Mixed Gas Equilibrium Adsorption on Zeolites and Energetic

molecular rearrangement in the cavity volume, which results from the difference in the fluid-solid .... maximum of 10 molecules and there are only 3 m...
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Langmuir 2002, 18, 3567-3577

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Mixed Gas Equilibrium Adsorption on Zeolites and Energetic Heterogeneity of Adsorption Volume E. A. Ustinov† and D. D. Do* Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia Received October 4, 2001. In Final Form: December 17, 2001 A model for binary mixture adsorption accounting for energetic heterogeneity and intermolecular interactions is proposed in this paper. The model is based on statistical thermodynamics, and it is able to describe molecular rearrangement of a mixture in a nonuniform adsorption field inside a cavity. The Helmholtz free energy obtained in the framework of this approach has upper and lower limits, which define a permissible range in which all possible solutions will be found. One limit corresponds to a completely chaotic distribution of molecules within a cavity, while the other corresponds to a maximum ordered molecular structure. Comparison of the nearly ideal O2-N2-zeolite NaX system at ambient temperature with the system of O2-N2-zeolite CaX at 144 K has shown that a decrease of temperature leads to a molecular rearrangement in the cavity volume, which results from the difference in the fluid-solid interactions. The model is able to describe this behavior and therefore allows predicting mixture adsorption more accurately compared to those assuming energetic uniformity of the adsorption volume. Another feature of the model is its ability to correctly describe the negative deviations from Raoult’s law exhibited by the O2-N2-CaX system at 144 K. Analysis of the highly nonideal CO2-C2H6-zeolite NaX system has shown that the spatial molecular rearrangement in separate cavities is induced by not only the ionquadrupole interaction of the CO2 molecule but also the significant difference in molecular size and the difference between the intermolecular interactions of molecules of the same species and those of molecules of different species. This leads to the highly ordered structure of this system.

1. Introduction Statistical thermodynamics is one of the most effective means in the development of theory of equilibrium adsorption in zeolites. Assuming that separate cavities form a grand canonical ensemble of quasi-independent subsystems, Bakaev1 was the first who applied the grand partition function for the analysis of pure component adsorption isotherms. This approach was further extended by introducing an equation of state for molecules in a cavity analogous to the van der Waals equation,2,3 which allowed expression of the Helmholtz free energy in explicit form and extension of this theory to adsorption of binary mixtures. However, better predictive ability can only be achieved by accounting for energetic heterogeneity of the adsorbent. In particular, analysis of binary mixture adsorption frequently reveals negative deviations from Raoult’s law in the adsorbed phase behavior, which is usually attributed to energetic nonhomogeneity. This has motivated many investigators to turn to simpler approaches such as the generalization of the Langmuir model. The common feature of these approaches is the introduction of a distribution function for the heat of adsorption. For each patch of active sites of the same adsorption energy, the extended Langmuir equation is usually applied.4 The overall adsorption is then simply the sum of adsorptions of all patches.5-7 Despite the high * To whom correspondence should be addressed. † On leave from Saint Petersburg State Technological Institute (Technical University), 26 Moskovsky Prospect, St. Petersburg 198013, Russia. (1) Bakaev, V. A. Dokl. Akad. Nauk SSSR 1966, 167, 369. (2) Ruthven, D. M.; Loughlin, K. F.; Holbarow, K. A. Chem. Eng. Sci. 1973, 28, 701. (3) Ruthven, D. M. AIChE J. 1976, 22, 753. (4) Markham, E. C.; Benton, E. F. J. Am. Chem. Soc. 1931, 53, 497. (5) Myers, A. L. AIChE J. 1983, 29, 691.

correlative ability of those approaches, they all suffer from many limitations, for example, neglecting lateral interactions among adsorbed molecules. Such a limitation has been removed in the case of pure component adsorption.8,9 However, extension of such an approach to multicomponent adsorption systems has not been straightforward. Recently,10,11 we established a correspondence between the statistical thermodynamic approach and the phenomenological theory of ideal adsorbed solution (IAS).12 The IAS is based on the assumption that Raoult’s law holds along the lines of constant spreading pressure. It was shown that this correspondence is possible only in the special case of a uniform potential field inside a cavity and when the isotherm of one component can be obtained from that of the another component when the pressure P is replaced by RP. Such a special case is rare in practice. In general, the energetic heterogeneity and the difference in polarity of molecules of different species lead to their spatial rearrangement with decreasing entropy, internal energy, and Helmholtz free energy. This results in negative deviations from Raoult’s law. Therefore, it is essential to characterize the energetic heterogeneity from pure component systems,13,14 and such energetic heterogeneity is then used in the prediction of the mixture adsorption (6) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397. (7) Do, D. D. Adsorption 1999, 5. (8) Shekhovtseva, L. G.; Fomkin, A. A.; Bakaev, V. A. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1987, 36, 2176. (9) Shekhovtseva, L. G.; Fomkin, A. A. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1990, 39, 867. (10) Ustinov, E. A.; Polyakov, N. S.; Stoekli, F. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1998, 47, 1873. (11) Ustinov, E. A.; Polyakov, N. S. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1999, 48, 1059. (12) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (13) Ustinov, E. A.; Klyuev, L. E. Adsorption 1999, 5, 331. (14) Ustinov, E. A. Russ. J. Phys. Chem. 1999, 73, 1836.

10.1021/la011514u CCC: $22.00 © 2002 American Chemical Society Published on Web 03/27/2002

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isotherm on the same adsorbent. Preliminary results were shown to be quite promising.15-17 In this paper, we analyze three binary systems. The first system is the oxygen-nitrogen mixture adsorption on zeolite NaX18 at 304 K. This system is a representation of ideal systems. The second system chosen is a representative of significantly nonideal mixtures: O2-N2zeolite 10X19 at low temperature (144 K). The oxygennitrogen mixture in itself is nearly ideal, but due to the strong quadrupole interaction of nitrogen with ions in the zeolite cavity the adsorption system of O2-N2-10X is essentially nonideal. The third adsorption system is CO2C2H6-zeolite NaX.20,21 Carbon dioxide has a strong quadrupole moment, and this contributes to the nonideality of this system. Furthermore, the nonideal behavior is also due to the difference in molecular size and to the intrinsic nonideality of the mixture of CO2-C2H6 in the bulk phase. 2. Model To develop a model accounting for intermolecular interactions as well as adsorption heterogeneity, we consider the following adsorption mechanism. Molecules of the same or different species inside a pore form a dense three-dimensional molecular cluster resulting from intermolecular interactions among adsorbate molecules. Each molecule once having entered the cavity will have different probabilities in occupying the available locations within the cavity. The limiting case of this is that there is only one definite location where that molecule can reside, meaning that the probability of occupation of that definite location is 1, while that of all other locations is 0. This limiting case is the mechanism that we shall consider in this paper. For a given total number of molecules in a pore, there is only one configuration of spatial arrangement. Other configurations would have a negligible probability. As a consequence, the number of locations occupied by molecules always equals the total number of molecules. Some of these locations are associated with active centers of the adsorbent, whereas the others are essentially points of stable equilibrium close to other molecules. Thus, the concept of a location is more general than that of a site in other multisite models. In the latter case, it is implied that there is a definite number of sites, each of which can be either filled or empty. In this approach, each new location is associated with the introduction of a molecule. If a pore contains, say, 3 molecules, there are only 3 sites (or, more generally, locations) where the molecules are located. These molecules can exchange their places, and it does not matter whether they belong to the same component or different components. If the pore can accommodate up to a maximum of 10 molecules and there are only 3 molecules in the pore, there are only 3 locations. The fourth location (15) Ustinov, E. A.; Polyakov, N. S. Bull. Acad. Sci. USSR, Div. Chem. Sci. 2000, 49, 1011. (16) Ustinov, E. A.; Vashchenko, L. A.; Katalnikova, V. V. Russ. J. Phys. Chem. 2000, 74, 1862. (17) Ustinov, E. A. Multicomponent Adsorption on Zeolites and State of Adsorbed Phase. Adsorption Science and Technology: Proceedings of The Second Pacific Basin Conference on Adsorption Science and Technology; Do, D. D., Ed.; World Scientific: River Edge, NJ, 2000; pp 613-617. (18) Gorbunov, M. B.; Arkharov, A. M.; Gorbunova, N. A.; Kalinnikova, I. A.; Serpinski, V. V. VINITI, 1984, No. 1600-84 Dep., 9-54 (in Russian). (19) Danner, R. P.; Wenzel, L. A. AIChE J. 1969, 15, 515. (20) Dunne, J. A.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1996, 12, 5896. (21) Dunne, J. A.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1997, 13, 4333.

Ustinov and Do

is assigned only when the next molecule has entered the pore. The mechanism of pore filling is viewed as progressively counting the number of locations of molecules, irrespective of which component they belong to. The only molecule in a pore occupies location 1. Two molecules occupy locations 1 and 2, and so forth. The first molecule interacts only with an active center of the solid. The second molecule could interact with the solid and the first molecule. Similarly, each next molecule could interact with some of previous molecules and the adsorbent. The interaction energy of the additional molecule with the surface may decrease in magnitude due to the progressive occupation of a lower energy position, but its interaction with neighboring molecules could overcompensate for such a decrease. Hence, the total energy of interactions may increase with loading. To simplify the mathematical description, we suppose that coordinates of each location are independent of the total number of molecules in a pore and the compositions of the cluster. Furthermore, we assume that the behavior of a molecule depends only on the component it belongs to and the location where this molecule is situated. Hence, the behavior of a molecule on the kth location does not depend on the total number of molecules in the pore (which can be greater than or equal to k), the composition, and the way molecules are adsorbed. Such a representation, despite its simplicity, allows accounting for the interaction of the molecule with other molecules and the adsorbent. In our model, the interactions between adsorbed molecules are not restricted to nearest neighbors, but rather they can involve many molecules within the cluster. As will be seen later, the model allows us to account for energetic heterogeneity of pore volume, which results in a nonuniform mutual distribution of different molecules in the case of mixture adsorption. This is the main advantage of this model. Finally, in this model, zeolites are assumed to be represented as a set of identical statistically independent cavities. Consequently, this set of cavities forms the grand canonical ensemble. In the future, this model may be extended to activated carbons on the basis of accounting for pore size distribution. Mathematical expressions of this model can be obtained by different but equivalent ways. The most general way is the combinatorial calculus. Details of this approach can be found in our previous papers.15-17 In this paper, we consider a simpler approach, which not only leads to the same results but also gives a highly effective algorithm for predicting binary mixture adsorption from adsorption data of pure components on the same adsorbent. This approach is a kinetic approach similar to that of Langmuir, that is, equality of rates of adsorption and desorption at equilibrium. The following assumptions are made in the kinetic model: (1) The probability of adsorption of a molecule of component j in a pore is proportional to the partial pressure of this component in the bulk phase, pj. (2) The probability of desorption of a molecule belonging to component j from the pore containing k molecules is proportional to Kj,k if that molecule occupies the kth location. Otherwise, the desorption of this molecule is not feasible. 2.1. Case of Pure Component Adsorption. Here we consider adsorption of pure component j. Since the rates of adsorption and desorption are equal to each other at equilibrium, one can write the following recurrence equation:

Kj,iωi ) ωi-1pj

i ) 1, 2, ...

(1)

Mixed Gas Equilibrium Adsorption on Zeolites

Langmuir, Vol. 18, No. 9, 2002 3569

Here ωi is the probability of the event that a cavity contains i molecules; Kj,i is a parameter depending on temperature and energy of adsorption of the ith location. Equation 1 is a first-order difference equation. Its solution for ωi is i

ωi ) ω0pij exp(-

∑ ln Kj,k)

(2)

k)1

The factor ω0 denotes the probability of the event when there are no molecules in the cavity. This factor can be determined by accounting for the normalization condition:

∑i ωi ) 1

(3)

Combining eqs 2 and 3 yields the solution for the probability of the event, that a cavity contains i molecules, in terms of partition function and grand partition function:

ωi ) ξ-1Qiλi

(4)

Qi ) exp(-Fi/kBT)

(5)

where

m

ξ)

λiQi ∑ i)0 i

∑ ln Kj,k k)1

Fi ) iµ° + ∆Fi ) iµ° + kBT

(1) (1) (2) K1,i+jωi,j ) (ωi-1,j + ωi-1,j )p1

λ ) exp(µ/kBT) Here, Qi is the partition function of a cluster containing i molecules, ξ is the grand partition function of the cavity subsystem, Fi is the Helmholtz free energy of the molecular cluster containing i molecules, T is the temperature, kB is Boltzmann’s constant, and m is the maximum number of molecules that can be accommodated in the cavity. The variable µ, as seen from the comparison of eqs 2 and 4, is the chemical potential:

µ ) µ°(T) + kBT ln pj

m

iλiQi ∑ i)1

(6)

The above adsorption equilibrium equation is capable of describing many different forms of isotherms. It comes as no surprise that we have obtained the same equation as that derived from the grand canonical distribution, because the same assumption concerning the determination of the Gibbs ensemble was used.14-16 Nevertheless, it would make no sense to present one more approach in deriving the same adsorption equation if there were not

(7)

(2) (1) (2) ) (ωi,j-1 + ωi,j-1 )p2 K2,i+jωi,j

If we do not make any distinction regarding which molecule is situated on location number i + j, then the probability ωi,j of existence of a molecular cluster consisting of i molecules of component 1 and j molecules of component (1) (2) + ωi,j . Hence, by combining this 2 is given by ωi,j ) ωi,j equation with eq 7, we get

ωi,j ) K1,i+j-1ωi-1,jp1 + K2,i+j-1ωi,j-1p2

where µ°(T) is the standard value of the chemical potential. This equation coincides with that for the ideal gas, which results from assumption 1 that the probability of adsorption of a molecule is proportional to the bulk phase pressure. This assumption can be relaxed by replacing the pressure by the fugacity for nonideal gases. The amount adsorbed is proportional to the mean number of molecules in a cavity and the total number of cavities. Thus, if a0 is the specific total number of cavities, the specific amount adsorbed at equilibrium is given by

a ) a0ξ-1

the aim to generalize this approach to multicomponent adsorption. In fact, the mechanism under consideration provides a perfect means to extend the analysis to the case of mixture adsorption. 2.2. Binary Mixture Adsorption. Following the kinetic approach used in the analysis of pure component adsorption, we can now consider the process of evolution (devolution) of a cluster that contains i molecules of component 1 and j molecules of component 2. For such a cluster, there are two distinct possibilities. In the first case, location number k ) i + j is occupied by a molecule of component 1. The probability of the event for such a (1) cluster is denoted as ωi,j . In accordance with assumption 2, the probability of desorption of a molecule of component 2 equals zero, and hence the cluster may reduce its size only as a result of the desorption of the first component (2) to denote the molecule. Similarly, one can write ωi,j probability of the event of a similar cluster containing i molecules of component 1 and j molecules of component 2, but location number k ) i + j is occupied by a molecule of component 2. The cluster of this type can shrink owing to the desorption of that molecule. On the other hand, the cluster of type 1 exists as a result of adsorption of a molecule of the first component in the cavity containing a cluster of (i - 1) molecules of the first component and j molecules of the second component. The probability of such an event is proportional (1) (2) + ωi-1,j ). Similarly, the cluster of type 2 is to p1(ωi-1,j (1) + generated at a rate which is proportional to p2(ωi,j-1 (2) ωi,j-1). Thus, for the case of binary adsorption the kinetic approach leads to following recurrence equations:

(8)

The set of parameters K1,i+j and K2,i+j is previously determined from pure component isotherms. Equation 8 suggests that ωi,j is proportional to pi1pj2. It then allows us to replace ωi,j with ω0pi1pj2Qi,j, where Qi,j is the partition function of a cluster containing i molecules of component 1 and j molecules of component 2, and ω0 is the probability of the event that the cavity does not contain any molecules. Taking this into account, eq 8 can be readily transformed into the following equation written in terms of the partition functions:

Qi,j ) K1,i+j-1Qi-1,j + K2,i+j-1Qi,j-1

(9)

This recurrence equation can be used to determine the partition function Qi,j, subject to the boundary conditions Q0,0 ) 1 and Qi,-1 ) Q-1,j ) 0. The system of recurrence equations (eq 9) presents a very simple and fast algorithm of calculation of all values Qi,j. Once that has been done, the amounts adsorbed for both components can be calculated by the following equations:

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∑i ∑j iQi,jpi1pj2

a1 ) a0ξ-1

-1

a2 ) a0ξ

∑i ∑j

Ustinov and Do

jQi,jpi1pj2

∑i ∑j Qi,jpi1pj2

i Ci+j )

(11)

2.3. Main Thermodynamic Relations. The partition function Qi,j is related to the Helmholtz free energy as follows:

Qi,j ) exp(-∆Fi,j/kBT)

1

n

exp(xi) g n exp ∑xi ∑ ni)1 i)1

The number of terms in eq 15 is basically the number of ways that molecules of component 1 occupy i positions and molecules of component 2 occupy j positions:

where the grand partition function is given by

ξ)

( )

n

(10)

(i + j)! i!j!

Hence, (max) ) ∆Fi,j e ∆Fi,j

(12)

i!j!

(i + j)!

∑ ∆FI(i,j) - kBT ln (i + j)!I⊂M,|I|)i

i!j! (17)

Substituting this expression into the system of recurrence equations (eq 9), one can obtain the following equations for the change of Helmholtz free energy ∆Fi,j of a molecular cluster consisting of i and j molecules of components 1 and 2, respectively:

Accounting for eq 16, this inequality can be simplified as shown below:

∆Fi,j ) -kBT ln[K1,i+j-1 exp(-∆Fi-1,j/kBT) +

These equations may be considered as a criterion of ideal mixture behavior as this corresponds to the case of equiprobable distribution of molecules over the locations. In other words, this situation would occur if an adsorbent were energetically homogeneous. The variables Fi+j,0 and F0,i+j are the Helmholtz free energies of i + j molecules of pure component 1 and pure component 2, respectively. It was recently shown13 that this equation is closely connected to the postulate of the ideal adsorbed solution theory12 that Raoult’s law holds along the lines of constant spreading pressure. The mixing entropy for this case of equiprobable distribution of molecules is given by

K2,i+j-1 exp(-∆Fi,j-1/kBT)] (13) The parameters K1,i+j and K2,i+j can be first determined from pure component isotherms by least-squares fitting. Their relationship with the change in the Helmholtz free energy is given below: i

∆Fi,0 ) kBT

∑ ln K1,k

(14)

k)1

(max) ) ∆Fi,j

i∆Fi+j,0 + j∆F0,i+j (i + j)! - kBT ln (18) i+j i!j!

j

∆F0,j ) kBT

∑ ln K2,k

(i + j)! Smix ) kB ln i!j!

k)1

If the second subscript equals zero, ∆Fi,0 is the Helmholtz free energy of a cluster containing i molecules of component 1, and vice versa, when the first subscript equals zero, ∆F0,j is for a cluster containing j molecules of component 2. The recurrence eq 13 can be readily solved for ∆Fi,j, subject to the boundary condition of eq 14. Finally, the Helmholtz free energy of a molecular cluster of arbitrary composition can be written in the following abbreviated form (see the Appendix):

∆Fi,j ) -kBT ln

∑

exp(-∆FI(i,j)/kBT)

(15)

I⊂M,|I|)i

Here, I is the set of locations occupied by i molecules of the first component (|I| ) i) and M (M ) 1:(i + j)) is the set of locations which are occupied by all molecules in the cluster. It is evident that I ⊂ M. ∆FI(i,j) is the Helmholtz free energy of a cluster having i molecules of the first component residing at the locations as defined by the set I and j molecules of the second component at positions as defined by the set M - I.

∆FI(i,j) )

(∆Fk,0 - ∆Fk-1,0) + ∑(∆F0,k - ∆F0,k-1) ∑ k∈I k∉I

(16)

Equations 15 and 16 allow us to derive an upper limit of the Helmholtz free energy. For an arbitrary set of variables x1, x2, ..., xn, the following inequality is valid:

(19)

Equation 15 also allows us to obtain an estimate for the minimum Helmholtz free energy: (min) ∆Fi,j ) min{∆FI(i,j)} - kBT ln[(i + j)!/(i!j!)]

(20)

This minimum of Helmholtz free energy corresponds to a hypothetical situation. Since the Helmholtz free energy is the difference between the internal energy and the entropy multiplied by temperature, its decrease can be possible with either a decrease of the internal energy or an increase of the entropy. The decrease of the internal energy is associated with a molecular rearrangement with formation of a spatially ordered structure, while an increase of the entropy results from a spatial disordering of the molecular structure. Of course, processes of ordering and disordering cannot occur simultaneously. Since the change of the internal energy is the predominant factor compared to the entropy change, an ordered structure always yields a lower value of the Helmholtz free energy than a disordered one. However, it is difficult to find a priori the relative contributions of these two factors in the free energy. For this reason, to estimate the minimal value of the free energy we consider a hypothetical unrealistic situation, when the internal energy achieves its lowest value due to the spatial ordering and the entropy approaches its highest value as a result of disordering. The Helmholtz free energy of systems of binary mixtures would fall between the two curves corresponding to the

Mixed Gas Equilibrium Adsorption on Zeolites

Figure 1. Isotherms of oxygen (filled points) and nitrogen (empty points) on zeolite NaX at 303.8 K (ref 18). Solid lines are the correlation by the approach.

two limits of the Helmholtz free energy as described by eqs 18 and 20. For a given total pressure in the bulk phase, there is a point of intersection of the curves describing the dependency of selectivity on composition. One curve corresponds to the chaotic distribution, while the other corresponds to the maximum ordered molecular structure. In the small neighborhood of this intersection, any thermodynamically correct theories would provide excellent predictions of mixture adsorption equilibrium. Exact calculation of the Helmholtz free energy from eqs 15 and 16 leads to the results, which describe the molecular rearrangement due to a nonuniform adsorption potential within a cavity. To some extent, the potential of the model is limited, because it implies that behavior of each molecule at a definite location in the mixture adsorption is the same as it is in the case of pure component adsorption. Nevertheless, despite this simplification the model is able to describe the nonideality caused by the energetic heterogeneity. Equations 15 and 16, derived by this kinetic approach, are general because they can be derived from other approaches.14-16 Obviously, the set of values of Helmholtz free energy at all possible molecular combinations is sufficient to calculate mixture isotherms. Furthermore, it may disclose information concerning the differences in solid-fluid interactions for different molecules, which will be elaborated in section 3. 3. Results 3.1. Oxygen and Nitrogen Mixture on Zeolite NaX at Room Temperature. Adsorption of O2, N2, and their mixtures on zeolite NaX was investigated by a volumetric method as described in ref 18. In the case of mixture adsorption, 12 series were carried out and in each series the total amount of one of the components in the measuring cell remained constant while known quantities of the other component were added to the cell. The first component redistributed itself between the sample of zeolite and the free volume of the adsorption cell with the addition of the second component. Measurements for pure component adsorption and binary mixture adsorption were carried out in the pressure range from 0.1 to 60 bar. The calculated value of a0 for zeolite NaX was 0.597 mol kg-1. Figure 1 shows the adsorption isotherms of oxygen and nitrogen at 303.8 K. These isotherms were analyzed by using eq 6. The dependence of the molar Helmholtz free energy change on the number of molecules in a cavity was calculated by eq 5, where parameters Kj,k were ap-

Langmuir, Vol. 18, No. 9, 2002 3571

Figure 2. Molar Helmholtz free energy of oxygen (filled points) and nitrogen (empty points) as a function of the number of molecules in a cavity of zeolite NaX at 303.8 K.

Figure 3. Interrelation between the molar fraction of oxygen and total pressure in the gas phase in different series at 303.8 K for the data taken from ref 18: (filled points) series with a constant total amount of nitrogen in the measuring cell; (empty points) series with a constant total amount of oxygen in the measuring cell.

proximated by a polynomial of the 6th degree in the following form: 6

ln Kj,k )

bj,lkl ∑ l)1

(21)

The coefficients bj,l (j )1 for component 1 and j ) 2 for component 2) of the polynomial were found by the leastsquares fitting of eq 6 against the pure component data. This set of parameters Kj,k was then used for calculating the partition function Qi,j for mixture adsorption by eq 9. The dependence of the Helmholtz free energy on the number of molecules in a cavity is presented in Figure 2. The reference state is chosen at the same temperature and the pressure of 1 bar. The Helmholtz free energy of nitrogen is less than that of oxygen, which is due to the stronger gas-solid interaction for nitrogen. The interrelation between the total pressure and the composition in the bulk phase for each of the 12 series is shown in Figure 3. The curves with filled points represent the series where the total amount of nitrogen in the measuring cell remained constant and the pressure in the bulk phase was increased by adding pure oxygen. On the other hand, the curves with open points represent

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Ustinov and Do

Figure 4. Dependence of the molar fraction of oxygen in the adsorbed phase on total pressure in the gas phase in different series at 303.8 K (ref 18): (filled points) series with a constant total amount of nitrogen in the measuring cell; (empty points) series with a constant total amount of oxygen in the measuring cell. Solid lines are calculated by the theory.

Figure 6. Dependence of the adsorbed phase composition on that of the bulk phase along a constant pressure of 50 bar in the gas phase for the system O2-N2-zeolite NaX. T ) 303.8 K. Points are interpolated by the experimental data using curves presented in Figures 3 and 4. The solid line is calculated by the theory.

Figure 5. Dependence of the amount adsorbed of oxygen on total pressure in the gas phase in different series at 303.8 K for the system O2-N2-zeolite NaX: (filled points) series with a constant total amount of nitrogen in the measuring cell; (empty points) series with a constant total amount of oxygen in the measuring cell. Solid lines are calculated by the theory.

Figure 7. Effect of the adsorbed phase composition on selectivity at a temperature of 303.8 K and a total pressure of 50 bar. Subscripts 1 and 2 relate to oxygen and nitrogen, respectively. The solid line is calculated by the theory. The dashed line corresponds to the case of chaotic distribution (ideal adsorbed phase). The dashed-dotted line corresponds to completely ordered mutual distribution of O2 and N2 in cavities.

the series with a constant total amount of oxygen in the cell. Figure 4 presents the dependencies of the adsorbed phase composition on the total pressure in the bulk phase. The solid lines in the figure are calculated by the model, and we see that the predictions agree reasonably well with the experimental data. Similar excellent agreement is observed in the prediction of the adsorbed amount of oxygen on the total pressure of different series as shown in Figure 5. In the case of binary mixture adsorption, each variable is a function of the other two variables. Therefore, to analyze the dependence of any two variables we keep the third variable constant. For example, when we study the dependence of the adsorbed phase composition on the bulk phase composition, we keep the total pressure constant. This dependence can be obtained by interpolation of the data presented in Figures 3-5. Points obtained by interpolation and the predicted curve at 50 bar are plotted in Figure 6, where we see that nitrogen is more favorable in adsorption. This is in agreement with the lower Helmholtz free energy for nitrogen as shown in Figure 2.

The parameter that is most sensitive to experimental errors and is an indicator of the predictive ability of a theory is the selectivity:

S2,1 )

x2/y2 a2p1 ) x1/y1 a1p2

(22)

Figure 7 displays the dependence of the selectivity on the adsorbed phase composition at a constant total pressure of 50 bar. The solid line is the prediction obtained from this theory. The dashed line corresponds to the case of absolutely chaotic arrangement of nitrogen and oxygen molecules in the cavity. Such a chaotic arrangement would occur if the adsorption field in the cavity were uniform or the interaction energies of O2 and N2 with zeolite were the same. The dashed-dotted line is plotted for the case of maximum ordered structure in the cavity. For example, if there are 3 molecules of nitrogen and 5 molecules of oxygen in the cavity, then nitrogen molecules would occupy locations 1-3 because of their stronger adsorbing ability. Other combinations would not be permissible in the case

Mixed Gas Equilibrium Adsorption on Zeolites

Figure 8. Probability of a N2 molecule to be on different locations for the system O2-N2-zeolite NaX at 303.8 K (ref 18). The total number of molecules in the cavity is equal to 10. Each curve corresponds to constant molecular composition. Number of N2 molecules: (b) 1; (9) 3; (2) 5; (O) 7; (0) 9.

of maximum ordered structure. One can see from the figure that the region bounding possible values of selectivity is quite narrow. For example, when the mole fraction of O2 in the adsorbed phase is 20% the selectivity must fall between 1.3 and 1.8. It is also seen from the figure that all experimental points fall into the region bounded by the two curves (excepting one point, which may be attributed to experimental errors). The prediction of the model (solid line) is quite close to the curve corresponding to equiprobable distribution of all molecules (dashed line). This is supported with the plot of the probability of nitrogen versus the location number as shown in Figure 8. Each curve in this figure corresponds to a constant number of nitrogen molecules. One can see that the probability for a nitrogen molecule to be in the first location is slightly higher than those at other locations. This is due to the fact that the first location is associated with cations residing in the cavity of the zeolite, and nitrogen is a stronger adsorbing species. However, this variation of the probability with location is not significant because the temperature is relatively high for this system. 3.2. Oxygen and Nitrogen Mixture on Zeolite 10X at 144 K. It is interesting to compare the previous system with a similar system of O2-N2-zeolite 10X but at a much lower temperature. The data were taken from the paper of Danner and Wenzel.19 Measurements were carried out at a constant total pressure of 1 bar. The calculated value of a0 was 0.497 mol kg-1. Isotherms for oxygen and nitrogen at 144 K are plotted in Figure 9 in logarithmic scale. Results of the analysis of these isotherms with eq 6 are presented in Figure 10, where the dependence of the molar Helmholtz free energy on the number of molecules in a cavity is plotted for oxygen and nitrogen. This dependence is then required in the prediction of mixture adsorption using eqs 9-12. Comparison of the predictions and the experimental data in the form of the dependence of the adsorbed phase composition on the bulk phase composition at a constant total pressure of 1 bar is shown in Figure 11. The agreement between the experimental data and the predicted curve is excellent despite the nonideality of this system. We shall elaborate on this nonideality below. Figure 12 presents the dependence of the selectivity on the adsorbed phase composition, keeping the total pressure in the bulk phase constant. As one can see, the agreement between prediction and experimental points is very good

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Figure 9. Isotherms of oxygen (filled points) and nitrogen (empty points) on zeolite 10X at 144 K (ref 19). Solid lines are the correlation by the approach.

Figure 10. Molar Helmholtz free energy of oxygen (filled points) and nitrogen (empty points) as a function of the number of molecules in a cavity. Calculations were made by processing the isotherms presented in Figure 9.

Figure 11. Composition of the adsorbed phase as a function of the gas-phase composition for the system O2-N2-zeolite 10X (ref 19) at 144 K and a total pressure of 1 bar. The solid line is the prediction by the theory.

even with the large change in the selectivity with composition (note that the logarithmic scale is used for selectivity). The dashed line is plotted for the hypothetical case of equal probability of all possible combinations of O2 and N2 distributions in a cluster. Such a situation would be realized if the adsorption field were uniform and very

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Figure 12. Selectivity as a function of adsorbed phase composition at a total pressure of 1 bar for the same system as in Figure 11. The solid line is the prediction by the theory. All experimental points and values calculated by any theory must be in the region bounded by the dashed and dashed-dotted lines. The dashed line corresponds to the case of equal probability for all combinations of molecular distribution over locations, which could be realized in a homogeneous adsorption field. The dashed-dotted line corresponds to minimal values of the Helmholtz free energy for the maximum ordered molecular distribution.

close to that implicitly assumed in the IAS theory, though the IAS usually describes slightly nonideal systems as if they were completely ideal.13 The dashed-dotted line in the figure is calculated for the case of maximum ordered molecular structure of a cluster. More exactly, this is the case when the Helmholtz free energy for each particular combination (for a definite number of molecules in the cluster) would be equal to its lowest possible value. The range where experimental points can be found is wider in comparison with the previous case. The two limiting curves intersect at a mole fraction of about 0.65. Interestingly, the predicted curve also passes through this point, which is required for thermodynamic consistency. We finally note that all experimental points fall within the range bounded by these two limits. To estimate quantitatively the degree of nonideality of the system, we plot the difference between the actual molar Helmholtz free energy and that for the ideal system (excess Helmholtz free energy) versus the adsorbed phase composition. The excess Helmholtz free energy is given by E (max) F h i,j ) ∆F h i,j - ∆F h i,j

(23)

The ideal system corresponds to the case of equal probability of molecular combination in the cluster. Figure 13 displays the dependence of the excess Helmholtz free energy on the number of oxygen molecules in a cluster of 10 molecules. Negative deviations from the ideal system are due to the spatial molecular ordering resulting from the nonuniform adsorption field. This molecular ordering leads to a decrease in entropy, which would increase the Helmholtz free energy if the internal energy remained constant. However, the decrease of the Helmholtz free energy with the molecular ordering is due to the fact that the internal energy associated with this rearrangement decreases more significantly than the entropy does. Figure 14 is an illustration of the molecular rearrangement in a cavity volume. The total number of molecules in the cavity in this case is 10. Each curve corresponds to a fixed molecular composition. The curves in this figure suggest that the probability of a nitrogen molecule is

Figure 13. Excess Helmholtz free energy in a cavity of zeolite 10X for different relations of numbers of O2 and N2 molecules at the same total number of molecules equal to 10.

Figure 14. Probability of a N2 molecule to be on different locations for the system O2-N2-zeolite 10X at 144 K (ref 19). The total number of molecules in the cavity is equal to 10. Each curve corresponds to constant molecular composition. Number of N2 molecules: (b) 1; (9) 3; (2) 5; (O) 7; (0) 9.

associated with a definite location. For example, the lowest curve corresponds to the case when there is one molecule of nitrogen and nine molecules of oxygen in the cavity. One can see that despite the fraction of N2 being only 10%, the probability to find a N2 molecule at the first location is 47%. The first location is the position where the first molecule will reside, and this suggests that this location is associated with a cation in the cavity. Due to strong ion-quadrupole interaction in the case of mixture adsorption, nitrogen tends to displace an oxygen molecule in the proximity of this cation. The probability to find a N2 molecule on location number 10 is negligibly small being only 0.5%. This feature is also observed for all other curves in the figure. Nitrogen always tends to the first location, which can definitely be ascertained to cations. Such kind of analysis provides deeper insight into the mechanism of mixture adsorption on zeolites. 3.3. Carbon Dioxide and Ethane Mixture on Zeolite NaX. The last highly nonideal system of CO2-C2H6NaX20,21 will be analyzed in the framework of this theory. In this case, the mixture adsorption was investigated at constant loadings of ethane. In contrast to nonpolar ethane, the carbon dioxide molecule is known to have a strong quadrupole moment. Isotherms of these components are presented in Figure 15. Analysis of pure component adsorption and heat of adsorption from the

Mixed Gas Equilibrium Adsorption on Zeolites

Figure 15. Isotherms of carbon dioxide at 31.4 °C (filled points) and ethane at 32.4 °C (empty points) on zeolite NaX (ref 20). Solid lines are the correlation by the approach.

Figure 16. Molar Helmholtz free energy of CO2 (filled points) and C2H6 (empty points) as a function of the number of molecules in a cavity. Calculations were made by processing the isotherms presented in Figure 15.

statistical thermodynamic standpoint was described earlier.13 In particular, it was shown that the entropy of ethane decreases with loading, which suggests the spatial structuring due to the closer proximity of molecules and the increasing role of repulsive forces. In contrast to this, the entropy of carbon dioxide remains nearly constant with loading, which prompts us to conclude that a molecular cluster has a highly ordered structure due to intermolecular interactions even if there are only a few molecules in the cavity. The behavior of the internal energy also has a substantial difference. For ethane adsorption, the internal energy decreases with the number of molecules in a cavity, suggesting the stronger intermolecular interactions. In the case of carbon dioxide adsorption, such interactions also increase, but the internal energy decreases because ion-quadrupole interactions drop sharply. Furthermore, ethane and carbon dioxide significantly differ in size. From the calculation, the maximum number of CO2 molecules in a cavity is 10, while only 7 molecules of ethane can be accommodated in the cavity. Analysis of the pure component experimental data with eq 6 is shown in Figure 16 where the molar Helmholtz free energy is plotted against the number of molecules in a cavity for ethane and carbon dioxide. The lower Helmholtz free energy of ethane suggests that ethane is a stronger adsorbing species.

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Figure 17. Change of selectivity with adsorbed phase composition at a low preloading of ethane (1.43 mol kg-1) in the system CO2-C2H6-NaX at 29.4 °C (ref 21). All comments are the same as in Figure 12.

The analysis of mixture adsorption is similar to that carried out for the previous two systems. Experimental measurements for mixture adsorption were carried out at a slightly different temperature from that for pure component adsorption. This temperature effect is taken into consideration by applying the Clapeyron equation to the information of the isosteric heat of adsorption as a function of loading.20 Results of the prediction of selectivity as a function of the adsorbed phase composition are presented in Figure 17 for the case of low preloading of ethane (1.43 mol kg-1). One can see that experimental points fall within the range bounded by the two curves plotted for the case of absolutely random molecule distribution over locations (dashed-dotted line) and for the case of maximum molecular ordering (dashed line). The experimental data points lie close to the case of maximum ordering. This can be explained as follows. Carbon dioxide molecules having a strong quadrupole moment displace ethane molecules near the cations, and the probability for this to happen is very high. Thus, the question is why the model fails to predict well the selectivity. In our opinion, there are two possible reasons. The first one is that the difference in molecular size is not accounted for in this version of the model. Molecules of smaller size have a greater ability to substitute the larger ones,22 which makes the molecular cluster more ordered. The other reason is apparently that the CO2-C2H6 mixture is nonideal in itself, not only due to the difference in solidfluid interactions. Figure 18 displays the dependence of the selectivity versus the composition of the adsorbed phase for a relatively higher preloading of ethane (2.57 mol kg-1). Again, we see that the experimental values of the selectivity are in the region bounded by the two curves corresponding to the ordered and disordered structures. The prediction is not as good as in the case of oxygennitrogen mixture adsorption in zeolite 10X but better than that for the ideal mixture (dashed-dotted line). 4. Conclusion A new approach for predicting equilibrium binary mixture adsorption on zeolites is considered. The approach is based on the assumption that cavities of a zeolite form a grand canonical ensemble of quasi-independent subsystems. To calculate the Helmholtz free energy of (22) Ustinov, E. A. Adsorption 2000, 6, 195.

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where Qi,j is the partition function for the molecular cluster consisting of i molecules of component 1 and j molecules of component 2. The boundary conditions are Q0,0 ) 1 and Qi,-1 ) Q-1,j ) 0. Let us sequentially derive the partition function for different compositions at a fixed total number of molecules. For i + j ) 2, eq A2 yields

Q2,0 ) A1A2

(A3)

Q1,1 ) A1B2 + B1A2 Q0,2 ) B1B2 For the total number of molecules i + j ) 3, one can obtain from eqs A2 and A3 Figure 18. Change of selectivity with adsorbed phase composition at a high preloading of ethane (2.57 mol kg-1) in the system CO2-C2H6-NaX at 28.94 °C (ref 21). All comments are the same as in Figure 12.

mixtures from those for pure components, a simple kinetic algorithm has been suggested instead of the timeconsuming combinatorial calculus as done earlier.15-17 The model accounts for the energetic heterogeneity and the difference in interaction of molecules with cations. The main outcome of the approach is that the information on the energetic heterogeneity of zeolite cavities is extracted from the analysis of pure component isotherms and it can then be used in the prediction of binary mixture adsorption. We also have derived expressions for the boundaries of a region within which experimental points as well as theoretical curves can be found. These boundaries correspond to the limiting cases of ordering and disordering of molecular clusters in the cavity volume. The approach was applied to three systems, one nearly ideal (O2-N2-zeolite NaX at 304 K) and two highly nonideal systems. The first nonideal system is O2-N2zeolite 10X at 144 K, for which the nonideality is caused only by the difference in the fluid-solid interaction. For example, a molecule of one component may have a quadrupole or dipole moment, while a molecule of the other component does not have it. This kind of nonideality is described well by the theory. The second nonideal system of CO2-C2H6 mixture adsorption on zeolite NaX exhibits more complex nonideality. This nonideality can be partly related to the difference in molecular size, which this approach does not take into account. Nevertheless, the theory correctly predicts the boundaries of the permissible region for selectivity. Acknowledgment. Support from the Australian Research Council is gratefully acknowledged.

Here, we present the derivation of the Helmholtz free energy for a binary mixture. For simplicity, the following notations are introduced:

(A1)

Q2,1 ) A1A2B3 + A1B2A3 + B1A2A3 Q1,2 ) A1B2B3 + B1A2B3 + B1B2A3 Q0,3 ) B1B2B3 Analogously, for the total number of molecules i + j ) 4 the partition functions are given by

Q4,0 ) A1A2A3A4

(A5)

Q3,1 ) A1A2A3B4 + A1A2B3A4 + A1B2A3A4 + B1A2A3A4 Q2,2 ) A1A2B3B4 + A1B2A3B4 + A1B2B3A4 + B1A2A3B4 + B1A2B3A4 + B1B2A3A4 Q1,3 ) A1B2B3B4 + B1A2B3B4 + B1B2A3B4 + B1B2B3A4 Q0,4 ) B1B2B3B4 In the general case, the partition function can be expressed as a sum and the number of terms in that sum is i Ci+j )

(i + j)! i!j!

Each term can be associated with a chain of (i + j) molecules distributed over (i + j) locations. This allows us to write the general expression for the partition function as follows:

Qi,j )

Bk ∑ ∏Ak∏ k∉I

(A6)

Here, I is the set of locations occupied by i molecules of the first component, and M is the set of all locations occupied by molecules of both components. For pure component adsorption, eq A6 is reduced to the following equation: i

Bk ) K2,k-1

Qi,0 )

Then the recurrence eq 9 can be rewritten as follows:

Qi,j ) Ai+jQi-1,j + Bi+jQi,j-1

(A4)

I⊂M,|I|)i k∈I

Appendix

Ak ) K1,k-1

Q3,0 ) A1A2A3

(A2)

Ak ∏ k)1 j

Q0,j )

Bk ∏ k)1

(A7)

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Hence, accounting for eq 12 for Ak and Bk we obtain

Ak ) Qk,0/Qk-1,0 ) exp[-(∆Fk,0 - ∆Fk-1,0)/kBT] (A8) Bk ) Q0,k/Q0,k-1 ) exp[-(∆F0,k - ∆F0,k-1)/kBT] Finally, the Helmholtz free energy can be expressed as follows:

∆Fi,j ) -kBT ln

∑

exp(-∆FI(i,j)/kBT) (A9)

I⊂M,|I|)i

where

∆FI(i,j) )

(∆Fk,0 - ∆Fk-1,0) + ∑(∆F0,k - ∆F0,k-1) ∑ k∈I k∉I

(A10)

LA011514U