Intracrystalline mean

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]. 

For an estimate of the correlation between t he intracrystalline mean life time and the system properties, the exchange curve between the (labelled) molecules of a single-file system and the (unlabelled) surroundings ( tracer exchange curve ) may be assumed to be determined by a single dimensionless parameter... 

The basic principles of both the nmr pulsed field gradient and nmr tracer desorption techniques are presented. It is shewn that these methods allow the measurement of the coefficients of intracrystalline and of long-range self-diiiusion as well as or the intracrystalline mean life times of the adsorbate molecules. By combining this information, a unique possibility for the direct proof of the existence of surface barriers is provided. The nmr methods are applied to study the transport properties of adsorbate molecules in zeolite NaX, NaCaA and ZSlf-5, with particular emphasis on the existence of surface barriers. 

Covering temperatures from -140 up to 200 C and chain lengths from one to six carbon atoms, the intracrystalline mean life times are found to coincide with the values oi rV 1 r calculated from the nmr self-diffusion coefficients. This clearly indicates that molecular exchange is controlled by intracrystalline self-diffusion, and that for the considered adsorbate-adsorbent systems there are no perceptible surface barriers. 

In the present case, the intracrystalline mean life times should be compared with a value of Ti lrou ,f = 0,007 ms, which results from inserting the mean crystallite radius and the intracrystalline self-diffusion coefficient (- 10 nrs ) at the given temperature and concentration (15) into Equation 10. Since in the nmr measurement is found to be much less than 0.2 ms,... 

Table I. Comparison of the Intracrystalline Mean Life Times t and the Quantities linu.0 1, calculated on the Basis of the Coefficients of Intracrystalline Self-Diffusion for Aromatic Compounds in Zeolite NaX...

Figure t gives a comparison of the intracrystalline mean life times of methane in ZSK-5 type monocrystais alter diiferent coxing times, and the values of t 1, calculated from the... 

Figure 6 Intracrystalline mean life times t,<. > and Tini,, "1" (<4 ) for methane at 296 K and a sorbate concentration of 3 molecules per channel intersection in H ZSM-5 coked by n-hexane (full symbols and mesitylene (open symbols in dependence on the time on stream (Reproduced with permission from Ref. 6. Copyright 1 >S7 Butterworth ...

Since for a large variety of samples the shape of the zeolite crystallites tends to be even closer to that of a sphere, in general, Eq. (2) represents a reasonable estimate. It is used again in Sec. II.C., where another (microscopic) method to determine intracrystalline mean lifetimes is introduced. 

Figure 6 provides a comparison between measured spectra and theoretical spectra calculated under the assumption that the adsorption/desorption process is controlled by either intracrystalline diffusion (Fig. 6a) or external transport resistances such as surface barriers (Fig. 6b). For simplicity in the calculations, the crystallites have been assumed to be of nearly spherical shape with a concentration-independent transport diffusivity Dj or surface permeability a, respectively. Values of the intracrystalline mean lifetime are therefore given by...

If molecular exchange is controlled by intracrystalline diffusion, then the intracrystalline mean lifetime is given by Eq. (2), where it is assumed that the crystallites may be approximated by spheres (Sec. II.A.). Clearly, coincides with the directly measured Tj ,ra if desorption is controlled by intracrystalline diffusion. If, however, the rate of molecular exchange is additionally reduced by transport resistances at the crystallite boundary (so-called surface barriers), Tji,ra may be much greater than ... 

Figure 22 gives a comparison of the results of NMR tracer desorption studies and self-diffusion measurements on short chain length paraffins in zeolite NaX [48]. For illustration, the complete tracer desorption curves are also given at selected temperatures. Covering the range from — 140 to 200°C and chain lengths from one to six carbon atoms, the intracrystalline mean lifetimes are found to coincide with values of calculated via Eq. (2) from the NMR self-diffusion co-... 

Table 5 Comparison of Values for the Intracrystalline Mean Lifetime Tj , and for the Quantity, Calculated on the Basis of the Intracrystalline Diffusivities via...

Figure 23 shows results from a systematic study dealing with the influence of a hydrothermal pretreatment of granulated zeolite NaCaA on the three main transport parameters accessible by PFG NMR the coefficients of intracrystalline and long-range diffusion, as well as the intracrystalline mean lifetimes [145,... 

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.)...

Figure 26 compares values for the intracrystalline mean lifetime and Tin,ra " for methane in ZSM-5 type crystallites after different coking times and of the values of [145,187]. Depending on the applied coking compound,... 

From Eq. (2), the measured diffusivities may be used to determine the mean lifetime of the reactant and product molecules within the individual crystallites under the assumption that the molecular exchange is exclusively controlled by intracrystalline diffusion. These values, being of the order of 30 ms, are found to agree with the real intracrystalline mean lifetime directly determined by NMR tracer desorption studies (208], so that any influence of crystallite surface barriers may be excluded. From an analysis of the time dependence of the intracrystalline concentration of the reactant and product molecules, the intrinsic reaction time constant is found to be on the order of 10 s. This value is much larger than the intracrystalline mean lifetimes determined by PFG NMR, and thus any limiting influence of mass transfer for the considered reaction may be excluded. In agreement with this conclusion, the size of the applied crystallites was found to have no influence on the conversion rates in measurements with a flow reactor (208]. 

Effective mean free path in the intercrystalline space Tortuosity factor Intracrystalline mean life time... 

A plot of the relative intensity of the broad constituent versus the observation time (i.e. the separation between the two field gradient pulses) contains information which is analogous to that of a tracer exchange experiment between a particular crystallite containing e.g. labelled molecules and the unlabelled surroundings. Therefore, this way of analysis of PFG NMR data of zeolitic diffusion has been termed the NMR tracer desorption technique [60]. The first statistical moment ( time constant ) of the NMR tracer desorption curve represents the intracrystalline mean lifetime Tintra of the molecules under study. 

The quantitative information provided by PPG NMR about the existence of additional transport resistances on the external surface of the zeolite crystallites (surface barriers) results from a comparison of the values for the intracrystalline mean life time determined directly (viz. Tintra) by an analysis of the time dependence of the spin-echo attenuation (and, hence, of the propagator), and determined indirectly (viz. tP ) from the intracrystalline diffusivity on the assumption that molecular exchange between different crystallites is controlled by intracrystalhne diffusion. On the additional assumption that the shape of the crystallites may be approximated by spheres with a mean square radius (R ) one has in the latter case [87,103]... 

A representation of the values for the intracrystalline mean life time in parallelepipeds with varying edge lengths may be found in [104]. 

Fig. 8 Values for the intracrystalline mean life time tintra ( M) and the quantity (A,A) for methane in H-ZSM5 which has been coked by n-hexane (filled symbols) and mesitylene (open symbols) as a function of the coking time (methane concentration 12 molecules per unit cell measuring temperature 296 K). From [116] with permission...

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. 

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