Single-file reaction

Figture 16 Effectiveness factor of single-file reaction plotted against the generalized Thiele modulus as defined by Eq. (42). (From Ref. 162.)... 

Fig. 5 Concentration profiles of the molecules of species B within the single-file system under stationary conditions and comparison with the dependence to be expected for ordinary diffusion (broken line, Eq. 29). The quantity 2L(klDy (the Thiele modulus 0) in Eq. 29 has been chosen to coincide with the generalized Thiele modulus (cf. Eq. 30) of the single-file reaction for k = 1.27 x 10 (0 = 2.77). z denotes the distance from the middle of the file and L = NX its length. From [1] with permission...

Fig. 6 Effectiveness factor rj = of single-file reaction plotted as a function of the generalized Thiele modulus. From [1] with permission...

In zeolites Na X and ZSM-5, the self-diffusion coefficients were found to decrease with increasing concentration while for zeolite NaCa A they are essentially constant The highest diffusivities were observed in zeolite Na X. This is in agreement with the fact that due to the internal pore structure the steric restrictions of molecular propagation in zeolite Na X are smaller Aan those in Na Ca A and ZSM-5 (94). Mass transfer and chemical reaction in zeolite channels in which the individual molecules cannot pass each other (single-file... 

FIGURE 10.24 Model of the prevention of secondary reactions by single-file diffusion in tnicroporous membranes. (Adapted from Lange, C., Storck, S Tesche, B and Maier, W.F., J. Catal, 175, 280, 1998.)... 

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... 

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. 

As in the case of tracer exchange, quantitative information about the correlated effect of transport and catalytic reactions in single-file systems has... 

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... 

This dependence is represented by the dotted line in Fig. 6. In a first attempt to systematize the simulation results of molecular reaction and diffusion in single-file systems, a generalized Thiele modulus has been introduced [1]. Combining Eq. 25 and Eq. 23, the Thiele modulus may be expressed in the alternative notation... 

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

The benefit of the analytical treatment presented thus far for the calculation of the characteristic functions of the single-file system is only limited by the increasing complexity of the joint probabilities and the related master equations. This treatment, however, has suggested a most informative access to the treatment of systems subjected to particle exchange with the surroundings and to internal transport and reaction mechanisms [74,75]. Summing over all values (Ji = 0 and 1 and, subsequently, over all sites i, Eq. 31 may be transferred to the relation Eq. 34... 

It should be emphasized, however, that likely in none of these studies was the zeolite material of such an ideal structure as implied in data analysis. In this respect, experimental studies with artificially created single-file systems [127-129] may provide a much higher rehability of the pre-supposed structural features. A treatise on the substantial deviations of the real structure, with particular emphasis on the consequences for ideal host systems for single-file diffusion as evidenced by optical techniques, is given in [95]. Irrespective of these limitations, however, a number of peculiarities of catalytic reactions in zeolites with one-dimensional channel systems are most Ukely to be attributed to the special conditions of molecular transport and molecular arrangement imder single-file conditions. 

Microporous membranes in general (pore diameter < 2 nm), and zeolite membranes in particular, have pores whose dimensions are similar to those of many molecules. This means that often molecules cannot pass each other in a restrictive pore medium, and single file diffusion occurs. Such a molecular queuing (see Figure 11.24) may provide a new scenario for avoiding secondary reactions, that is, to increase selectivity in consecutive reaction networks with a valuable intermediate... 

A consequence of single-file diffusion is that catalysis in one-dimensional microporous systems becomes diffusion limited for crystal sizes much smaller than for three-dimensional systems. Hence equilibration of different reaction products within the zeolite micropores may be expected to occur most rapidly in one-dimensional microporous systems. 

An important consequence of single-file diffusion to the overall rate of a catalytic reaction is that the overall rate of conversion may show a maximum as a function of micropore filling (increasing reactant pressure) Whereas at low micropore filling a reaction may show no diffusional limitation, with increasing micropore filling the rate of diffusion decreases strongly because of collective motion. This is illustrated in Fig. 4.43. 

Figure 4.43. (a) The rate of a monomolecular zeolite-catalyzed reaction as a function of pressure (no diffusional limitation), (b) The rate of the monomolecular reaction that is diflfusional limited (no single-file diffusion), (c) The rate of the monomolecular reaction that is diffusional limited (single-file diffusion). 

Fig. 28. Median finger reaction times (divided by 10), directly observed cycle times and directly observed cycle numbers of subject P12, n=4 100 (v22lP12-l,2,6,7 and v22rP12-1,2,6,7). The doCT and doCN are measured by help of the sum file, the median finger reaction times (mFRTs) are depicted for each single file...

This database contains 337 000 single-step reactions (1991-1996 see Section 4.5) and is updated with more than 50000 reactions per year from over 150 journals (but no patents). While the in-house version of Chemlnform RX is a reaction-based file, the public STN version is like CASRE-ACT, a document-based file with similar search facilities. 

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