and non-electrolytes has been studied. The chemical shifts and relaxation times Ti and T2 of He and Xe nuclei of noble gas atoms as well as those of Li and Cs nuclei of isoelectronic Li" and Cs" ions have been measured in aqueous solutions by Mazitov et Ruso et al. studied the self-association of weakly selfassociating propranolol hydrochloride in aqueous electrolyte solutions by... 

The theory presented here resolves itself into a generalization of the well-known Nemst-Einstein equation D = 22T/C>) to several components and optional concentration characteristics. The cases of two- and three-component mixtures are treated in detail. The latter case is also shown to be of interest in treating self diffusion in a binary mixture, a system which results from letting two components become diffusionally identical although still distinguishable. 

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]... 

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