Intracrystalline self-diffusion coefficient

The validity of eqn. (3) for determining the intracrystalline self-diffusion coefficients from uptake data has been shown for the sorption of benzene by... 

The concept of transport resistances localized in the outermost regions of NS crystals was introduced in order to explain the differences between intracrystalline self-diffusion coefficients obtained by n.m.r methods and diffusion coefficients derived from non-equilibrium experiments based on the assumption that Intracrystalline transport is rate-limiting. This concept has been discussed during the past decade, cf. the pioneering work [79-81] and the reviews [2,7,8,23,32,82]. Nowadays, one can state that surface barriers do not occur necessarily in sorption uptake by NS crystals, but they may occur if the cross-sections of the sorbing molecular species and the micropore openings become comparable. For indication of their significance, careful analysis of... 

Figure 5) NMR intracrystalline self-diffusion coefficient Di ( ) and NMR desorption diffusivity Dd of methane sorbed in ZSM5 with increasing coking time, (reproduced with the permission from reference 18)...

As Fig. 25 shows, the intracrystalline self-diffusion coefficient of methane in ZSM-5 is between coefficients in zeolites NaCaA and NaX (5,71,114,115,). This order can be interpreted in terms of the minimum apertures of the zeolite channels, which are approximately 0.45, 0.55, and 0.75 nm for 5A, ZSM-5, and X-type zeolites. Due to the hydrophobic nature of ZSM-5, the mobility of water in ZSM-5 considerably exceeds the mobility in zeolites NaA and NaX. A change in the Si02/Al203 ratio of ZSM-5 does not alter the self-diffusion coefficient of methane. On the contrary, for water in ZSM-5 an increase in the self-diffusion coefficients with decreasing A1 concentrations in the framework is indicated. 

A reduction of the intracrystalline self-diffusion coefficient, Dintra, of a probe molecule (e.g., methane) as measured by PFG NMR has to be regarded as proof of the existence of coke, modifiers, etc. in the volume phase of the zeolite crystal. 

Fig. 34. Intracrystalline self-diffusion coefficient Dimm (A) and effective seif-diffusion coefficient ( ) of methane in HZSM-5 modified by impregnation with H3PO4. The starting self-diffusion coefficient is 8.9 X 10" m s" (6 CH4 per u.c., 293 K) 130).

Flo. 37. NMR intracrystalline self-diffusion coefficient Dm, (a) and effective self-diffusivity Dcir ( ) of methane in HZSM-5 crystals that were coked for different times by n-hexane cracking (131-133). Before loading with methane (9.2 CHa per u.c.), the coked ZSM-5 crystals were carefully outgassed at 623 K and 10 Pa. The remaining carbonaceous residues were defined as coke. Amounts of coke after different times on stream 1 h, 0.8 wt% C 2 h, 1.3 wt% C 6 h, 3.2 wt% C 16 h, 4.8 wt% C. The starting self-diffusion coefficient is 8.1 x 10" m s . ... 

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,... 

Fig. 3 Types of concentration dependence of the intracrystalline self-diffusion coefficient [20]...

Figure 8 shows the self-diffusion coefficients, Db, of methane, ethane and propane as a function of soibate loading at various temperatures. These corrected diffusion coefficients were calculated from Equation 12 using diffusion coefficients obtained fi m both sorption and desorption half-cycles. These diffusion coefficients were identical. The corrected, Db, diffusion coefficients in Figure 8 are <5 smaller than the intracrystalline self-diffusion coefficients measured directly by NMR which are also included in this figure and are in close agreement with the corrected diffusion coefficients obriuned by the full FR method. 

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... 

The absence of additional diffusion barriers at the crystal surface (coke, modifiers, etc.) can be assumed if the experimentally determined mean intracrystalline lifetimes, Timra (measured by TD NMR), and the corresponding calculated data, rSitra [according to Eq. (10)], coincide. The rSim data are calculated by using the self-diffusion coefficients, Dimra (measured by PFG NMR), and crystal radii, R, assuming the adsorption/desorption process to be diffusion controlled. 

If the values of the effective self-diffusion coefficients, D rf [calculated from the complete xit) curves in TD NMR experiments, assuming diffusion-limited uptake (52)] are below the corresponding intracrystalline data, Dintra (measured directly by PFG NMR), the existence of additional mass transfer resistances in a layer near or on the outer surface of the zeolite crystals is indicated. 

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... 

KaX than in smaii-port zeolites such as NaCaA. For hydrocarbons in HaX it has been found that the coefficients of intracrystalline self-diffusion decrease with increasing concentration (concentration dependences of type I and II (6).i. By contrast, for not toe high gas phase concentrations (i.e., as long as molecular transfer in the intercrystalline space proceeds by Knudsen diffusion)... 

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. 

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 28 Values for the coefficients of intracrystalline self-diffusion (A) and for the apparent diffusivity determined from NMR tracer desorption curves ( 1) at 293 K...

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.)...

Fig. 16 Coefficient of intracrystalline self-diffusion of methane in zeolite (o) Na75,Ca-X, >) Na30,Ca-X, and ( ) Na-X determined by PFG NMR. The numbers inserted in the symbols indicate the concentration in molecules per supercage. From [152] with permission...

Fig. 18 Apparent coefficients of intracrystalline self-diffusion of n-hexane as observed by time- and space-resolved PFG NMR in a bed of zeolite Na-X with restricted ( ) and unrestricted ( ) sorbate supply in dependence on the sorbate concentration. The real diffusivities open symbols) were calculated from these values by using the correspondence presented by Fig. 3. The full line with the indicated error bars represents the range of intracrystalline diffusivities as observed in previous PFG NMR studies with closed sample tubes. From [163] with permission...

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.)...

---