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Molecular transport through assemblages of microporous particles
Molecular
Transport through Assemblages of Microporous Particles JiiRG KARGER
NMR-Labor
der Sektion Physik der Karl-Marx-Universitit, Linntstrasse 5, German Democratic
DDR-7010
Leipzig,
Republic
AND
MILAN Heyrovsky
Institute
KO&RIK
ARLETTE
AND
ZIKANOVA
of Physical Chemistry and Electrochemistry, Czechoslovak CSSR-12138 Prague 2, Machova 7, Czechoslovakia
Academy
of Sciences,
Received December 12, 1980; accepted March 25, 1981 Steady-state diffusion studies and the NMR pulsed field gradient technique are introduced as equivalent methods for determining long-range diffusivities. Theoretical methods for mass transfer in heterogeneous media are reconsidered and adapted to adsorbate-adsorbent systems. Comparison with the experimental data for n-butane diffusion in NaX zeolites shows that long-range diffusivities are predominantly determined by intercrystalline transport. Estimates of the absolute value of the temperature dependence and of the influence of the pressure of compaction are in agreement with this conclusion. INTRODUCTION
much larger than the crystallite diameters (“long-range” diffusion) does not depend therefore on the intracrystalline diffusivities. However, it can be shown by the application of the NMR pulsed field gradient technique (g-lo), that for a number of zeolitic adsorbate-adsorbent systems the coefficients of intracrystalline diffusion are considerably higher than previously assumed (lo- 12) and that the above assumption is not valid. In view of this conclusion it is worthwhile to reconsider, whether the influence of intracrystalline transport on long-range diffusivities is indeed negligible. A distinct functional relationship between long-range and intracrystalline diffusivities would be of twofold significance: (i) In many cases, uptake on the individual crystallites is too fast to allow a determination of intracrystalline diffusivities. This difficulty could be circumvented by uptake measurements on the
With the application of molecular sieve crystals (1, 2) to numerous technical processes (3), the study of transport phenomena through assemblages of microporous particles was stimulated not only by scientific interest but also by the requirement of practical application. Most theoretical work on diffusion in assemblages of molecular sieve crystals circumvents the difficulties of the mathematical treatment of compound media by assuming that the efficiency of molecular transport in the interior of the crystallites is much less than in the intercrystalline space. In this case, the surface concentration over an individual crystallite is constant (only depending on time and on the position of the crystallite in the assemblage) and inter- and intracrystalline transport can be separated from each other (4-7). Molecular transport over distances 240 0021-9797/81/l 10240-10$02.00/O Copyright 0 1981 by Academic Press, Inc. Au rights of reproduction in any form reserved.
Journal of Colloid and Interface
Science, Vol. 84, No. I, November
1981
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diffusivities through assemblages of microporous particles. Maintaining fixed sorbate concentrations at the boundaries, the longrange diffusivity Di.,. is obtained from the particle flux density j and from the gradient of overall concentration c by the equation direction
diffusion flux
of a
FIG. 1. Scheme of the experimental arrangement for determining long-range diffusivities through assemblages of microporous crystallites under steadystate conditions. Different hatching indicates different sorbate concentration.
crystallite assemblage. Moreover, in this case, also, steady-state techniques could be applied for which a number of disturbing influences significant for transient experiments (such as due to the finite rates of adsorption heat dissipation and adsorbate supply (13)) can be excluded. (ii) Previous comparative studies between the diffusivities obtained directly by the NMR pulsed field gradient technique and those from uptake experiments (12, 14) showed similar trends in the characteristic dependences but discrepancies up to five orders of magnitude in the absolute values. Taking into account that the uptake experiments have been carried out with zeolite assemblages, this result could be explained by the relationship mentioned above. It is the aim of the paper to give an estimate of the dependence of the long-range diffusivity on intracrystalline diffusion. EXPERIMENTAL
Figure 1 shows the arrangement of a steady state experiment for determining the
j = -Dl,,.dcldx.
111
Overall concentrations are referred to volume elements much larger than the crystallites and can easily be determined from the microscopic concentrations in the adsorbate (c,) and gaseous phases (c,) and the volume fraction p of the dispersed phase by the relation c = pc, + (1 - p)cp = PC,.
PI
Under the assumption that the thermodynamic factor d In c,/d In c, is equal to one and that any bulk (Poiseuille) flow due to pressure gradients in the intercrystalline pore system can be neglected (which holds exactly for sufficiently low sorbate concentrations and for tracer counter flow experiments), the coefficients of diffusion and of self-diffusion coincide (15, 16). Consequently, the quantity Dl.r., as introduced by Eq. [l] should be accessible also from selfdiffusion measurements, i.e., from the study of the velocity of molecular redistribution under uniform concentration over the sample. In the last few years, the NMR pulsed field gradient technique (8) has become a valuable tool for studying self-diffusion in adsorbate-adsorbent systems (9, 10). This technique can directly determine the molecular mean square displacements (i-‘(A)) during a time interval A. Via Einstein’s equation (?(A))
= 6DA,
[31
one may obtain the coefficients of both intracrystalline (( rf’) 112< crystallite radius R) and long-range (( J2)1ip + R) self-diffusion With the assumption following Eq. [2], the latter quantity coincides with D,,,,. nmml of Colioid and lnrerface Science-, Vol. 84, No. 1, November
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KARGER,
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This versatility in the information about molecular transport is the main advantage of the NMR pulsed field gradient technique in comparison with the traditional “macroscopic” methods (8- 10). Therefore we have applied this latter technique in our experiments. The measurements have been carried out with a 60-MHz pulse spectrometer, specially built in the NMR Laboratory of the Physics Department of the Karl Marx University, Leipzig (17). As described in detail previously (9, 18), mean-square displacements were determined from recording the spinecho height for different field gradient intensities. For attaining observation times, A, up to 200 msec, the stimulated echo was used (19). We applied as an adsorbent NaX zeolite crystallites with mean diameters of about 16 and 5 pm, synthesized by idanov and Samulevii:, Leningrad. The measurements were carried out in 7-mm-diameter sealed sample tubes, containing the adsorbate-adsorbent system. Activation of the zeolite crystals in the sample tubes prior to the introduction of the adsorbate (n-butane, purity > 99%) was accomplished by evacuating the samples at 400°C and 5 x lop3 Pa for 20 hr. The amount sorbed was determined gravimetrically and volumetrically and checked afterward by the intensity of the NMR signal of the sealed samples (18). In all our experiments we tried to realize sorbate concentrations of about 80 mg butane per 1 g activated adsorbent. This corresponds to a relative pore-filling factor of about 0.5. The mean error in the sorbate concentrations determined in the above mentioned way is of the order of 10 mg g-l maximum (18). With many zeolite samples one can find that the crystallite packing density decreases with decreasing crystallite size. In our experiments, the volume fractions p of the dipersed phases were about 0.5 and 0.25 for the large and for the small crystallites, respectively. In one case, prior Journal of Colloid and Interface
Science, Vol. 84, No. 1, November
1981
I,
AND ZIKANOVA
FIG. 2. NMR data for butane self-diffusion in an NaX zeolite assemblage with a mean crystallite diameter of about 16 pm and for a sorbate concentration of about 80 mg g-r, open symbols: values for D,rr as defined by Eq. [3] for observation times A = 50 msec (V), 100 msec (A), 200 msec (0), full symbols: estimates of the long-range diffusivities from the values of Derf in the vicinity of the region of restricted diffusion, dashed line: intracrystalline diffusivities taken from Ref. (20).
to sample preparation the zeolite material of a small-crystallite sample was compacted by a pressure of 2.5 MPa, thus realizing also a dipersed phase fraction of about 0.5. Self-diffusion
Measurements
Figures 2 and 3 show the effective diffusion coefficients obtained by the NMR pulsed field gradient technique for the given observation times on the basis of Eq. [3]. For comparison, the coefficients of intracrystalline self-diffusion obtained in previous experiments (20) are included. It has been shown by Riekert and co-workers that these values are in satisfactory agreement with uptake experiments (21). The effective diffusion coefficients can be readily attributed to the regions of “intercrystalline” diffusion (( r2)li2 > R; T 2 -40°C in Fig. 2, T 2 -60°C in Fig. 3) and of “restricted” diffusion (9, 10). According to the theory of restricted diffusion (9, 11, 22), in the latter case the spin-echo attenuation should
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long-range diffusivities obtained for different observation times. In the following, we discuss to what extent the long-range diffusivities are determined by intracrystalline transport. THEORETICAL
FIG. 3. Temperature dependence ofD,, for butane in a loose assemblage (0) and in a pressed compact (0) of NaX zeolite crystals with a mean diameter of about 5 pm for a sorbate concentration of about 80 mg g-l, dashed line: intracrystalline diffusivities taken from Ref. (20).
As a first approximation, we represent the molecular mean-square displacement due to long-range diffusions as the sum of molecular mean-square displacements in the intraand intercrystalline spaces:
(v”(A)) = (%.,(A)) + (%tra@>). 81 Since the total lifetime Ainter in the intercrystalline space is related to the relative number Pinter of molecules in the intercrystalline space @inter < 1) by the condition
be the same as for a system of molecules moving freely with a diffusion coefficient
Ainter = Pinter A,
D restr. = RY5A.
via Einstein’s relation [3], Eq. [7] can be easily transformed into
[41
It is demonstrated in Figs. 2 and 3 that the experimental data and the theoretical estimates on the basis of Eq. [4] satisfactorily agree with each other. For displacements slightly exceeding the crystal dimensions, the NMR data contain contributions of both intra- (restricted) and long-range diffusion. In a first approximation, we assume the molecular displacements inside and outside the crystallites to be uncorrelated. In this case, the total mean-square displacement is the sum of the mean-square displacements due to restricted diffusion and due to long-range diffusion outside of the regarded crystal. Via Einsteins relation [3] we obtain the estimate
D Ix.
[71
= PinterDinter + (v’?ntra(A))/6A,PI
where Dint,y denotes the self-diffusion coefficient in the intercrystalline space. The migration path in the intracrystalline space (i.intra) is influenced by the energetic steps molecules have to overcome when leaving the crystallites (9, 10). For sufficiently high temperatures, the thermal energy of the molecules is high enough, so that the molecules easily can pass these resistances. In this case we have (GLra(A))l6A
= Dintra-
[91
For lower temperatures, however, molecules will be reflected repeatedly at the D eff = DL,, + Drestr., [51 crystallite boundaries, so ?intra will be considerably smaller than in an infinite, which we have applied to the representation homogeneous medium. In the mathematical in Fig. 2. It becomes evident that this expressions for spin-echo attenuation due to approximation satisfactorily describes the diffusion in a two-region system (9, 10, 23), transition from restricted to intercrystalline this fact was taken into account by introdiffusion. As required from theory, there ducing an effective coefficient D:,$,, of is no significant difference between the intracrystalline diffusion. This coefficient is Journal of Coiloid and Interface
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1981
244
KARGER,
characterized
KOCIRIK,
AND
by the inequality
and
Dintra 2 DCra 2 Drestr.,
UOI
where the equation D$,., = Dintra(Drestr,) was assumed to hold for sufficiently high (low) temperatures. Since the activation energy of PinterDinter is larger than that for Dintra (9, 10, 23, 24), the second term on the right-hand side of Eq. [8] becomes negligible for sufficiently high temperatures, so that indeed long-range diffusivity is determined by intercrystalline transport. With decreasing temperatures, however, the second term, though decreasing in its absolute value, increases relatively to the first one. In the limits of the considerations given until now, a further quantification of the influence of intercrystalline transport on long-range diffusion is impossible. For an estimate of the longrange diffusivities we have to start, therefore, from the complete set of diffusion equations under stationary conditions: div grad c, = 0,
ZIKANOVA
div grad c, = 0
[l I]
D, = WY,
our real system can be transfered into a hypothetical system with the respective concentrations c,(,) and diffusivities Daccj. It is apparent that this transformation does not lead to any change in the diffusion fluxes. The set of equations describing the diffusion behavior in this system can be determined by inserting these definitions into Eqs. [ 1 l] - [ 131. The resulting equations div grad c, = 0,
= D, 2
an
,
1121 must [I31
at the boundary between adjacent subregions. a/&r denotes the derivative normal to the phase boundary, and y stands for the Henry constant. Whereas the coefficient Dtnter, generally applied in NMR diffusion studies (9, 10, 23, 24) is defined on the basis of Einstein’s relation [3] for rms displacements much larger than the dimensions of the crystallites, D, denotes the microscopic (bulk) diffusivity in the intercrystalline space. Obviously, both quantities are related to each other by the equation
Dinter= Ds&,
cc = YCg Journal of Colloid and Interface
[I41
factor.
[W
Science, Vol. 84, No. 1, November
1981
[17] [I81
c, = c, u91 are of the form treated in a number of models and approximations in theoretical studies of mass transfer in heterogeneous media (25, 26). All estimates of long-range diffusivities, starting from the Eqs. [ 17]-[ 191 are based on Eq. [ 11, in which, for our hypothetical system, the concentration, c = pc, + (1 - p)cc = c,,
ca = wycg
where 7 denotes the tortuosity By the definitions
div grad c, = 0
Dintra % +ac,, on
with the matching conditions Dintra 2
[I61
ml
be inserted, the latter equality being a consequence of Eq. [19]. However, as we have learned from Eq. [2], in real systems of concentrations c, and cg, overa concentrations and with it also the corresponding concentration gradients are by a factor p < 1 smaller, than given by Eq. [20]. Hence, for maintaining the same fluxes, the total concentration gradients in the real system are smaller by this factor p than in the theoretical system of concentrations c, and c,. According to Eq. [ 11, the long-range diffusivities of the mathematical models must be multiplied therefore by an additional factor l/p to provide the corresponding quantities of the real adsorbateadsorbent system. In our estimates, we regarded the dependence of the normalized long-range diffusivity Dl,,,lD, on the ratio v = DintralDc of the diffusivities in the adsorbate and in
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the continuous phases for the dispersed phase fractions p = 0.25 and 0.5. Such representations allowed a direct comparison with our experimental data. For the mathematical analysis we used the customary models for diffusion through dilute suspensions of (spherical) crystallites in a continuous phase as formulated by
Maxwell (27) and developed by Rayleigh (28), Runge (29) and Meredith and Tobias (30), and for parallel-series arrangements of diffusional resistances as proposed by Jefferson et al. (31) and by Cheng and Vachon (32). Since details may be found in the original papers as well as in reviews by Barrer (25) and Crank (26), we will only present the relevant equations:
Dilute Suspension of Crystallites
Maxwell
(27): 1
D 1.r. D,
=F-
3
Pll
[(2 + v)l(l - v)] + p ;
Rayleigh (28)/Runge (29): 1
D 1.r. D,
Meredith D 1.r. D,
3 [(2 + v)l(l - v)] + p - 0.523[(1 - v)l(4/3) - .>lp’O’” ’
=,-
P21
and Tobias (30): 1
=T
3 + 1.227[(3 - 3v)l(4 + ~v)]P~‘~
-
~231
[(2 + v)l(l - v) +p + 0.409[(6 + 3u)l(4 + ~v)IP~‘~ - 0.90613 - 3u)l(4 + 3~)lp~O’~ Series-Parallel
Jefferson, 5
Witzell, Sibbett (31): ’
DC=;
Formulae
_
1 21p-1’3
P
+
0.513{1.61~-~/~
.
-213
- 2 + (1 - llv)l[v/(v
- 1) In v - l]}
; WI
Cheng and Vachon (32):4 B1’2
-1
x tan-l {-[B(l - v)]l[l
- B(l - v)]}“”
+ 1
B
1
,
v<
; WI
v>
>
Bl’”
x ln 1 + {IIW - I)141 + B(v - 1W2 + 1 1 - {[B(v - l)]/[l
+ B(v - l)]}“”
B
-1 1 ,
WI
with B = (3~12)~‘~. A model of molecular transport in heterogeneous media directly devoted to adsorbateadsorbent systems, recently has been proposed by Jury (33, 34). In contrast to previous Journal ojColloid
and Inferface
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1981
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KARGER,
KO&l?IK,
AND ZIKANOVA
calculations for adsorbate-adsorbent systems (4-7), his formulation includes the possibility that the crystallite boundary concentration is not constant over the surface of the individual crystallite:
1 + [( 1 - *)/2(9
- !P1)l In { [( 1 - T1)( 1 + ?)I/[( 1 - Y)(l + *‘,)I}
+ q/( 1 - W) I
1271
with qr
=
m31 those for intracrystalline
p”3,
!Pl = [l + (V”
- l)lV]“2.
WI
It becomes evident from the representation of the Eqs. [21]-[27] in Fig. 4, that the estimates of the long-range diffusivities though being obtained from quite different starting points, show similar trends. They can be regarded therefore as a reasonable basis for an analysis of the experimental results. DISCUSSION
It can be seen from Fig. 4 that the longrange diffusivities may be increased by a factor of 10 or more due to the influence of the intracrystalline mass transfer. That means, that the long-range diffusivities through assemblages of small- and largeport zeolite crystallites may differ by more than one order of magnitude, though both the intercrystalline diffusivities and the Henry constants might be identical. Such an effect should be observable, e.g., for paraffin diffusion in Na-Ca- A zeolites of different calcium content (35, 36). In the following we will compare the influence of intracrystalline transport on long-range diffusion to that of intercrystalline migration. In our experiments we could increase the temperature up to those temperatures, for which the coefficients for long-range diffusivities are clearly above 4 In the corresponding equation on p. 280 of Ref. (26) the correct definition of the quantity E there introduced, must be D, + B(Dd - D,) instead of D, - B(Dd Journal of
- D,). Colioid and Interface
Science, Vol. 84, No. 1, November
1981
diffusion. According to our considerations following Eq. [8], in this temperature region D1,,,is practically independent of intracrystalline transport and coincides with PinterDintere In order to find out, up to which extend D1,,,may depend on Dintrafor lower temperatures, we have marked in Figs. 5a-c hypothetical curves for D,, which were chosen in such a way that the theoretical long-range diffusivities calculated from them came into the region of the experimental values. The calculations were carried out by using the geometrical mean of the functional dependences of Fig. 4, regarding the three approximations for dilute suspensions of crystallites (Eqs. [2 l] - [23]) as one case only. It can be found by an analysis of the adsorption isotherms that for sorbate concentrations of 80 mg g-l, the gas phase concentration in the considered temperature interval is so small that mutual encounters of the molecules are negligible (37). Hence, molecular transport in the gas phase proceeds by Knudsen diffusion. The temperature dependence of D, = c,/c, .D, is therefore predominantly given by that of C&,, and D, should follow an Arrhenius dependence with an activation energy coinciding with the isosteric heat of adsorption (23, 37). It can be seen from our estimates of D ,.r., that the additional influence of intracrystalline diffusion, increasing with decreasing temperature, leads to a deviation from the Arrhenius dependence. On the other hand, corresponding deviations in the experimental data are-if existent at
MOLECULAR
FIG. 4. Ratio D,,,/D, of long-range over continuous phase diffusivities in dependence on the normalized intracrystalline diffusivity v = Dintra/Dc for dipersed phase fractions p = 0.5 (upper representation) and p = 0.25 (lower representation) according to the mathematical models of Maxwell (-), Rayleigh Runge (- - -), Meredith and Tobias (- -), Jefferson et al. (-,-), Cheng and Vachon (-..-), and Jury (-,-). The differences between the first three models for p = 0.25 (total range) as well as for p = 0.5 and I) 5 2 are within the uncertainty of the drawing.
all- within the uncertainty of our measurements, as indicated by the error bars in our representations. One can conclude, therefore, that in comparison with the experiments, the applied models will not underestimate the influence of intracrystalline transport. But even this influence is minor compared to that of intercrystalline diffusion. While in all our representations Di,,, is varying with D, (cPin+rDinter) over approximately four orders of magnitude, the influence of Dintra can affect D,,,, by a factor of about 10 at most. The chance for determining Dintra from D,.,. by this influence is rather small, therefore. In any case, it can be excluded, that this influence could lead to coinciding tendencies in the coefficients of long-range and intracrystalline diffusion over several orders of magnitude. This conclusion increases the significance of alternative explanations (38-40) for the analogous trends in uptake and NMR measurements on intracrystalline diffusion mentioned above (12, 14). Besides these general considerations, the present system provides three more arguments for the conclusion that long-range
247
TRANSPORT
diffusivities are predominantly determined by intercrystalline transport: (i) The activation energy of 44 + 4 kJ mole-l coincides with the values for the adsorption heats (42-44 kJ mole-‘) as taken from the literature (43). (ii) A priori estimates (41, 42) of Dl.r., using the relation D,,,, = PinterDi,tra,, lead to theoretical values of the order of magnitude of the experimental data. As an example, the values for T = 253 K are presented: With Knudsen’s formula Dinter = VA/~ (X = R) and v = (SkTh~rn)~~ = 310 m see-l, one obtains Din+r = 2.6 x 1O-4 rnz set-l, (small crystallites) and 8.2 x 1O-4 m” set-’ (large crystallites). From the adsorption isotherms (43, 44) one obtains for the gas pressure in equilibrium with the sorbate concentration of 80 mg g-l at 253 K, a value of about 2 Pa. Using the ideal gas law, one obtains Pinter -L 5 x 10eT (p = 0.5) and =15 x 10e7 (p = 0.25). Thus the long-range diffusivities are estimated to be PinterDinter I- 4 x IO-‘” rnz see-’ (large crystallite and unpressed small crystallite sample) and ~1.5 x 10-l” m’ set-’ (pressed small crystallite sample). These values do not differ by more than one order of magnitude from the experimental data (12 x 10-l” m’ set-‘, 4 x 10-l” m’ set-* and 1.2 x lo-l0 mz set-‘, respectively). (iii) If there were some significant influence of the intracrystalline diffusion on D l.r., this influence should be increased by the pressing procedure. However, in our experiments we only observe a parallel shift by a factor of about 3 to lower values. This decrease is readily explained by the fact that the increase in the crystallite phase fraction due to the pressing procedure leads to a decrease in Pinter by just this factor of 3. APPENDIX:
B C
Journal of
NOTATION
abbreviation used [=(3p/2)“‘] total concentration Co/hid and
Interface
in Eq.
Science, Vol. 84, No. I, November
[25]
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248
KARGER,
KOCIRIK,
AND ZIKANOVA
‘1 3 b
I -
5 103K /r
0
2 c
3 -
c 5 103K/7
6
FIG. 5. Theoretical temperature dependence for Dr.,. (-), calculated on the basis of the representations of Fig. 4 (geometric mean) from the experimental data for intracrystalline diffusion (- - -) and hypothetical values for D, (-). The values of D, were determined by the requirement that they follow an Arrhenius dependence and lead to theoretical values of Dr.,, in the region of the experimental data (area between the dotted lines). (a) butane/NaX (16 km, p = 0.5), cf. Fig. 2; (b) butane/NaX (S pm, loose assemblage, p i= 0.29, cf. Fig. 3, upper representation; (c) butane/NaX (5 pm, pressure compacted material, p = OS), cf. Fig. 3, lower representation.
concentration in the adsorbate phase concentration in the gas phase cg D (self-) diffusion coefficient (= diffusivity) diffusivity in the continuous phase DC of the hypothetic system ( =DgIy) D eff effective diffusivity bulk (Knudsen) diffusivity in the Dg intercrystalline space intercrystalline diffusivity, as deDinter fined by Einstein’s relation [3] for rms displacement B R (=Dgh) intracrystalline diffusivity Dintra D 1.r. effective diffusivity for rms displacements P R D restr. effective diffusivity for restricted diffusion (=R’/5A) diffusion flux density .i k Boltzmann factor m molecular mass volume fraction of the dispersed P phase relative number of molecules in Pinter the intercrystalline space R mean crystallite radius
ca
Journal of
CoNoid and
Interface Science,
Vol. 84, No. 1, November
1981
T V X
Y
A
A V
molecular displacement during A vector sum of molecular displacements in the intracrystalline space during A temperature molecular velocity space coordinate Henry constant (~c,/c~ 1equilibrium) observation time total life time in the intercrystalline space during observation time A molecular mean free path in the intercrystalline space normalized intracrystalline diffusivity (ZDintralDe) tortuosity factor abbreviation used in Eq. [271 [ep1’3]
abbreviation used in Eq. [-( 1 + (9-Z - l)lv)“2]
[271
ACKNOWLEDGMENTS The authors gratefully acknowledge the stimulus afforded by discussions with Professor H. Pfeifer.
MOLECULAR Thanks are also due to Professors D. M. Ruthven and S. H. Jury for valuable comments. REFERENCES 1. Breck, D. W., “Zeolite Molecular Sieves.” Wiley, New York, 1974. 2. Pfeifer, H., Kristall. Technik. 11, 577 (1976). 3. Uytterhoeven, J. B., Progr. Colloid Polymer Sci. 65, 233 (1978). 4. Ruckenstein, E., Vaidyanathan, A. S., and Youngquist, G. R., Chem. Eng. Sci. 26, 130 (1971). 5. KoCii-ik, M., and Zikanova, A., 2. Phys. Chem. (Leipzig) 250, 250 (1972). 6. Dubinin, M. M., Erashko, I. I., Kadlec, O., Ulin, V. I., Volos&rk, A. M., and Zolotarev, P. P., Carbon 13, 193 (1975). 7. Garg, D. R., and Ruthven, D. M., Chem. Eng. Sci. 28, 791 (1973). 8. Stejskal, E. O., and Tanner, J. E., J. Chem. Phys. 42, 288 (1965). 9. Pfeifer, H., in “N.M.R.-Basic Principles and Progress,” Vol. 7, p. 53. Springer, Berlin, 1972. 10. Pfeifer, H., Phys. Rep. (Phys. Left. C) 26, 293 (1976). 11. Karger, J. and Caro, J., J. Colloid Interface Sci. 52, 623 (1975). 12. Karger, J., and Caro, J., J. Chem. Sot. Faraday I 73, 1363 (1977). 13. Biilow, H., Karger, J., KoCirik, M., and VolosEuk, A. M., 2. Chem. 21, 175 (1981). 14. Ruthven, D. M., ACS Symp. Series, No. 40, 320 (1977). 15. Ash, R., and Barrer, R. M., Surface Sci. 8, 461 (1967). 16. Karger, J., Surface Sci. 36, 797 (1973). 17. Heink, W., Wiss. Z. Karl-Marx-Univ., Leipzig 23, 453 (1971). 18. Heink, W., Karger, J., and Walter, A., Exp. Techn. Physik 26, 161 (1978). 19. Tanner, J. E., J. Chem. Phys. 52, 2523 (1970). 20. Karger, J., Pfeifer, H., Rauscher, M., and Walter, A., J. Chem. Sot. Faraday I76, 717 (1980). 21. Doelle, H.-F., and Riekert, L., ACS Symp. Series, No. 40, 401 (1977).
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22. Tanner, J. E., and Stejskal, E. O., J. Chem. Phys. 49, 1768 (1968). 23. Karger, J., Z. Phys. Chemie (Leipzig) 248, 27 (1971). 24. Barrer, R. M., in “Characterization of Porous Solids,” p. 155. Society of Chemical Industry, London, 1979. 25. Barrer, R. M., in “Diffusion in Polymers,” p. 165. Academic Press, London/New York, 1968. 26. Crank, J., “The Mathematics of Diffusion,” p. 266. Oxford Univ. Press (Clarendon), London, 1975. 27. Maxwell, C., “Treatise on Electricity and Magnetism,” Vol. 1, p. 365. Oxford Univ. Press, London, 1873. 28. Rayleigh, Lord, Phil. Mag. 34, 481 (1892). 29. Runge, I., 2. Techn. Phys. 6, 61 (1925). 30. Meredith, R. E., andTobias, C. W., J. Appl. Phys. 31, 1270 (1960). 31. Jefferson, T. B., Witzell, 0. W., and Sibbett, W. L., 2nd. Eng. Chem. 50, 1589 (1958). 32. Cheng, S. C., and Vachon, R. I., Int. J. Heat Muss Trunsfer 12, 149 (1969). 33. Jury, S. H., Canad. J. Chem. Eng. 55, 538 (1977). 34. Jury, S. H., J. Franklin Inst. 305, 79 (1978). 35. Ruthven, D. M., Canad. J. Chem. 52, 3523 (1974). 36. Caro, J., Karger, J., Finger, G., Pfeifer, H., and Schollner, R., Z. Phys. Chem. (Leipzig) 257, 903 (1976). 37. Karger, J., and SamuleviE, N. N., Z. Chem. 18, 155 (1978). 38. Ruthven, D. M., Graham, A. M., and Vavlitis, A., in “Proceedings, 5th Conference on Molecular Sieves, Naples, 1980.” 39. Ruthven, D. M., Lee, Lap-Keung, and Yucel, H., AIChE J. 26, 16 (1980). 40. Yucel, H., and Ruthven, D. M., J. Colloid Interface Sci. 74, 186 (1980). 41. Karger, J., and Volkmer, P.,Z. Phys. Chem. (Leipzig) 261, 900 (1980). 42. Karger, J., Rauscher, M., and Torge, H., J. Colloid Interface Sci. 76, 525 (1980). 43. Stach, H., Thesis (Prom. B), Academy of Sciences of the German Democratic Republic, Berlin, 1977. 44. Geyer, W., Thesis (Prom. A), Leipzig, 1976.
Journal of Cdoid
and Inrerfice
Science, Vol. 84, No. 1, November
1981
Transport through Assemblages of Microporous Particles JiiRG KARGER
NMR-Labor
der Sektion Physik der Karl-Marx-Universitit, Linntstrasse 5, German Democratic
DDR-7010
Leipzig,
Republic
AND
MILAN Heyrovsky
Institute
KO&RIK
ARLETTE
AND
ZIKANOVA
of Physical Chemistry and Electrochemistry, Czechoslovak CSSR-12138 Prague 2, Machova 7, Czechoslovakia
Academy
of Sciences,
Received December 12, 1980; accepted March 25, 1981 Steady-state diffusion studies and the NMR pulsed field gradient technique are introduced as equivalent methods for determining long-range diffusivities. Theoretical methods for mass transfer in heterogeneous media are reconsidered and adapted to adsorbate-adsorbent systems. Comparison with the experimental data for n-butane diffusion in NaX zeolites shows that long-range diffusivities are predominantly determined by intercrystalline transport. Estimates of the absolute value of the temperature dependence and of the influence of the pressure of compaction are in agreement with this conclusion. INTRODUCTION
much larger than the crystallite diameters (“long-range” diffusion) does not depend therefore on the intracrystalline diffusivities. However, it can be shown by the application of the NMR pulsed field gradient technique (g-lo), that for a number of zeolitic adsorbate-adsorbent systems the coefficients of intracrystalline diffusion are considerably higher than previously assumed (lo- 12) and that the above assumption is not valid. In view of this conclusion it is worthwhile to reconsider, whether the influence of intracrystalline transport on long-range diffusivities is indeed negligible. A distinct functional relationship between long-range and intracrystalline diffusivities would be of twofold significance: (i) In many cases, uptake on the individual crystallites is too fast to allow a determination of intracrystalline diffusivities. This difficulty could be circumvented by uptake measurements on the
With the application of molecular sieve crystals (1, 2) to numerous technical processes (3), the study of transport phenomena through assemblages of microporous particles was stimulated not only by scientific interest but also by the requirement of practical application. Most theoretical work on diffusion in assemblages of molecular sieve crystals circumvents the difficulties of the mathematical treatment of compound media by assuming that the efficiency of molecular transport in the interior of the crystallites is much less than in the intercrystalline space. In this case, the surface concentration over an individual crystallite is constant (only depending on time and on the position of the crystallite in the assemblage) and inter- and intracrystalline transport can be separated from each other (4-7). Molecular transport over distances 240 0021-9797/81/l 10240-10$02.00/O Copyright 0 1981 by Academic Press, Inc. Au rights of reproduction in any form reserved.
Journal of Colloid and Interface
Science, Vol. 84, No. I, November
1981
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diffusivities through assemblages of microporous particles. Maintaining fixed sorbate concentrations at the boundaries, the longrange diffusivity Di.,. is obtained from the particle flux density j and from the gradient of overall concentration c by the equation direction
diffusion flux
of a
FIG. 1. Scheme of the experimental arrangement for determining long-range diffusivities through assemblages of microporous crystallites under steadystate conditions. Different hatching indicates different sorbate concentration.
crystallite assemblage. Moreover, in this case, also, steady-state techniques could be applied for which a number of disturbing influences significant for transient experiments (such as due to the finite rates of adsorption heat dissipation and adsorbate supply (13)) can be excluded. (ii) Previous comparative studies between the diffusivities obtained directly by the NMR pulsed field gradient technique and those from uptake experiments (12, 14) showed similar trends in the characteristic dependences but discrepancies up to five orders of magnitude in the absolute values. Taking into account that the uptake experiments have been carried out with zeolite assemblages, this result could be explained by the relationship mentioned above. It is the aim of the paper to give an estimate of the dependence of the long-range diffusivity on intracrystalline diffusion. EXPERIMENTAL
Figure 1 shows the arrangement of a steady state experiment for determining the
j = -Dl,,.dcldx.
111
Overall concentrations are referred to volume elements much larger than the crystallites and can easily be determined from the microscopic concentrations in the adsorbate (c,) and gaseous phases (c,) and the volume fraction p of the dispersed phase by the relation c = pc, + (1 - p)cp = PC,.
PI
Under the assumption that the thermodynamic factor d In c,/d In c, is equal to one and that any bulk (Poiseuille) flow due to pressure gradients in the intercrystalline pore system can be neglected (which holds exactly for sufficiently low sorbate concentrations and for tracer counter flow experiments), the coefficients of diffusion and of self-diffusion coincide (15, 16). Consequently, the quantity Dl.r., as introduced by Eq. [l] should be accessible also from selfdiffusion measurements, i.e., from the study of the velocity of molecular redistribution under uniform concentration over the sample. In the last few years, the NMR pulsed field gradient technique (8) has become a valuable tool for studying self-diffusion in adsorbate-adsorbent systems (9, 10). This technique can directly determine the molecular mean square displacements (i-‘(A)) during a time interval A. Via Einstein’s equation (?(A))
= 6DA,
[31
one may obtain the coefficients of both intracrystalline (( rf’) 112< crystallite radius R) and long-range (( J2)1ip + R) self-diffusion With the assumption following Eq. [2], the latter quantity coincides with D,,,,. nmml of Colioid and lnrerface Science-, Vol. 84, No. 1, November
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KARGER,
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This versatility in the information about molecular transport is the main advantage of the NMR pulsed field gradient technique in comparison with the traditional “macroscopic” methods (8- 10). Therefore we have applied this latter technique in our experiments. The measurements have been carried out with a 60-MHz pulse spectrometer, specially built in the NMR Laboratory of the Physics Department of the Karl Marx University, Leipzig (17). As described in detail previously (9, 18), mean-square displacements were determined from recording the spinecho height for different field gradient intensities. For attaining observation times, A, up to 200 msec, the stimulated echo was used (19). We applied as an adsorbent NaX zeolite crystallites with mean diameters of about 16 and 5 pm, synthesized by idanov and Samulevii:, Leningrad. The measurements were carried out in 7-mm-diameter sealed sample tubes, containing the adsorbate-adsorbent system. Activation of the zeolite crystals in the sample tubes prior to the introduction of the adsorbate (n-butane, purity > 99%) was accomplished by evacuating the samples at 400°C and 5 x lop3 Pa for 20 hr. The amount sorbed was determined gravimetrically and volumetrically and checked afterward by the intensity of the NMR signal of the sealed samples (18). In all our experiments we tried to realize sorbate concentrations of about 80 mg butane per 1 g activated adsorbent. This corresponds to a relative pore-filling factor of about 0.5. The mean error in the sorbate concentrations determined in the above mentioned way is of the order of 10 mg g-l maximum (18). With many zeolite samples one can find that the crystallite packing density decreases with decreasing crystallite size. In our experiments, the volume fractions p of the dipersed phases were about 0.5 and 0.25 for the large and for the small crystallites, respectively. In one case, prior Journal of Colloid and Interface
Science, Vol. 84, No. 1, November
1981
I,
AND ZIKANOVA
FIG. 2. NMR data for butane self-diffusion in an NaX zeolite assemblage with a mean crystallite diameter of about 16 pm and for a sorbate concentration of about 80 mg g-r, open symbols: values for D,rr as defined by Eq. [3] for observation times A = 50 msec (V), 100 msec (A), 200 msec (0), full symbols: estimates of the long-range diffusivities from the values of Derf in the vicinity of the region of restricted diffusion, dashed line: intracrystalline diffusivities taken from Ref. (20).
to sample preparation the zeolite material of a small-crystallite sample was compacted by a pressure of 2.5 MPa, thus realizing also a dipersed phase fraction of about 0.5. Self-diffusion
Measurements
Figures 2 and 3 show the effective diffusion coefficients obtained by the NMR pulsed field gradient technique for the given observation times on the basis of Eq. [3]. For comparison, the coefficients of intracrystalline self-diffusion obtained in previous experiments (20) are included. It has been shown by Riekert and co-workers that these values are in satisfactory agreement with uptake experiments (21). The effective diffusion coefficients can be readily attributed to the regions of “intercrystalline” diffusion (( r2)li2 > R; T 2 -40°C in Fig. 2, T 2 -60°C in Fig. 3) and of “restricted” diffusion (9, 10). According to the theory of restricted diffusion (9, 11, 22), in the latter case the spin-echo attenuation should
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long-range diffusivities obtained for different observation times. In the following, we discuss to what extent the long-range diffusivities are determined by intracrystalline transport. THEORETICAL
FIG. 3. Temperature dependence ofD,, for butane in a loose assemblage (0) and in a pressed compact (0) of NaX zeolite crystals with a mean diameter of about 5 pm for a sorbate concentration of about 80 mg g-l, dashed line: intracrystalline diffusivities taken from Ref. (20).
As a first approximation, we represent the molecular mean-square displacement due to long-range diffusions as the sum of molecular mean-square displacements in the intraand intercrystalline spaces:
(v”(A)) = (%.,(A)) + (%tra@>). 81 Since the total lifetime Ainter in the intercrystalline space is related to the relative number Pinter of molecules in the intercrystalline space @inter < 1) by the condition
be the same as for a system of molecules moving freely with a diffusion coefficient
Ainter = Pinter A,
D restr. = RY5A.
via Einstein’s relation [3], Eq. [7] can be easily transformed into
[41
It is demonstrated in Figs. 2 and 3 that the experimental data and the theoretical estimates on the basis of Eq. [4] satisfactorily agree with each other. For displacements slightly exceeding the crystal dimensions, the NMR data contain contributions of both intra- (restricted) and long-range diffusion. In a first approximation, we assume the molecular displacements inside and outside the crystallites to be uncorrelated. In this case, the total mean-square displacement is the sum of the mean-square displacements due to restricted diffusion and due to long-range diffusion outside of the regarded crystal. Via Einsteins relation [3] we obtain the estimate
D Ix.
[71
= PinterDinter + (v’?ntra(A))/6A,PI
where Dint,y denotes the self-diffusion coefficient in the intercrystalline space. The migration path in the intracrystalline space (i.intra) is influenced by the energetic steps molecules have to overcome when leaving the crystallites (9, 10). For sufficiently high temperatures, the thermal energy of the molecules is high enough, so that the molecules easily can pass these resistances. In this case we have (GLra(A))l6A
= Dintra-
[91
For lower temperatures, however, molecules will be reflected repeatedly at the D eff = DL,, + Drestr., [51 crystallite boundaries, so ?intra will be considerably smaller than in an infinite, which we have applied to the representation homogeneous medium. In the mathematical in Fig. 2. It becomes evident that this expressions for spin-echo attenuation due to approximation satisfactorily describes the diffusion in a two-region system (9, 10, 23), transition from restricted to intercrystalline this fact was taken into account by introdiffusion. As required from theory, there ducing an effective coefficient D:,$,, of is no significant difference between the intracrystalline diffusion. This coefficient is Journal of Coiloid and Interface
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1981
244
KARGER,
characterized
KOCIRIK,
AND
by the inequality
and
Dintra 2 DCra 2 Drestr.,
UOI
where the equation D$,., = Dintra(Drestr,) was assumed to hold for sufficiently high (low) temperatures. Since the activation energy of PinterDinter is larger than that for Dintra (9, 10, 23, 24), the second term on the right-hand side of Eq. [8] becomes negligible for sufficiently high temperatures, so that indeed long-range diffusivity is determined by intercrystalline transport. With decreasing temperatures, however, the second term, though decreasing in its absolute value, increases relatively to the first one. In the limits of the considerations given until now, a further quantification of the influence of intercrystalline transport on long-range diffusion is impossible. For an estimate of the longrange diffusivities we have to start, therefore, from the complete set of diffusion equations under stationary conditions: div grad c, = 0,
ZIKANOVA
div grad c, = 0
[l I]
D, = WY,
our real system can be transfered into a hypothetical system with the respective concentrations c,(,) and diffusivities Daccj. It is apparent that this transformation does not lead to any change in the diffusion fluxes. The set of equations describing the diffusion behavior in this system can be determined by inserting these definitions into Eqs. [ 1 l] - [ 131. The resulting equations div grad c, = 0,
= D, 2
an
,
1121 must [I31
at the boundary between adjacent subregions. a/&r denotes the derivative normal to the phase boundary, and y stands for the Henry constant. Whereas the coefficient Dtnter, generally applied in NMR diffusion studies (9, 10, 23, 24) is defined on the basis of Einstein’s relation [3] for rms displacements much larger than the dimensions of the crystallites, D, denotes the microscopic (bulk) diffusivity in the intercrystalline space. Obviously, both quantities are related to each other by the equation
Dinter= Ds&,
cc = YCg Journal of Colloid and Interface
[I41
factor.
[W
Science, Vol. 84, No. 1, November
1981
[17] [I81
c, = c, u91 are of the form treated in a number of models and approximations in theoretical studies of mass transfer in heterogeneous media (25, 26). All estimates of long-range diffusivities, starting from the Eqs. [ 17]-[ 191 are based on Eq. [ 11, in which, for our hypothetical system, the concentration, c = pc, + (1 - p)cc = c,,
ca = wycg
where 7 denotes the tortuosity By the definitions
div grad c, = 0
Dintra % +ac,, on
with the matching conditions Dintra 2
[I61
ml
be inserted, the latter equality being a consequence of Eq. [19]. However, as we have learned from Eq. [2], in real systems of concentrations c, and cg, overa concentrations and with it also the corresponding concentration gradients are by a factor p < 1 smaller, than given by Eq. [20]. Hence, for maintaining the same fluxes, the total concentration gradients in the real system are smaller by this factor p than in the theoretical system of concentrations c, and c,. According to Eq. [ 11, the long-range diffusivities of the mathematical models must be multiplied therefore by an additional factor l/p to provide the corresponding quantities of the real adsorbateadsorbent system. In our estimates, we regarded the dependence of the normalized long-range diffusivity Dl,,,lD, on the ratio v = DintralDc of the diffusivities in the adsorbate and in
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the continuous phases for the dispersed phase fractions p = 0.25 and 0.5. Such representations allowed a direct comparison with our experimental data. For the mathematical analysis we used the customary models for diffusion through dilute suspensions of (spherical) crystallites in a continuous phase as formulated by
Maxwell (27) and developed by Rayleigh (28), Runge (29) and Meredith and Tobias (30), and for parallel-series arrangements of diffusional resistances as proposed by Jefferson et al. (31) and by Cheng and Vachon (32). Since details may be found in the original papers as well as in reviews by Barrer (25) and Crank (26), we will only present the relevant equations:
Dilute Suspension of Crystallites
Maxwell
(27): 1
D 1.r. D,
=F-
3
Pll
[(2 + v)l(l - v)] + p ;
Rayleigh (28)/Runge (29): 1
D 1.r. D,
Meredith D 1.r. D,
3 [(2 + v)l(l - v)] + p - 0.523[(1 - v)l(4/3) - .>lp’O’” ’
=,-
P21
and Tobias (30): 1
=T
3 + 1.227[(3 - 3v)l(4 + ~v)]P~‘~
-
~231
[(2 + v)l(l - v) +p + 0.409[(6 + 3u)l(4 + ~v)IP~‘~ - 0.90613 - 3u)l(4 + 3~)lp~O’~ Series-Parallel
Jefferson, 5
Witzell, Sibbett (31): ’
DC=;
Formulae
_
1 21p-1’3
P
+
0.513{1.61~-~/~
.
-213
- 2 + (1 - llv)l[v/(v
- 1) In v - l]}
; WI
Cheng and Vachon (32):4 B1’2
-1
x tan-l {-[B(l - v)]l[l
- B(l - v)]}“”
+ 1
B
1
,
v<
; WI
v>
>
Bl’”
x ln 1 + {IIW - I)141 + B(v - 1W2 + 1 1 - {[B(v - l)]/[l
+ B(v - l)]}“”
B
-1 1 ,
WI
with B = (3~12)~‘~. A model of molecular transport in heterogeneous media directly devoted to adsorbateadsorbent systems, recently has been proposed by Jury (33, 34). In contrast to previous Journal ojColloid
and Inferface
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1981
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KARGER,
KO&l?IK,
AND ZIKANOVA
calculations for adsorbate-adsorbent systems (4-7), his formulation includes the possibility that the crystallite boundary concentration is not constant over the surface of the individual crystallite:
1 + [( 1 - *)/2(9
- !P1)l In { [( 1 - T1)( 1 + ?)I/[( 1 - Y)(l + *‘,)I}
+ q/( 1 - W) I
1271
with qr
=
m31 those for intracrystalline
p”3,
!Pl = [l + (V”
- l)lV]“2.
WI
It becomes evident from the representation of the Eqs. [21]-[27] in Fig. 4, that the estimates of the long-range diffusivities though being obtained from quite different starting points, show similar trends. They can be regarded therefore as a reasonable basis for an analysis of the experimental results. DISCUSSION
It can be seen from Fig. 4 that the longrange diffusivities may be increased by a factor of 10 or more due to the influence of the intracrystalline mass transfer. That means, that the long-range diffusivities through assemblages of small- and largeport zeolite crystallites may differ by more than one order of magnitude, though both the intercrystalline diffusivities and the Henry constants might be identical. Such an effect should be observable, e.g., for paraffin diffusion in Na-Ca- A zeolites of different calcium content (35, 36). In the following we will compare the influence of intracrystalline transport on long-range diffusion to that of intercrystalline migration. In our experiments we could increase the temperature up to those temperatures, for which the coefficients for long-range diffusivities are clearly above 4 In the corresponding equation on p. 280 of Ref. (26) the correct definition of the quantity E there introduced, must be D, + B(Dd - D,) instead of D, - B(Dd Journal of
- D,). Colioid and Interface
Science, Vol. 84, No. 1, November
1981
diffusion. According to our considerations following Eq. [8], in this temperature region D1,,,is practically independent of intracrystalline transport and coincides with PinterDintere In order to find out, up to which extend D1,,,may depend on Dintrafor lower temperatures, we have marked in Figs. 5a-c hypothetical curves for D,, which were chosen in such a way that the theoretical long-range diffusivities calculated from them came into the region of the experimental values. The calculations were carried out by using the geometrical mean of the functional dependences of Fig. 4, regarding the three approximations for dilute suspensions of crystallites (Eqs. [2 l] - [23]) as one case only. It can be found by an analysis of the adsorption isotherms that for sorbate concentrations of 80 mg g-l, the gas phase concentration in the considered temperature interval is so small that mutual encounters of the molecules are negligible (37). Hence, molecular transport in the gas phase proceeds by Knudsen diffusion. The temperature dependence of D, = c,/c, .D, is therefore predominantly given by that of C&,, and D, should follow an Arrhenius dependence with an activation energy coinciding with the isosteric heat of adsorption (23, 37). It can be seen from our estimates of D ,.r., that the additional influence of intracrystalline diffusion, increasing with decreasing temperature, leads to a deviation from the Arrhenius dependence. On the other hand, corresponding deviations in the experimental data are-if existent at
MOLECULAR
FIG. 4. Ratio D,,,/D, of long-range over continuous phase diffusivities in dependence on the normalized intracrystalline diffusivity v = Dintra/Dc for dipersed phase fractions p = 0.5 (upper representation) and p = 0.25 (lower representation) according to the mathematical models of Maxwell (-), Rayleigh Runge (- - -), Meredith and Tobias (- -), Jefferson et al. (-,-), Cheng and Vachon (-..-), and Jury (-,-). The differences between the first three models for p = 0.25 (total range) as well as for p = 0.5 and I) 5 2 are within the uncertainty of the drawing.
all- within the uncertainty of our measurements, as indicated by the error bars in our representations. One can conclude, therefore, that in comparison with the experiments, the applied models will not underestimate the influence of intracrystalline transport. But even this influence is minor compared to that of intercrystalline diffusion. While in all our representations Di,,, is varying with D, (cPin+rDinter) over approximately four orders of magnitude, the influence of Dintra can affect D,,,, by a factor of about 10 at most. The chance for determining Dintra from D,.,. by this influence is rather small, therefore. In any case, it can be excluded, that this influence could lead to coinciding tendencies in the coefficients of long-range and intracrystalline diffusion over several orders of magnitude. This conclusion increases the significance of alternative explanations (38-40) for the analogous trends in uptake and NMR measurements on intracrystalline diffusion mentioned above (12, 14). Besides these general considerations, the present system provides three more arguments for the conclusion that long-range
247
TRANSPORT
diffusivities are predominantly determined by intercrystalline transport: (i) The activation energy of 44 + 4 kJ mole-l coincides with the values for the adsorption heats (42-44 kJ mole-‘) as taken from the literature (43). (ii) A priori estimates (41, 42) of Dl.r., using the relation D,,,, = PinterDi,tra,, lead to theoretical values of the order of magnitude of the experimental data. As an example, the values for T = 253 K are presented: With Knudsen’s formula Dinter = VA/~ (X = R) and v = (SkTh~rn)~~ = 310 m see-l, one obtains Din+r = 2.6 x 1O-4 rnz set-l, (small crystallites) and 8.2 x 1O-4 m” set-’ (large crystallites). From the adsorption isotherms (43, 44) one obtains for the gas pressure in equilibrium with the sorbate concentration of 80 mg g-l at 253 K, a value of about 2 Pa. Using the ideal gas law, one obtains Pinter -L 5 x 10eT (p = 0.5) and =15 x 10e7 (p = 0.25). Thus the long-range diffusivities are estimated to be PinterDinter I- 4 x IO-‘” rnz see-’ (large crystallite and unpressed small crystallite sample) and ~1.5 x 10-l” m’ set-’ (pressed small crystallite sample). These values do not differ by more than one order of magnitude from the experimental data (12 x 10-l” m’ set-‘, 4 x 10-l” m’ set-* and 1.2 x lo-l0 mz set-‘, respectively). (iii) If there were some significant influence of the intracrystalline diffusion on D l.r., this influence should be increased by the pressing procedure. However, in our experiments we only observe a parallel shift by a factor of about 3 to lower values. This decrease is readily explained by the fact that the increase in the crystallite phase fraction due to the pressing procedure leads to a decrease in Pinter by just this factor of 3. APPENDIX:
B C
Journal of
NOTATION
abbreviation used [=(3p/2)“‘] total concentration Co/hid and
Interface
in Eq.
Science, Vol. 84, No. I, November
[25]
1981
248
KARGER,
KOCIRIK,
AND ZIKANOVA
‘1 3 b
I -
5 103K /r
0
2 c
3 -
c 5 103K/7
6
FIG. 5. Theoretical temperature dependence for Dr.,. (-), calculated on the basis of the representations of Fig. 4 (geometric mean) from the experimental data for intracrystalline diffusion (- - -) and hypothetical values for D, (-). The values of D, were determined by the requirement that they follow an Arrhenius dependence and lead to theoretical values of Dr.,, in the region of the experimental data (area between the dotted lines). (a) butane/NaX (16 km, p = 0.5), cf. Fig. 2; (b) butane/NaX (S pm, loose assemblage, p i= 0.29, cf. Fig. 3, upper representation; (c) butane/NaX (5 pm, pressure compacted material, p = OS), cf. Fig. 3, lower representation.
concentration in the adsorbate phase concentration in the gas phase cg D (self-) diffusion coefficient (= diffusivity) diffusivity in the continuous phase DC of the hypothetic system ( =DgIy) D eff effective diffusivity bulk (Knudsen) diffusivity in the Dg intercrystalline space intercrystalline diffusivity, as deDinter fined by Einstein’s relation [3] for rms displacement B R (=Dgh) intracrystalline diffusivity Dintra D 1.r. effective diffusivity for rms displacements P R D restr. effective diffusivity for restricted diffusion (=R’/5A) diffusion flux density .i k Boltzmann factor m molecular mass volume fraction of the dispersed P phase relative number of molecules in Pinter the intercrystalline space R mean crystallite radius
ca
Journal of
CoNoid and
Interface Science,
Vol. 84, No. 1, November
1981
T V X
Y
A
A V
molecular displacement during A vector sum of molecular displacements in the intracrystalline space during A temperature molecular velocity space coordinate Henry constant (~c,/c~ 1equilibrium) observation time total life time in the intercrystalline space during observation time A molecular mean free path in the intercrystalline space normalized intracrystalline diffusivity (ZDintralDe) tortuosity factor abbreviation used in Eq. [271 [ep1’3]
abbreviation used in Eq. [-( 1 + (9-Z - l)lv)“2]
[271
ACKNOWLEDGMENTS The authors gratefully acknowledge the stimulus afforded by discussions with Professor H. Pfeifer.
MOLECULAR Thanks are also due to Professors D. M. Ruthven and S. H. Jury for valuable comments. REFERENCES 1. Breck, D. W., “Zeolite Molecular Sieves.” Wiley, New York, 1974. 2. Pfeifer, H., Kristall. Technik. 11, 577 (1976). 3. Uytterhoeven, J. B., Progr. Colloid Polymer Sci. 65, 233 (1978). 4. Ruckenstein, E., Vaidyanathan, A. S., and Youngquist, G. R., Chem. Eng. Sci. 26, 130 (1971). 5. KoCii-ik, M., and Zikanova, A., 2. Phys. Chem. (Leipzig) 250, 250 (1972). 6. Dubinin, M. M., Erashko, I. I., Kadlec, O., Ulin, V. I., Volos&rk, A. M., and Zolotarev, P. P., Carbon 13, 193 (1975). 7. Garg, D. R., and Ruthven, D. M., Chem. Eng. Sci. 28, 791 (1973). 8. Stejskal, E. O., and Tanner, J. E., J. Chem. Phys. 42, 288 (1965). 9. Pfeifer, H., in “N.M.R.-Basic Principles and Progress,” Vol. 7, p. 53. Springer, Berlin, 1972. 10. Pfeifer, H., Phys. Rep. (Phys. Left. C) 26, 293 (1976). 11. Karger, J. and Caro, J., J. Colloid Interface Sci. 52, 623 (1975). 12. Karger, J., and Caro, J., J. Chem. Sot. Faraday I 73, 1363 (1977). 13. Biilow, H., Karger, J., KoCirik, M., and VolosEuk, A. M., 2. Chem. 21, 175 (1981). 14. Ruthven, D. M., ACS Symp. Series, No. 40, 320 (1977). 15. Ash, R., and Barrer, R. M., Surface Sci. 8, 461 (1967). 16. Karger, J., Surface Sci. 36, 797 (1973). 17. Heink, W., Wiss. Z. Karl-Marx-Univ., Leipzig 23, 453 (1971). 18. Heink, W., Karger, J., and Walter, A., Exp. Techn. Physik 26, 161 (1978). 19. Tanner, J. E., J. Chem. Phys. 52, 2523 (1970). 20. Karger, J., Pfeifer, H., Rauscher, M., and Walter, A., J. Chem. Sot. Faraday I76, 717 (1980). 21. Doelle, H.-F., and Riekert, L., ACS Symp. Series, No. 40, 401 (1977).
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22. Tanner, J. E., and Stejskal, E. O., J. Chem. Phys. 49, 1768 (1968). 23. Karger, J., Z. Phys. Chemie (Leipzig) 248, 27 (1971). 24. Barrer, R. M., in “Characterization of Porous Solids,” p. 155. Society of Chemical Industry, London, 1979. 25. Barrer, R. M., in “Diffusion in Polymers,” p. 165. Academic Press, London/New York, 1968. 26. Crank, J., “The Mathematics of Diffusion,” p. 266. Oxford Univ. Press (Clarendon), London, 1975. 27. Maxwell, C., “Treatise on Electricity and Magnetism,” Vol. 1, p. 365. Oxford Univ. Press, London, 1873. 28. Rayleigh, Lord, Phil. Mag. 34, 481 (1892). 29. Runge, I., 2. Techn. Phys. 6, 61 (1925). 30. Meredith, R. E., andTobias, C. W., J. Appl. Phys. 31, 1270 (1960). 31. Jefferson, T. B., Witzell, 0. W., and Sibbett, W. L., 2nd. Eng. Chem. 50, 1589 (1958). 32. Cheng, S. C., and Vachon, R. I., Int. J. Heat Muss Trunsfer 12, 149 (1969). 33. Jury, S. H., Canad. J. Chem. Eng. 55, 538 (1977). 34. Jury, S. H., J. Franklin Inst. 305, 79 (1978). 35. Ruthven, D. M., Canad. J. Chem. 52, 3523 (1974). 36. Caro, J., Karger, J., Finger, G., Pfeifer, H., and Schollner, R., Z. Phys. Chem. (Leipzig) 257, 903 (1976). 37. Karger, J., and SamuleviE, N. N., Z. Chem. 18, 155 (1978). 38. Ruthven, D. M., Graham, A. M., and Vavlitis, A., in “Proceedings, 5th Conference on Molecular Sieves, Naples, 1980.” 39. Ruthven, D. M., Lee, Lap-Keung, and Yucel, H., AIChE J. 26, 16 (1980). 40. Yucel, H., and Ruthven, D. M., J. Colloid Interface Sci. 74, 186 (1980). 41. Karger, J., and Volkmer, P.,Z. Phys. Chem. (Leipzig) 261, 900 (1980). 42. Karger, J., Rauscher, M., and Torge, H., J. Colloid Interface Sci. 76, 525 (1980). 43. Stach, H., Thesis (Prom. B), Academy of Sciences of the German Democratic Republic, Berlin, 1977. 44. Geyer, W., Thesis (Prom. A), Leipzig, 1976.
Journal of Cdoid
and Inrerfice
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1981