Effective single-file diffusion

It has been demonstrated by numerical simulations [9] that, with this definition, eq. 2 provides a reasonable order-of-magnitude estimate of the effectiveness factor also in the case of single-file diffusion. While in the case of ordinary diffnsion the intracrystalline mean life time may be easily correlated with the crystal size and the internal mobility [11], similar analytical expressions for single-file diffusion have not been established. The rule-of-thumb given in Ref. [10] on the basis of a few first numerical simulations turned out to be of rather limited validity in recent more refined considerations [12]. 

With this more general equation, representation of the effectiveness factor in terms of the Thiele modulus also becomes possible for single-file diffusion. As an example. Fig. 16 shows the result of a computer simulation of diffusion and reaction within a single-file system consisting of A = 100 sites for different occupation numbers in comparison with the dependence... 

These simple model calculations illustrate the dramatic effect of micropore filling on catalytic performance. Since dilution by pentane does not increase the rate of hexane in Mordenite. The experiments by Spivey and Bryant indicate that diffusion has to be explicitly considered. It is likely that not only the higher heat of adsorption of alkanes in Mordenite compared to Faujasite, but ultimately the one dimensional stracture and resulting single file diffusion are responsible for the unique behavior of the Mordenite catalyst. 

Effect of single-file diffusion on the hydroisomerization of 2,2-dimethylbutane on platinum loaded H-mordenite... 

The effect of single-file diffusion limitation on the rate of an irreversible first order catalytic reaction was studied both theoretically and experimentally. A rate equation was derived using a relation between the effective diffusion constant and the concentration of adsorbed molecules under single-file conditions which is valid on the time scale of catalytic reactions. The hydroisomerization of 2,2-dimethylbutane on platinum loaded large crystallites of H-Mordenite was used as a test reaction to verify the theoretical results. 

Single file diffusion, effect on catalysis 28-0-05 K-01 Singlet molecular oxygen 27-P-ll... 

Effect of single-file diffusion on the hydroisomerization of 2,2-dimethylbutane on platinum loaded H-mordenite F.J.M.M. de Gauw, J. van Grondelle and R.A. vanSanten... 

Before the introduction of measuring techniques such as pulsed field gradient (PEG) NMR ([14,16,45], pp. 168-206) and quasielastic neutron scattering (QENS) [49,50], which are able to trace the diffusion path of the individual molecules, molecular diffusion in adsorbate-adsorbent systems has mainly been studied by adsorption/desorption techniques [ 16]. In the case of singlefile systems, adsorption/desorption techniques cannot be expected to provide new features in comparison to the case of normal diffusion [51,52]. In adsorption/desorption measurements it is irrelevant whether or not two adjacent molecules have exchanged their positions. But it is this effect which makes the difference between normal and single-file diffusion. 

Most importantly, combining the expressions for the intracrystalline mean life time Eq. 23 and the effective self-diffusivity (Eq. 19), in the case of singlefile diffusion the mean time of molecular exchange is found to scale with L, rather than with the dependence typical of normal diffusion. Therefore, under the conditions of single-file diffusion, the exchange rate with zeoUte crystallites decreases even more pronounced with increasing crystal sizes as in the case of normal diffusion. 

Thiele moduli. These also appear in the smaller area below the single-file profile curve for /c = 1.27 x 10 in Fig. 5, in comparison to the area under the profile resulting from the conditions of normal diffusion with the same Thiele modulus. By comparing single-file diffusion with ordinary diffusion the prevailing effect is clearly the dramatic enhancement of rintra and hence of the Thiele modulus, leading to a correspondingly dramatic reduction of the effectiveness factor. 

Fig. 9 Probability distribution function (p r) (a) and effectiveness factor rj k) (b) corresponding to the tracer exchange curves in the limiting cases of dominating single-file diffusion, normal diffusion and surface barriers as a function of the quotient of r and Tintra- From [74] with permission...

After a longer time, single-file diffusional time dependence will disappear because diffusion will be controlled by the center of mass motion of the correlated molecules. The characteristic of single-file diffusion that remains is the concentration dependence and an increased effective mass, which now has to be taken as the center of mass of the collectively moving particles (ruc.m.), proportional to the length of the zeolite porel ... 

If molecules can pass one another in the pore, then an increased pore occupation may increase the diffusion rates since molecule-molecule interactions may be weaker than moleculi zeolite channel interactions. When molecules cannot pass one another, the phenomenon known as single file diffusion may arise. In single file diffusion, which tends to occur in one dimensional porous systems, the large reduction in diffusion rate is represented by an effective diffusion constant proportional to the center of mass of the file of molecules occluded in the zeolite channel. 

Buttry, D. A. and Anson, F. C. 1983 Effects of electron exchange and single-file diffusion on charge propagation in nation films containing redox couples. J. Am. Chem. Soc. 105 685-689. 

SO far only been attained by Monte Carlo simulations. Figure 5 illustrates the situation due to the combined effect of diffusion and catalytic reaction in a single-file system for the case of a monomolecular reaction A B [1]. For the sake of simplicity it is assumed that the molecular species A and B are completely equivalent in their microdynamic properties. Moreover, it is assumed that in the gas phase A is in abimdance and that, therefore, only molecules of type A are captured by the marginal sites of the file. Figure 5 shows the concentration profile of the reaction product B within the singlefile system imder stationary conditions. A parameter of the representation is the probabiUty k that during the mean time between two jump attempts (t), a molecule of type A is converted to B. It is related to the intrinsic reactivity k by the equation... 

In complete agreement with the fact that the product molecules in a singlefile system are prevented from leaving the system by their file neighbors, the concentration profiles in the single-file cases show a much more pronounced tendency of accumulation of the reaction products in the file center than in the case of normal diffusion. Under stationary conditions, the effective reactivity k is related to the intrinsic reactivity k by the equation... 

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