Langmuir 1999, 15, 6091-6102
6091
Heterogeneous Three-Site Lattice Model for Adsorption of Aromatics in ZSM-5 Zeolites: Temperature Dependence of Adsorption Isotherms† J. Narkiewicz-Michałek, P. Szabelski, and W. Rudzin´ski* Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Skłodowska University, 20-031 Lublin, Poland
A. S. T. Chiang Department of Chemical Engineering, National Central University, Chung-Li, Taiwan, Republic of China 32054 Received October 22, 1998. In Final Form: March 18, 1999 The three-site lattice model of collective localized adsorption of aromatics in ZSM-5 zeolites, presented in our previous publications, is extended by taking into account the effects of energetic heterogeneity of the sites of the same type. The appropriate theoretical equations are derived and used for simultaneous description of the experimental adsorption isotherms and heats of adsorption of benzene and p-xylene in silicalite at 303 K. It is shown that taking into account this additional level of heterogeneity leads to a much better description of both the adsorption isotherms and the related heats of adsorption in these systems. The extended model also allows one to predict correctly the adsorption isotherms of benzene and p-xylene in silicalite at different temperatures using the parameters found at one temperature.
Introduction MFI-type zeolite is one of the most versatile and valuable zeolites in modern hydrocarbon processing. A wide range of applications including catalytic and adsorptive processes has been proposed based on the size and configuration of its pore structure. In particular, the adsorption of aromatic compounds in ZSM-5, and its aluminum-deficient structural analog silicalite, has been extensively studied in the recent years. Various experimental methods including isotherm measurements,1-7 isosteric heat8-10 or calorimetry,11-13 gas chromatography,14,15 detailed X-ray or neutron powder diffraction,16-21 or NMR22,23 have been † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998. * To whom the correspondence should be addressed. Fax/phone: 48(81)5375685. E-mail:
[email protected].
(1) Anderson, J. R.; Foger, K.; Mole, T.; Rajadyaksha, R. A.; Snaders, J. V. J. Catal. 1979, 58, 114. (2) Jacobs, P. A.; Beyer, H. K.; Valyon, J. Zeolites 1981, 1, 161. (3) Wu, P.; Debebe, A.; Ma, Y. H. Zeolites 1983, 3, 118. (4) Lohse; U.; Fahlke, B. Chem. Tech. (Leipzig) 1983, 35, 350. (5) Choudhary, V. R.; Srinivasan, K. R. J. Catal. 1986, 102, 328. (6) Stach, H.; Lohse, U.; Thamm, H.; Schirmer, W. Zeolites 1986, 6, 74. (7) Guo, C. J.; Talu, O.; Hayhurst, D. T. AIChE J. 1989, 35, 573. (8) Pope, C. G. J. Phys. Chem. 1984, 88, 6312. (9) Pope, C. G. J. Phys. Chem. 1986, 90, 835. (10) Richards, R. E.; Rees, L. V. C. Zeolites 1988, 8, 35. (11) Thamm, H. J. Phys. Chem. 1987, 91, 8. (12) Thamm, H. Zeolites 1987, 7, 341. (13) Stach, H.; Thamm, H.; Janchen, J.; Fiedler, K.; Schirmer, W. In Proceedings of the 6th International Conference on Zeolites; Bisio, A., Olson, D. H., Eds.; Butterworths: London, 1984; p 225. (14) Lechert, H.; Schweitzer, W. In Proceedings of the 6th International Conference on Zeolites; Bisio, A., Olson, D. H., Eds.; Butterworths: London, 1984; p 210. (15) Choudhary, V. R.; Srinivasan, K. R. Chem. Eng. Sci. 1987, 42, 382. (16) Mentzen, B. F.; Vigne-Maeder, F. Mater. Res. Bull. 1987, 22, 309. (17) Mentzen, B. F.; Bosselet, F. Mater. Res. Bull. 1988, 23, 227. (18) Sacerdote, M.; Bosselet, F.; Mentzen, B. F. Mater. Res. Bull. 1990, 25, 593.
used. The results reveal some unusual adsorption characteristics of the MFI/aromatic systems, associated with the site heterogeneity and the possibility of a combined zeolite-adsorbate phase transition. This unusual behavior is the consequence of both the peculiar framework structure of MFI zeolites and the strong interactions induced by the tight fit-situation. At least three different locations for adsorption have been suggested by considering the topology of the MFI framework. Site heterogeneity in silicalite was first reported in the calorimetric studies of Thamm.11,12 The adsorption heat exhibited a strange jump at intermediate loading for benzene, ethylbenzene, and toluene as well as for some alkanes. This could only be explained in terms of a rapid change in the state of the admolecules. A phase transition in the ZSM-5/p-xylene system was confirmed.10,24 Talu et al.7,25 outlined the phase boundaries of ZSM-5/benzene and ZSM-5/p-xylene systems with isotherms measured at different temperatures, but the appearance of a phase transition in the benzene/silicalite system is still an open question. Using a deuterium NMR, Portsmouth and Gladden22 proved that the adsorbed benzene molecules were indeed under different environments before and after the step on the adsorption isotherm. More recently, the powder diffraction studies of Mentzen et al.16-21 of the MFI/ aromatics systems revealed the presence of several types of zeolite/admolecule complexes. In many cases,19-21 the (19) Mentzen, B. F. Mater. Res. Bull. 1992, 27, 831. (20) Sacerdote, M.; Mentzen, B. F. Mater. Res. Bull. 1993, 28, 767. (21) Mentzen, B. F.; Peronnet, M. S.; Be´rar, J.; Lefebvre, F. Zeolites 1993, 13, 485. (22) Portsmouth, R. L.; Gladden, L. F. J. Chem. Soc., Chem. Commun. 1992, 512. (23) Reischman, P. T.; Schmitt, K. D.; Olson, D. H. J. Phys. Chem. 1988, 92, 5165. (24) Olson, D. H.; Kokotalio, G. T.; Lawton, S. L.; Meier, W. M. J. Phys. Chem. 1981, 85, 2238. (25) Talu, O.; Guo, C. J.; Hayhurst, D. T. J. Phys. Chem. 1989, 93, 7294.
10.1021/la9814900 CCC: $18.00 © 1999 American Chemical Society Published on Web 08/03/1999
6092 Langmuir, Vol. 15, No. 18, 1999
location of the admolecule was identified by comparing model simulation with the refined diffraction data. A variety of theoretical approaches, based on either computer simulation26-33 or use of methods of statistical thermodynamics,6,7,34-40 have been proposed to model these adsorption systems. Talu and co-workers7,25 launched a two-patch model of collective mobile adsorption with the surface phase transition. They assumed that there exist two types of homogeneous patches in the silicalite structure which correspond to large elliptical and smaller circular pores. To describe local equilibrium on each patch, they used the Hill-de Boer equation. While considering the p-xylene adsorption in ZSM-5, Pan and Mersmann35,36 have proposed a model of localized adsorption on two independent subsystems of sites, corresponding to the channel intersections and zigzag channels. Adsorption in the channel intersections was described by the Langmuir isotherm, while the adsorption in the zigzag channels was modeled by the isotherm corresponding to the quasi-chemical approximation for cooperative adsorption. The interactions between the molecules in the neighboring zigzag channels were assumed to be mediated by structural relaxations in the zeolite itself occurring when p-xylene is adsorbed at high loadings in ZSM-5 zeolite. Pan and Mersmann proposed also a construction of the hysteresis loop for p-xylene on the basis of their model. The idea to treat the whole adsorption system as a threedimensional cooperative system has been accepted by Lee and co-workers.37,38 They used a lattice gas model with three kinds of adsorption sites located at the channel intersections (I), the straight channels (S), and the zigzag channels (Z), as indicated in Figure 1, and showed equivalence of their lattice gas model to an Ising model. The authors postulated next that the sharp increase in the adsorption isotherms is due to the phase transition in the adsorbed phase predicted by this model. They used their three-site lattice gas model to fit the experimental isotherms of benzene and p-xylene adsorbed in silicalite. To arrive at the phase transition in the adsorbed phase, they were forced to assume certain values of the adsorption energies s, z, and i and some interrelations of the gassolid and gas-gas parameters to be fulfilled. However, the calculated isosteric heat of adsorption could not be seen as reproducing well the qualitative features of the experimental heats of adsorption reported by Thamm.11 The estimated energy of adsorption s was larger than i, (26) Pickett, S. D.; Nowak, A. K.; Thomas, J. M.; Cheetham, A. K. Zeolites 1989, 9, 123. (27) Li, J.; Talu, O. J. Chem. Soc., Faraday Trans. 1993, 89, 1683. (28) Grauert, B.; Fiedler, K. Adsorpt. Sci. Technol. 1989, 6, 191. (29) Schroder, K. P.; Sauer, J. Z. Phys. Chem. (Leipzig) 1990, 271, 289. (30) Vigne-Maeder, F.; Jobic, H. Chem. Phys. Lett. 1990, 169, 31. (31) Talu, O. Mol. Simul. 1991, 8, 119. (32) Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1993, 97, 13742. (33) Snurr, R. Q.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1994, 98, 5111. (34) Stach, H.; Wendt, R.; Fiedler, K.; Grauert, B.; Janchen, J.; Spindler, H. In Characterization of Porous Solids; Unger, K. K., et al., Eds.; Elsevier: Amsterdam, The Netherlands, 1988; p 109. (35) Pan, D.; Mersmann, A. In Characterization of Porous Solids II; Rodriguez-Reinoso, F., et al., Eds.; Elsevier: Amsterdam, The Netherlands, 1991; p 519. (36) Pan, D.; Mersmann, A. Gas Sep. Purif. 1991, 5, 210. (37) Lee, C. K.; Chiang, A. S. T.; Wu, F. Y. AIChE J. 1992, 38, 128. (38) Lee, C. K.; Chiang, A. S. T. Proc. Sep. Topic Conf. AIChE, Miami 1992, 365. (39) van Koningsveld, H.; Tuinstra, F.; van Bekkum, H.; Jansen, J. C. Acta Crystallogr. 1989, B45, 423. (40) Fyfe, C. A.; Feng, Y.; Grondey, H.; Kokotailo, G. T. J. Chem. Soc., Chem. Commun. 1990, 1224.
Narkiewicz-Michałek et al.
Figure 1. Channel structure of silicalite. The straight channels are near-circular with dimensions 0.54 × 0.56 nm, while the sinusoidal channels are elliptical with dimensions 0.51 × 0.55 nm.
which would mean that the straight channel sites S are filled first. Meanwhile, the reported atomistic simulations26-33 suggest that channel intersections I are filled first. There are some experimental findings39,40 that seem to support such a view. None of the phenomenological models discussed above took the observed structure change of the zeolite framework into account. By the atom-atom grand canonical Monte Carlo (GCMC) or molecular dynamics (MD) simulations, such a framework change may be reflected in the model. However, the atom-atom simulations are very time consuming and require access to a supercomputer or a dedicated workstation. The situation is even worse for tight-fit systems such as aromatics in ZSM-5 (Li and Talu27 and Snurr et al.32). The lattice model, on the other hand, is less computationally intensive but requires a large number of energy parameters to be determined by data fitting. To circumvent this difficulty, Snurr et al.33 proposed recently a hierarchical atom/lattice strategy, in which detailed atom-atom simulations are used to obtain the required parameters for the lattice model. They tested this approach by considering the benzene/silicalite system. Two sets of parameters were obtained corresponding to the ortho and para symmetries of the silicalite framework. An additional empirical parameter, which still had to be fitted, was included to account for the free energy change upon the transformation of a framework. In their model the step seen in the experimental isotherms is essentially due to the change of the zeolite structure. In other words, in spite of the cooperative character of their model, the sharp increase in the adsorption isotherm is not due to a phase transition in the phase of the adsorbed molecules but to a transformation of the zeolite structure from the ortho to the para form. The theoretical isotherms calculated by means of their lattice model were compared with GCMC molecular simulations but were not fitted to the experimental adsorption isotherms. Although calculations were performed for four temperatures, the corresponding heats of adsorption were not calculated. Heats of adsorption were estimated by these authors only using the GCMC simulations for the changing zeolite structure, but the results of their simulations were not able to reproduce the experimental data well. As is well-known, the behavior of isosteric heats of adsorption is a much stronger test for a theory than the behavior of adsorption isotherms. So far, theoretical studies of aromatics adsorption in ZSM-5 have focused, almost exclusively, on the comparison between the calculated and experimental adsorption isotherms. Although experimental data on the coverage (loading) dependence of the isosteric heats of aromatics adsorption in ZSM-5 were reported as long as a decade
Temperature Dependence of Adsorption Isotherms
ago, they were largely ignored in the theoretical studies. In our opinion, a reliable theory of adsorption should lead to a simultaneous good fit of the experimental adsorption isotherms and the accompanying heats of adsorption. It should also predict the appearance of two steps which have been observed experimentally on the isotherm of benzene adsorption in silicalite at higher temperatures.38 As the reported simulations already provide a strong support for the three-site model, in our previous publication41 we proposed a modified version of this model which led us to a consistent theoretical description explaining the behavior of the experimental adsorption isotherms of benzene and p-xylene in silicalite, as well as the accompanying heats of adsorption. While looking for possible revisions and improvements of the three-site model, we rejected the concept that this is only the phase change in the zeolite structure, which might be a source of the step on adsorption isotherms. First, in the case of benzene it is only a hypothesis at present. Second, this hypothesis alone does not lead to predicting the two steps on the adsorption isotherm of benzene observed at higher loadings and temperatures.33 Thus, we have decided to investigate which agreement between the three-site lattice gas model and experiment could be obtained by neglecting the changes in the adsorption parameters, which might be caused by the possible phase transition in the zeolite structure. Following the X-ray diffraction data on the site location of benzene molecules in MFI zeolite and computersimulated atom-atom interactions reported by Mentzen et al.,18 we assumed that two equally probable orientations of benzene molecules adsorbed in the channel intersection are possible and that these orientations are characterized by different values of the adsorption energy . These forms, denoted by I1 and I2, were assumed to compete for occupying channel intersections.41 Though this additional assumption introduced to the three-site lattice model allowed us to predict the two steps on the adsorption isotherm of benzene in silicalite, it failed, however, to reproduce correctly the strong decrease observed in the heat of adsorption curve at low loadings of benzene molecules. Theory Looking for further improvements of the three-site model, we have decided to take into consideration another important factor that has not received enough attention so far. It was the energetic heterogeneity of the sites of the same kind.42 Zeolites are usually viewed as very regular crystallographic structures. The common existence of various structure defects has been known for a long time but has not received enough attention so far. Indeed, such defects should not affect strongly adsorption of small molecules in large cavities and channels. If, however, the dimensions of cavities and channels become comparable to the values of van der Waals interaction parameters, even small changes in the zeolite local dimensions may result in considerable changes in the gas-solid interactions. This will cause the appearance of a new level of surface heterogeneity. Not only the features of the sites S, Z, and I are different. The local imperfections in the silicalite structure will introduce an additional dispersion of the (41) Rudzin´ski, W.; Narkiewicz-Michałek, J.; Szabelski, P.; Chiang, A. S. T. Langmuir 1997,13, 1095. (42) Chiang, A. S. T.; Lee, C. K.; Rudzin´ski, W.; Narkiewicz-Michałek, J.; Szabelski, P. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces, Studies in Surface Science and Catalysis; Rudzin´ski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, The Netherlands, 1997; Vol. 104, p 519.
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nonconfigurational free energies, even for the same kinds of adsorption sites. It was Thamm11 himself who first emphasized that the strongly decreasing heats of aromatics adsorption at small surface coverages must be due to structure imperfections. Similar views were expressed by Talu et al.25 This can be seen in Figure 3. Our model, assuming that all of the sites of the same type have identical adsorption features, is not able to reproduce well the strong decrease in the initial part of the heat of adsorption curve. From the literature on the adsorption on heterogeneous surfaces,43,44 it is known that this decrease could be reproduced by assuming a certain dispersion of the nonconfigurational free energy values, Ej, on various kinds of adsorption sites j ) s, z, i1, i2. Because of the local distortions of the zeolite structure, Ej may vary when going from one site of the same type to another. Then, it seems reasonable to assume that these local distortions have a random nature. In such cases, the external force field acting on an adsorbed molecule, and created by its interactions with other molecules adsorbed nearby, should be a function of the average occupancy of sites of all types θjt (j ) s, z, i1, i2). To describe adsorbateadsorbate interactions, the mean-field theory37,38 has been applied. This approximation may not be very accurate near the phase boundary, but it is quite adequate away from it. In the case when all possible interactions between adsorbed molecules ωij are taken into account, usage of MF theory seems to be the only way to obtain analytical expressions for adsorption isotherms as well as for heats of adsorption. In such cases the local adsorption isotherms on different types of sites will take the following forms:
{( {( {( {(
λ exp Es + θs )
∑j ωsjθjt
1 + λ exp Es + λ exp Ez + θz )
∑j
)/ } )/ } )/ } )/ }
∑j
kT
(2)
ωzjθjt kT
{ {( ∑ )/ }}/ { {( ∑ )/ }
θi1 ) λ exp Ei1 +
(1)
ωsjθjt kT
∑j ωzjθjt
1 + λ exp Ez +
kT
ωi1jθjt kT
j
1 + λ exp Ei1 +
{(
ωi1jθjt kT +
j
λ exp Ei2 +
∑j ωi2jθjt
{ {( ∑ )/ }/ { {( ∑ )/ }
θi2 ) λ exp Ei2 +
)/ }} kT
(3)
)/ }}
(4)
ωi2jθjt kT
j
1 + λ exp Ei1 +
{(
ωi1jθjt kT +
j
λ exp Ei2 +
∑j ωi2jθjt
kT
where λ ) exp(µ/kT) is the absolute activity of the adsorbate, Ej ) j + kT ln fj, fj is the molecular partition function of the admolecules occupying the sites of type j, (43) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (44) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1989.
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Narkiewicz-Michałek et al.
and ωij is the interaction energy of a molecule adsorbed on site i with molecules adsorbed on the nearest sites j. Let us remark that the parameters Ej in the above equations are related to the nonconfigurational part of the free energy of adsorption per molecule. Let χj(Ej) denote the differential distribution of the number of sites j having adsorption energy Ej, among the j-type sites, normalized to unity. Traditionally, the surface energetic heterogeneity was viewed as the dispersion of values (adsorption potential values at the local gassolid minima), not affecting fj. Here we must consider the dispersion of Ej rather, because the local distortions may affect also the local movement of the adsorbed molecules, i.e., the molecular partition function of these molecules. This is because the dimensions of the channels of ZSM-5 and the adsorbed aromatics are comparable. Then, it seems natural to assume that the local distortions of the zeolite structure will result in a Gaussian-like dispersion of adsorption energies for all types of sites. If we assume that the dispersion of Ej is represented by the Gaussianlike function
χ(Ej) )
1 cj
{
Ej exp cj
[
}
{
θst )
[ {( [ {( [ {( [ {(
1 + λ exp E0s + λ exp E0z +
θzt )
}]
(5)
2
θi1t )
)/ }] )/ }] )/ }] )/ }]
∑j ωsjθjt
(6)
λ exp
+
( )
∂µ )0 ∂θt
∫θθ
tL
tG
µ(θt) dθt
γs
and
Fs )
(7)
γz
Fi1 )
kT
γi1
µs - µ
)
kT
1 γs
ln
E0s +
θst 1 - θst
-
µz - µ
)
kT µi1 - µ
1 γz
)
kT
ln
1 γi1
E0z
θzt 1 - θzt
ln
Fi2 )
γi2
ωi2jθjt kT
-
1 - θi1t - θi2t
(8)
µi2 - µ kT
)
1 γi2
ln
λ exp E0i2 +
1 - θi1t - θi2t E0i2 +
ωi1jθjt kT
γi1
+
j
λ exp E0i2 +
ωi2jθjt kT
j
where γj < 1 is identified with kT/cj.
(15)
- ln λ ) 0 (16)
-
- ln λ ) 0 (17)
Let Qj({θmt}) denote the molar differential heat of adsorption on the site of type j, at a certain set of the average surface coverages {θmt} (m ) s, z, i1, i2). It is given by
j
1 + λ exp E0i1 +
- ln λ ) 0
-
∑j ωi2jθjt
kT
γi2
ωi2jθjt kT
∑j ωzjθjt
∑j ωi1jθjt
θi2t
- ln λ ) 0 (14)
kT
θi1t
j
θi2t )
+
kT
j
λ exp E0i2 +
∑j ωsjθjt kT
E0i1 +
+
(13)
where subscripts G and L correspond to the gas and liquid phases, respectively. Now, let us investigate the related behavior of the heat of adsorption. To that purpose, we rewrite eqs 6-9 as follows:
Fz )
γz
ωi1jθjt kT
(12)
θtL - θtG
j
1 + λ exp E0i1 +
(11)
and next a nonphysical loop at still lower temperatures. Then, one has to perform the Maxwell construction to arrive at the physical isotherm, represented by a straight line fulfilling the conditions
γi1
ωi1jθjt kT
( )
∂2µ )0 ∂θt2
and
kT
{[ {( ∑ )/ }] }/ { [ {( ∑ )/ }] [ {( ∑ )/ }] } {[ {( ∑ )/ }] }/ { [ {( ∑ )/ }] [ {( ∑ )/ }] } E0i1
(10)
At appropriate values of the interaction parameters ωjm (j, m ) s, z, i), or in suitably low temperature, the µ/kT considered as a function of θt will show a critical point at which
γs
kT
∑j ωzjθjt
j ) i1, i2, s, z
µM ) µ(θtG) ) µ(θtL)
ωsjθjt kT
∑j ωzjθjt
1 + λ exp E0z +
∑j θjt
3
µM )
where E0j is the most probable value of Ej and cj is the heterogeneity parameter characterizing the width of the distribution function (5), the corresponding expressions for the average surface coverage of sites of j th type will take the following forms:43
∑j
1
θt )
E0j
Ej - E0j 1 + exp cj
λ exp E0s +
The overall adsorption isotherm θ obtained by solving the equation system (6)-(9) is then given by
γi2
(9)
Qj({θmt}) ) -k
(
)
µj - µ ∂ ∂(1/T) kT
{θmt}
Thus, Qj’s take the following explicit forms
(18)
Temperature Dependence of Adsorption Isotherms
Qs({θmt}) ) Qos + ζs ln Qz({θmt}) ) Qoz + ζz ln Qi1({θmt}) )
Qoi1
+ ζi1 ln
Qi2({θmt}) ) Qoi2 + ζi2 ln
θst 1 - θst θzt 1 - θzt
Langmuir, Vol. 15, No. 18, 1999 6095
+
∑j ωsjθjt
(19)
+
∑j
(20)
θi1t 1 - θi1t - θi2t θi2t 1 - θi1t - θi2t
where
ωzjθjt
+
∑j ωi1jθjt
(21)
+
∑j ωi2jθjt
(22)
where
d Qoj ) k
(
)
Ej0 + µo kT d(1/T)
(23)
and
ζj ) -k
d(1/γj)
(24)
d(1/T)
µo in eq 23 is the standard chemical potential of adsorbate molecules in the gaseous phase. An incremental increase in µ, dµ, will result in an incremental increase of θt, dθt, represented by
∑j
dθt )
( ) ∂θjt ∂µ
dµ
(25)
That incremental increase will be accompanied by a heat effect dQ
dQ )
∑j
Qj
( ) ∂θjt ∂µ
dµ
(26)
Thus, the overall (measured) differential heat of adsorption Q will be given by
∑Qj ( ∂µ ) ∂θjt
Q)
∑(
)
(27)
∂θjt ∂µ
The derivatives (∂θjt/∂µ) can be evaluated from the equation system (14)-(17). It can be done as follows. Let Gj denote Fj multiplied by kT. Thus
∂Gj ∂µ
) -1 +
∂Gj ∂θmt
∑ m ∂θ
mt
∂µ
)0
(28)
The derivatives (∂θmt/∂µ) are found by solving this system of four linear equations. Let Gm j denote the derivative
Gm j )
∂θm Dm ) ∂µ D
( ) ∂Gj ∂θmt
The solution of the equation system (28) reads
(29)
D)
(
Gss
Gzs
Gi1 Gi2 s s
Gsz
Gzz
Gi1 Gi2 z z
i2 Gsi1 Gzi1 Gi1 i1 Gi1 i2 Gsi2 Gzi2 Gi1 i2 Gi2
(30)
)
(31)
and Dm is obtained from D by replacing the mth column of the determinant D by the column of constants. From eqs 27 and 30 one can evaluate the heat of adsorption as a function of loading of zeolite channels. The above considerations are based on the so-called “heterogeneous three-site model”. Putting in the above equations γj ) 1 and ζj ) 0 (j ) i1, i2, s, z), one arrives at the equations corresponding to the so-called “homogeneous three-site model” discussed in our previous publication41 and describing the adsorption in the case when all adsorption sites of the same type have the same energy of adsorption. Results and Discussion In order to check the influence of the energetic heterogeneity effects, we decided to compare an ability of both our models (“homogeneous” and “heterogeneous”) to fit the experimental data. As a testing procedure, we chose the simultaneous fitting of the adsorption isotherms, isosteric heats of adsorption, and the temperature dependence of adsorption isotherms. To this purpose, we took for analysis the adsorption isotherms of benzene and p-xylene in silicalite at different temperatures reported by Lee et al.,45 along with the differential heats of adsorption measured by Thamm11 at 301 K. For the comparison with theoretical predictions and the regression of model parameters, good experimental isotherms over a wide temperature range are indispensable. It seems that the isotherms measured by Lee et al. fulfill this condition and may serve to verify our theory. They have used very large (180 × 40 × 40 µm3) and well-defined crystals of silicalite and kept the pressure steps as small as possible. They have also determined the isosteric heats of adsorption directly from the adsorption isotherms. As is well-known, changes in the heats of adsorption within small intervals of pore filling and/or temperature are not very accurately observed by this method. This is because of difficulties in measuring correct equilibrium pressures in the low pressure region and the averaging of experimental data for the deduction of heats of adsorption. The isosteric heats of adsorption calculated by them show a variation with loading which is completely different from that observed in the calorimetric experiment. The advantage of using the calorimetric data for the present work is, therefore, obvious. However, the fact that the adsorption and calorimetric experiments were carried out in different laboratories has to be taken into account while considering the obtained agreement between theory and experiment. For p-xylene only one step is observed in the adsorption isotherm at a loading of 4 molecules/unit cell at all temperatures. In the case of benzene adsorption, two such steps are observed: the first is at around 4 molecules/unit cell and the second one at around 6 molecules/unit cell. (45) Lee, C. K.; Chiang, A. S. T. J. Chem. Soc., Faraday Trans. 1996, 92, 3445.
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Table 1. Parameters Found by Fitting Simultaneously the Experimental Isotherms and Heats of Adsorption of Benzene and p-Xylene in Silicalite by the Equations Corresponding to the “Homogeneous Three-Site Model” (γj ) 1, ζj ) 0 for j ) i1, i2, s, z)a benzene p-xylene
benzene p-xylene
benzene p-xylene
Ki1 (Pa-l)
Ki2 (Pa-l)
Ks (Pa-l)
Kz (Pa-l)
ωzz (kJ/mol)
8.547 × 10-2 3.042 × 10-1
3.873 × 10-3 4.905 × 10-2
2.605 × 10-3 0.0
3.643 × 10-2 3.432 × 10-2
7.60 15.50
ωsz (kJ/mol)
ωsi2 (kJ/mol)
ωzi2 (kJ/mol)
ωsi1 (kJ/mol)
ωzi1 (kJ/mol)
-2.75 0.00
21.00 0.00
-30.30 5.50
-11.00 0.00
-14.00 -4.00
ωi1i1 (kJ/mol)
Qoi1 (kJ/mol)
Qoi2 (kJ/mol)
Qos (kJ/mol)
Qoz (kJ/mol)
0.00 8.20
56.00 76.00
105.00 84.00
69.00 0.00
70.00 87.00
a The parameters ω jm which were found to be zero for the both adsorbates are not listed. The Henry’s constant Kj is defined as exp{(Ej + µo)/kT}.
The sharp steps in the isotherms of aromatics adsorbed in silicalite at the loadings of above 4 molecules/unit cell have been observed by many authors.7-9,13,24,25 It seems, however, that Lee et al. were the first to report on the two steps on the experimental adsorption isotherms of benzene at higher temperatures. At the same time no hysteresis was found in the benzene and p-xylene adsorption isotherms. Such hystereses were reported earlier by Thamm,12 using much smaller crystals than those in Lee’s experiment. It seems, therefore, that the observed hystereses were due to desorption from the intraparticle pores. The differential heats of sorption of p-xylene exhibit only a small decline at the lowest loadings and then continuously increase, reaching a week maximum at a coverage of 4 molecules/unit cell. This maximum is followed by a plateau which extends up to a saturation capacity. The calorimetrically determined differential molar heats of adsorption of benzene on silicalite exhibit a much more complex variation with loading. They show a sharp decline at small loadings and then remain constant up to a coverage of 4 molecules/unit cell. With further rising adsorption, Q passes through a minimum, exhibits a relative maximum at a coverage of about 6 molecules/ unit cell, then reaches an absolute maximum, and upon approaching saturation capacity falls to a value close to that of the heat of condensation of the adsorbate in the liquid phase. A disadvantage of our lattice model is the large number of interaction parameters to be fitted. If, however, one looks at the complicated shape of the benzene and p-xylene isotherms and of the related heat of adsorption curves and realizes that we fit them simultaneously by using the same set of parameters, one can see a thin margin for an arbitrary choice of these parameters. As a matter of fact, we have performed numerous model calculations which proved that the calculated data are very sensitive to a particular choice of these parameters. The strategy of our calculations was as follows. First, we tried to fit simultaneously the experimental adsorption isotherms and heats of adsorption of benzene and p-xylene in silicalite at 303 K with the smallest possible number of ωjm (j, m ) i1, i2, s, z) parameters. Next, we used the parameters found for the adsorption isotherm at 303 K to predict the adsorption isotherms at different temperatures. To this purpose, the temperature dependence of the free energy of adsorption on different types of sites should be known. In the simplest case one may treat the parameters of lateral interactions ωjm, as temperature independent and assume the free energy change upon adsorption on a jth type of site, -RT ln Kj ) -(E0j + µo),
Table 2. Parameters rj Found by Fitting Our Equations for the “Homogeneous Three-Site Model” to the Experimental Isotherms of Adsorption of Benzene and p-Xylene in Silicalite at Different Temperaturesa
benzene p-xylene
benzene p-xylene
Ri1 (kJ/mol K)
Ri2 (kJ/mol K)
Rs (kJ/mol K)
Rz (kJ/mol K)
-0.187 -0.141
-0.237 -0.175
-0.199
-0.216 -0.208
i1 (kJ/mol)
i2 (kJ/mol)
s (kJ/mol)
z (kJ/mol)
50.51 39.85
58.03 45.33
45.49
57.18 54.41
a The potential energies of gas-solid interaction were calculated j from the Henry’s constants Kj at 303 K given in Table 1.
to be a linear function of temperature,
∆Aj ) -RT ln Kj ) -(E0j + µo) ) -oj - RjT (32) where the parameter Rj has a meaning of the entropy change upon adsorption on the site of jth type. Table 1 collects the values of the parameters found by computer, while fitting simultaneously the experimental adsorption isotherms and the corresponding isosteric heats of adsorption of benzene and p-xylene by the equations derived for the homogeneous three-site model. Having the values Kj and ωjm at 303 K, we had to find four additional best fit parameters Rj (j ) i1, i2, s, z) while fitting best the experimental adsorption isotherms of benzene and p-xylene in silicalite at different temperatures. The values of these parameters are given in Table 2. Before we comment on the values of the parameters found by the computer, we will first analyze the behavior of our adsorption system predicted by our equations for that particular set of parameters. Figure 2 shows graphically the agreement between the experimental adsorption isotherms and the theoretical ones calculated by using the parameters collected in Table 1. In Figure 3 the comparison between the experimental and theoretical heats of adsorption is presented. In Figure 4, in addition to the overall theoretical adsorption isotherms, also the contributions from the adsorption on various adsorption sites are shown. One can see that neither the total theoretical isotherm nor its composite isotherms on a particular kind of sites (in a particular configuration) shows loops, which could be associated with phase transitions. The sharp changes in the adsorption isotherm of benzene are due to rapid changes in the occupation of various adsorption sites. We have already called it the cooperative redistribution of
Temperature Dependence of Adsorption Isotherms
Figure 2. (A) Comparison of the experimental isotherm of p-xylene adsorption in silicalite (]) measured by Lee et al.45 at 303 K with the theoretical one (s) calculated for the homogeneous three-site model using the parameters collected in Table 1. (B) Comparison of the experimental isotherm of benzene adsorption in silicalite (b) measured by Lee et al.45 at 303 K with the theoretical one (s) calculated for the homogeneous three-site model using the parameters collected in Table 1.
adsorbed molecules. Our theoretical calculations confirm, thus, the long-shared feeling by many scientists that the adsorption of aromatics in silicalite is governed by a delicate balance between the adsorbate-solid and adsorbate-adsorbate interactions. Looking at Figure 4, one can see that at loadings smaller than 4 molecules/unit cell, adsorbed molecules fill mainly the channel intersections. At coverages above 4 molecules/ unit cell, we observe a sharp increase of adsorption in sinusoidal (or straight) channels, accompanied by a decrease and reorientation of molecules occupying channel intersections. For p-xylene only one redistribution of adsorbed molecules is observed, leading to a steep step on the adsorption isotherm. In the case of benzene at still higher loadings (above 6.5 molecules/unit cell), a second redistribution of adsorbed molecules takes place. At maximum loading of about 8 molecules/unit cell, all straight (or sinusoidal) channels and channel intersections are filled by benzene molecules. These redistributions of the adsorbed molecules are responsible for the two steps observed on the adsorption isotherm of benzene. The reason for using z(s) and s(z) in Figure 4 to denote the occupancy of Z and S sites is as follows: From a purely theoretical point of view, it is impossible to judge which of the calculated solid lines means the occupancy of Z sites, or S sites alternatively. This is because of the symmetry of eqs 1 and 2. Equation 2 can be obtained from
Langmuir, Vol. 15, No. 18, 1999 6097
Figure 3. Comparison between the experimental differential heats of adsorption of p-xylene (A) and benzene (B) in silicalite measured by Thamm11 at 301 K and the theoretical ones (s) calculated for the homogeneous three-site model using the parameters collected in Table 1.
eq 1 by replacing the index s by z, and vice versa. The discrimination between the calculated contributions must be made on some rational physical basis. In the case of benzene adsorption, the theoretical and experimental findings reported by Mentzen et al.18 are the basis for us. They argue that at the highest loadings of silicalite the adsorbed benzene molecules form onedimensional polymer-like structures in the straight channels. The formation of such one-dimensional polymers means full occupation of both I and S sites at the highest possible loading. Looking at Figure 4B, we can deduce that such occupation could really exist, provided that the solid line s(z) means the occupancy of sites S rather than of the Z sites. The symbol z(s) has a similar meaning. If, however, we interchange the interpretation of these theoretical isotherms, our conclusion that I and Z sites are fully covered at the highest loadings will be in agreement with the one drawn by Snurr et al.32,33 In Figure 4B one can see that at a loading of about 4 molecules/unit cell both a sudden increase of the z(s) and s(z) forms and a sudden disappearance of the form i1 take place even in the absence of the phase transition in the zeolite structure assumed so far. We can imagine, however, a situation when this sudden redistribution induces the silicalite phase transition which, in turn, promotes this sudden rearrangement. This view could be supported by the values of the parameters found by computer and collected in Table 1. One striking property observed is the high value of the
6098 Langmuir, Vol. 15, No. 18, 1999
Figure 4. (A) Occupation by p-xylene molecules of various adsorption sites in silicalite calculated for the homogeneous three-site model using the parameters collected in Table 1. The solid line rising sharply at small adsorbate pressures is the occupancy of I sites by p-xylene molecules being in the state i1, whereas the solid line rising sharply at the highest adsorbate pressures is the occupancy of I sites by p-xylene molecules being in the state i2. The solid lines denoted by z(s) and s(z) are the occupancies of the sites Z and S, or vice versa. The dashed line is the overall adsorption isotherm. (B) Occupation by benzene molecules of various adsorption sites in silicalite calculated for the homogeneous three-site model using the parameters collected in Table 1.
interaction (attraction) parameter ωzz for p-xylene and benzene adsorption. This would suggest a high positive cooperativity of adsorption of molecules adsorbed on Z sites. That means that the total energy of the molecules adsorbed on Z sites grows rapidly with the number of molecules adsorbed on these sites. However, the calculations show that adsorption on Z sites starts rapidly only when the surface loading exceeds 4 molecules/unit cell, i.e., when the phase transition in the zeolite structure can take place. One may, therefore, assume that this unusual positive cooperativity simulates, in fact, another factor leading to such an increase of adsorption on Z sites. This may well be a sudden increase of the adsorption energy Ez, induced by the zeolite phase transition. In the case of the argument that for benzene the phase change in the silicalite is only a hypothesis at present, we might offer another explanation following the arguments by Pan and Mersmann.35 They believe that a strong attraction exists between two molecules on the nearest Z sites, transmitted through the solid phase. Finally, a certain compromise between the views expressed by Pan and Mersmann and those launched by Snurr et al.33 also seems to be possible.
Narkiewicz-Michałek et al.
Figure 5. (A) Comparison between the experimental adsorption isotherms for p-xylene in silicalite measured by Lee et al.45 at different temperatures (0, 273 K; O, 283 K; ×, 293 K; ], 303 K; +, 323 K) and the theoretical ones calculated for the homogeneous three-site model using the parameters from Tables 1 and 2. (B) Comparison between the experimental adsorption isotherms for benzene in silicalite measured by Lee et al.45 at different temperatures (0, 273 K; O, 283 K; b, 303 K) with the theoretical ones calculated for the homogeneous three-site model using the parameters from Tables 1 and 2.
As we mentioned before, the behavior of theoretically predicted isosteric heats of adsorption is a much stronger test for the theory than the behavior of theoretical adsorption isotherms. This is because the behavior of the experimental heats of adsorption is much more sensitive to the nature of an experimental adsorption system. Thus, we believe that special attention should be given to the agreement between theoretical and experimental heats of adsorption. In the case of p-xylene, the agreement between theoretically predicted and experimentally measured heats of adsorption is quite good. However, the homogeneous threesite model is not able to predict correctly the rapid decrease of the differential heat of adsorption of benzene in silicalite observed experimentally by Thamm11 (see Figure 3B). Also, the minima occurring on the curve of the heat of adsorption at the intermediate and higher loadings are not properly reproduced. This is probably due to the fact that this model neglects the chemical and geometrical defects in the silicalite structure which cause the dispersion of the free energy of adsorption on the three kinds of sites. In Figure 5 the agreement between the experimentally and theoretically predicted adsorption isotherms at different temperatures is presented. One can see that the homogeneous three-site model correctly predicts the temperature dependence of the sharp steps appearing on
Temperature Dependence of Adsorption Isotherms
Langmuir, Vol. 15, No. 18, 1999 6099
Table 3. Parameters Found by Fitting Simultaneously the Experimental Isotherms and Heats of Adsorption of Benzene and p-Xylene in Silicalite at 303 K by the Equations Derived for the Heterogeneous Three-Site Modela Ki1 (Pa-1) benzene p-xylene
benzene p-xylene
benzene p-xylene
benzene p-xylene
10-2
4.53 × 3.29 × 10-1
Ki2 (Pa-1)
Ks (Pa-1)
10-3
2.95 × 3.42 × 10-2
10-3
6.23 × 1.27 × 10-7
Kz (Pa-1)
ωi1i1 (kJ/mol)
ωzz (kJ/mol)
1.33 × 10-2 6.46 × 10-2
3.00 8.00
10.00 12.00
ωsi1 (kJ/mol)
ωzi1 (kJ/mol)
ωsz (kJ/mol)
ωsi2 (kJ/mol)
ωzi2 (kJ/mol)
-17.13 0.00
-12.40 -5.00
-19.80 0.00
24.90 0.00
-5.41 7.00
γi1
γi2
γs
γz
ζi1 (kJ/mol)
ζi2 (kJ/mol)
0.99 1.00
0.86 0.93
0.93 0.97
0.90 1.00
-0.60 -0.40
-2.92 -2.30
ζs (kJ/mol)
ζz (kJ/mol)
Qoi1 (kJ/mol)
Qoi2 (kJ/mol)
Qos (kJ/mol)
Qoz (kJ/mol)
-2.72 0.00
-2.80 0.00
57.5 76.0
97.0 83.5
97.0 0.0
55.0 89.0
0 a The parameters ω jm which were found to be zero for both adsorbates are not listed. The Henry’s constant Kj is defined as exp{(Ej + µo)/kT}.
Figure 6. (A) Comparison of the experimental isotherm of p-xylene adsorption in silicalite (]) measured by Lee et al.45 at 303 K with the theoretical one (s) calculated for the heterogeneous three-site model using the parameters collected in Table 3. (B) Comparison of the experimental isotherm of benzene adsorption in silicalite (b) measured by Lee et al.45 at 303 K with the theoretical one (s) calculated for the heterogeneous three-site model using the parameters collected in Table 3.
the adsorption isotherms of p-xylene whereas in the case of benzene adsorption the agreement is not satisfactory. Thus, we decided to check what agreement would be obtained while using the equations derived for the heterogeneous three-site model. Table 3 collects the values of the parameters found by computer while fitting simultaneously the experimental isotherms and heats of adsorption for the investigated systems.
Figure 7. Comparison between the experimental differential heats of adsorption of p-xylene (A) and benzene (B) in silicalite measured by Thamm11 at 301 K and the theoretical ones (s) calculated for the heterogeneous three-site model using the parameters collected in Table 3.
Figures 6 and 7 show the comparison of the experimental adsorption isotherms and heats of adsorption with the theoretical ones, calculated for the heterogeneous threesite model using the parameters collected in Table 3. In Figure 8 the adsorption isotherms at different temperatures predicted by the model are presented together with the experimental data. One can see that taking into account the additional level of site heterogeneity due to the dispersion of Ej values improves somewhat the agreement between the theoreti-
6100 Langmuir, Vol. 15, No. 18, 1999
Figure 8. (A) Comparison between the experimental adsorption isotherms for p-xylene in silicalite measured by Lee et al.45 at different temperatures (0, 273 K; O, 283 K; ×, 293 K; ], 303 K; +, 323 K) with the theoretical ones calculated for the heterogeneous three-site model using the parameters from Tables 3 and 4. (B) Comparison between the experimental adsorption isotherms for benzene in silicalite measured by Lee et al.45 at three different temperatures (0, 273 K; O, 283 K; b, 303 K) with the theoretical ones calculated for the heterogeneous three-site model using the parameters from Tables 3 and 4.
cal and experimental adsorption isotherm of p-xylene at 303 K, especially in the region of low adsorbate pressures. Also the theoretical adsorption heat curve better reproduces the experimental data at the lowest loadings. However, the improvement is not significant, indicating that the energetic heterogeneity of the sites of the same type has a minor effect on the behavior of this system (the estimated values of the heterogeneity parameters γj are close to unity). The fact that the heats of adsorption of p-xylene increase from the very low loadings indicates that the molecules interact with each other already below 4 molecules/unit cell. This is confirmed by a large value of the energy of interaction between two p-xylene molecules adsorbed on the neighboring sites I, ωi1i1, predicted by both models. The strong sorbate-sorbate interaction in the system p-xylene/silicalite leading to the formation of double complexes has been identified recently from the double peaks in the differential thermogravimetry curves and thermal effects in the differential thermal analysis curves.46 It should be noted, however, that this strong attractive interaction between p-xylene molecules may mask to some extent the effects of the energetic heterogeneity of the adsorption sites in silicalite. As is wellknown, it is difficult to separate these two contributions (46) Long, Y.-C.; Jiang, H.-W.; Zeng, H. Langmuir 1997, 13, 4094.
Narkiewicz-Michałek et al.
Figure 9. Occupation by benzene molecules of various adsorption sites in silicalite at 273 K (A) and 303 K (B) calculated for the heterogeneous three-site model using the parameters collected in Table 3. The solid lines denoted by i1, i2, z(s), and s(z) are the occupancies of different adsorption sites by benzene molecules in silicalite. The dashed line is the overall adsorption isotherm.
to the total energy of adsorption because the energetic heterogeneity of the adsorbent results in a decrease of the differential energy of adsorption with coverage, whereas the energy of adsorbate-adsorbate interaction increases with adsorbed amount. At the same time, a much better agreement is observed between the experimental and theoretical heats of benzene adsorption (Figure 7B). The rapid decrease in the heat of adsorption at small surface coverages as well as the two local minima observed at the loadings of about 4 and 6.5 molecules/unit cell are better reproduced. These two minima on the adsorption heat curve appear at the loadings at which the sudden reorientation of adsorbed molecules takes place. From Figure 8 it follows that the temperature dependence of benzene adsorption is also better reproduced by the heterogeneous model than by the homogeneous one. Trying to understand what adsorption mechanism is behind that improvement, in Figure 9 we have displayed the contributions to the adsorption isotherm from benzene molecules adsorbed on various sites and in various configurations at two temperatures, 273 and 303 K. One can see that at both temperatures the overall adsorption isotherms exhibit two-step behavior; however, at the lower temperature the second step is less pronounced. From a comparison of Figures 4B and 9B, it follows that both our models predict the same location of benzene
Temperature Dependence of Adsorption Isotherms
Langmuir, Vol. 15, No. 18, 1999 6101 Table 4. Parameters rj Found by Fitting Our Equations for the “Heterogeneous Three-Site Model” to the Experimental Isotherms of Adsorption of Benzene and p-Xylene in Silicalite at Different Temperaturesa
benzene p-xylene
benzene p-xylene
Ri1 (kJ/mol K)
Ri2 (kJ/mol K)
Rs (kJ/mol K)
Rz (kJ/mol K)
-0.187 -0.150
-0.224 -0.183
-0.208
-0.220 -0.200
i1 (kJ/mol)
i2 (kJ/mol)
s (kJ/mol)
z (kJ/mol)
48.91 42.57
53.36 46.95
50.21
55.90 53.59
a The potential energies of gas-solid interaction were calculated j from the Henry’s constants Kj at 303 K given in Table 3.
Figure 10. Contributions Qcj to the total isosteric heat of adsorption of benzene in silicalite, calculated for the homogeneous three-site model using the parameters collected in Table 1 (A) and for the heterogeneous three-site model using the parameters collected in Table 3 (B). These contributions are denoted in the same way as the contributions to the total adsorption isotherm from various sites, shown in Figure 9. The black circles (b) are the experimental data reported by Thamm.
molecules at the low and highest adsorbate loadings. The difference occurs at the loadings between 4 and 6.5 molecules/unit cell. The model accounting for the energetic heterogeneity of the adsorption sites of the same type predicts that in this coverage region all zigzag (or straight) channels and most of the channel intersections are occupied, whereas the model neglecting this additional level of heterogeneity predicts the occupation of zigzag and straight channels. In the case of p-xylene adsorption, both homogeneous and heterogeneous models predict the same location of adsorbed molecules before and after a sudden jump on the adsorption isotherm. At low adsorbate pressures p-xylene molecules fill mainly the channel intersections in the location denoted by i1. At loadings higher than 4 molecules/unit cell, a rapid filling of zigzag channels accompanied by the reorientation of molecules adsorbed in the channel intersections starts. Figure 10 shows the contributions to the total isosteric heat of adsorption of benzene, from the heats of adsorption Qcj generated by the adsorption on various adsorption sites.
( ) ∑( )
Qj Qcj )
∂θj ∂µ ∂θj ∂µ
The additional index “c” in Qcj means that this is the “contribution” to the total heat of adsorption, from the molecules occupying sites j. Looking at Figure 10, one can see that the contributions to the total isosteric heat of adsorption predicted by the heterogeneous three-site model and coming from the forms Z, S, and I in the orientation denoted by i2 decrease at low loadings. Such a decrease of an adsorption heat curve is usually attributed to the adsorption on an energetically heterogeneous surface. The heterogeneity parameters (kT/ cj) for these forms listed in Table 3 are less than unity, and the temperature derivatives ζj defined in eq 24 are equal to -cj, indicating thus that cj’s are temperature independent. So, this is the dispersion of the free energy of adsorption of benzene molecules on these sites that is responsible for the rapid decrease of the heat of adsorption at very low loadings. On the other hand, the heterogeneity parameter obtained for the form i1 is very close to unity, and the derivative ζi1 is not equal to -ci1. This would indicate that the adsorption of benzene molecules in channel intersections at low loadings, i.e., in the state of benzene molecules denoted here by i1, is not sensitive to imperfections in the zeolite structure. After the initial sharp decline, the heat of adsorption of benzene molecules remains constant up to 4 molecules/unit cell, indicating that there is no interaction between molecules adsorbed on the neighboring I sites in the location denoted by i1. The predicted values of the parameter ωi1i1 characterizing interaction between the admolecules in this adsorption region for the homogeneous and heterogeneous models are equal to 0.0 and 3.0, respectively. These values are much lower than those obtained for the p-xylene/silicalite system in which strong interactions between adsorbed molecules are expected at loadings smaller than 4 molecules/unit cell. The high negative values of some interaction parameters obtained in both models may simulate the restrictions imposed on two adsorbate molecules to be adsorbed on the neighboring sites and/or the changes in the zeolite structure caused by the tight-fit situation which may promote the redistribution of the adsorbed molecules. Because of a large number of parameters involved in our model, their detailed discussion is difficult at present. We note, however, that the values of the energy and entropy of adsorption found by us and listed in Tables 2 and 4 are comparable with those found in the literature.32,33,45 Conclusions
(33)
An extended version of the three-site lattice model for adsorption of aromatics in ZSM-5 zeolites has been developed and used to predict semiquantitatively the adsorption isotherms and isosteric heats of adsorption of
6102 Langmuir, Vol. 15, No. 18, 1999
benzene and p-xylene in silicalite. When the experimental data are best fit by our theoretical expressions, it is possible to elucidate how sorbate siting within the zeolite varies with loading and how this affects the isotherms and accompanying isosteric heats of adsorption. The predictions of our model concerning the location of aromatic molecules at low and high loadings are in accordance with the experimental findings and computer simulations. Assuming an additional level of the energetic heterogeneity of the sites of the same type leads to a much better description of the adsorption heat curve and temperature dependence of the adsorption isotherms for benzene, whereas for p-xylene both the homogeneous and heterogeneous models give similar results. It is possible that
Narkiewicz-Michałek et al.
because of the higher mobility the smaller benzene molecules are much more sensitive to the structural defects in the silicalite crystal than the larger p-xylene molecules. It may be concluded that our model performs very well considering the complexity of the system analyzed. However, a detailed analysis of a large number of accurate experimental data is still needed to clarify the physical meaning of the parameters involved in the model. Acknowledgment. This research was supported by KBN Research Grant 3 T09A 015 14. LA9814900
