3202
J. Phys. Chem. 1994,98, 3202-3206
Encapsulation of Simple Gases in Zeolites Jong-Ho Yoon* Department of Industrial Chemistry, Kyungpook Sanup University, 55, Hyomokdong, Donggu, Taegu, 701- 702 Korea
M~w-WOO Huh Department of Textile Engineering, Kyungpook Sanup University, 55. Hyomokdong, Donggu, Taegu, 701 702 Korea
-
Received: September 13, 1993; In Final Form: January 4, 1994'
By application of the statistical theory of the radial distribution function and the three-dimensional lattice gas theory, an Ar encapsulation isotherm curve of dehydrated sodalite up to 1000 atm at 534 K and CH4, C2H4, Ar, and Kr encapsulation isotherm curves of dehydrated 40% potassium-modified zeolite A up to 4300 atm at 523 and 623 K were calculated. The gases used were approximated as Lennard-Jones gases. Comparison of these calculated data with known experimental data showed a good agreement, particularly in the systems involving gas molecules which have a spherical shape such as CH4, Ar, and Kr. However, the results from the system involving the nonspherical molecule C2H4 were not satisfactory. It is believed that this is mainly due to the anisotropy of the C2H4 molecule.
Introduction At high-temperature and elevated pressure conditions, some gaseous molecules may permeate cavitiesin certain zeolite crystals, such as zeolite A, zeolite X,and sodalite.lq2 If the crystals are quenched to room temperature and the pressure is subsequently reduced, they may contain much encapsulated gas molecules, which can be released only by heating the crystals again to high temperat~res.l-~Since the term encapsulation stands for a physical trap of molecules in a few-molecular-diameteredmicrocavity, it is believed that the molecules encapsulated are spatially distributed in the cavity.4J In fact, because of this space distributed property, encapsulation isotherm curves frequently differ in form from those of conventional adsorption isotherm curves such as the Langmuir or BET type isotherm curves based on the adsorption surface or la~er.69~ This may also account for the failure of many conventional adsorption methods6.' to predict and to correlate with an encapsulation isotherm curve. The uniform and known crystal structure of zeolites738 makes it possible to use a molecular-based approach to the theory of gas encapsulation. As shown if one regards a gaszeolite system as a three-dimensional (3D)lattice gas system9JO and applies the theory of gas rdf," the gas-encapsulationcapacity can be expressed using the molecular properties of the zeolite crystal and that of the gas encapsulate. Encouraged by the previous successof this method in predicting hydrogen encapsulation capacity of Cs-substituted zeolite A up to the moderately high-pressure range (up to -150 atm),495 its application was extended further to other nonpolar gas-zeolite encapsulation systems and to higher pressures. Examined were Ar-encapsulation capacities of sodalite up to 1000 atm at 534 K and CH4, C2H4, Ar, and Kr encapsulation capacities of 40% potassium-exchanged zeolite A up to 4300 atm at 523 and 623 K. Since very high pressure gas states were involved, a more realistic Lennard-Jones (LJ) molecular potential was used to account for the intermolecular interactions between the gas encapsulates. Theory As in our previous we express the gas-encapsulation capacity C of a zeolite as e Abstract
published in Aduance ACS Abstracts, February IS, 1994.
0022-3654/94/2098-3202$04.50/0
C = jlve = j [ p r g ( r ) 4 d d r + 11[ 1 - exp(-p/kT)]
(1)
where j is the total number of cavities in the system, N is the maximum number of gas molecules that can be encapsulated in a single cavity, and 0 is the fractional coverage of the cavity. p and g(r) stand for the number density and the rdf of the gas, respectively. In the integral term, r is the separation between the gas molecules and R is the free radius of the cavity. In the exponential term, p, 7,k, and T are the system pressure, the cavity volume, the Boltzmann constant, and the system temperature, respectively. The gas rdfs used were obtained by applying the perturbation theorydeveloped by Barker and Henderson (BH).12 It has been known that a gas rdf obtained through the BH perturbation theory is good enough to calculate various equilibriumphysical properties (such as pressure, free energy, ...) of a highly compressed gas or liquid system.1*J2 The BH perturbation theory uses the wellknown hard-sphere system13 for the reference of a gas rdf and the difference between the reference hard-sphere potential and the molecular potential as the perturbative force. One of the BH type LJ molecular gas rdf expressions derived by Yoon et al.I4 is g(r)
= A exp(-U(r)/kT)H(r& - r) +
where g(r), go(r), and D are the LJ gas rdf, the reference hardsphere rdf, and the diameter of the reference hard-sphere, respectively. H(r) is the Heaviside step function which is 0 for r < 0 and 1 for r > 0. The LJ molecular potential function U(r) has the form
where e and u stand for the potential depth and the intermolecular separation at which U(r) = 0, respectively. In eq 2, A is the weighting factor given by12
A = jorbgo(r) dr/l*exp(-U(r))/kTdr 0 1994 American Chemical Society
(4)
The Journal of Physical Chemistry, Vol. 98, No. 12, 1994 3203
Encapsulation of Simple Gases in Zeolites
TABLE 1: Lennard-Jones Potential Parameters and Reference Hard-Sphere Diameters
5000
D (A) gas
1014t(erg)“
u (A)”
CHI C2H4
1.891 2.830 1.712 2.247
3.82 4.23 3.42 3.85
Ar Kr
523 K
534K
623 K 3.61
4000
4.09 3.26
3.22 3.66
* E
4-
a
Given in ref 18.
where r,i, = 2 l k at which the LJ molecular potential becomes the minimum.
.? 3000
e
7
e
n
2000
Calculation The unit-cell composition of the dehydrated sodalite used is
N ~ ~ [ A ~ ~ S ~ ~ ~ Z ~ ] X N ~ Owhere H.(~ x=Z1X .04.2 ) H Each ~O, unit cell contains two &cages. The unit-cell composition of the 40% potassium-exchanged zeolite A used is K4.8Na7.2[(A102)12(SiO2)12].1 In this unit cell there are one a- and one &cages. Crystallographicstudies have shown that the free cavity diameter of &cage is -6.6 A and that of a-cage is -1 1.4 Both the a-cage and the 8-cage are known to have the ability to encapsulate gaseous CH4, C2H4, Ar, and Kr molecules of interest to us.ls2 Since Barrer and Vaughanz have compiled the experimental data of argon encapsulationcapacities of sodalite at pressures up to 1000 atm at 534 K and Breckl has obtained the experimental CH4, CzH4, Ar, and Kr encapsulation capacities of 40% potassiumexchanged zeolite A at pressures up to 4300 atm at temperatures of 523 and 623 K,calculations were performed to include these thermodynamic conditions for comparison. The hard-spherediameters to obtain the referencehard-sphere rdfs (go’s) of the gases were calculated by the BH method+
1000
Density/lOO (/A^3)
Figure 1. Density vs pressure curve of Kr at 623 K. 3.0
I
I
1
...:
173 atm
- : 1512 otm -.-
: 5037 a h
D = S,”{l- exp(-U(r)/kT)) d r Note that D is dependent on the temperature not on the gas density. In Table 1, the LJ parameters of the gases used and the hard-spherediameters of the gases calculated at the temperatures examined are shown. Since D is independent of density, once D at a temperature is determined it becomes possible to obtain g&) (thus, g(r)) at any desired density at the temperature.13 The necessary number densities of the gases at the temperatures and pressures used were determined through interpolation by utilizing the density vs pressure curves of the gases obtained with the well-known pressure equation:”
p = pkT - $c47rr3U’(r) g(r) dr where Ll’(r) is the first derivativeof the LJ molecular potential. A representative picture of density vs pressure curve is shown in Figure 1. Substituting the obtained reference hard-sphere rdfs and the determined densities into eq 2, we obtained the gas rdfs at the temperatures and pressures of interest. A representative picture of a gas rdf change with pressure obtained by this method is shown in Figure 2. The encapsulation capacity calculations were performed in the model system designed by us. In Figure 3 this model is shown. In this model a probe molecule lies at the center of the cavity and the neighboring molecules are spatially distributed around the probe molecule. Since molecules have a finite size, we substrated the excluded radius u/2 from the crystallographicallydetermined radius of the cavity. Also, to account for the penetration effect, which may originate from the penetration of the encapsulated gas molecules into the interstices intrinsically present in the aluminosilicate framework that constitutes the zeolite cavities, we included the penetration radius R,. It is believed that the higher the system temperature, the more significant the pene-
5
10
15
20
Seporotlon (A)
Figure 2. Kr radial distribution functions at 623 K.
tration effect. Thus, we may write the free radius of the cavity as
where R’ is the crystallographically determined cavity radius. However, it should be noticed when R’is smaller than the kinetic diameter of an encapsulated molecule, a restriction that no more than one molecule can enter the cavity (that is, g(r) = 0 in the cavity) emerges.* Since the kinetic diameters of the gases studied have a value between 3.5 and 4.0 A, which is bigger than the &cage cavity radius of 4 . 3 A, if one of the gases is encapsulated in the p-cages, the restriction comes into effect. Thus, for the calculation of a &cage gas encapsulation, one may use a more simplified gas encapsulation capacity expression:
where the subscript 8 stands for the 8-cage. Since only one gas molecule is allowed in a &cage, a gas encapsulation capacity of a @cage is independent of the gas rdf. At this point, we would
3204 The Journal of Physical Chemistry, Vol. 98, No. 12, 1994
Yoon and Huh 50
40
\
m
I I \
I
noiecue
0
h
E,
30
0
-6 P
0
0 20
0
In
0
--
\ \
R’ : a/2: : R :
5
0 Cavity Wall
0
Crystallographlc Radius Excluded Radius Penetration Radius Cavity Free Radius
01
2.88 3.85 5.19 7.78 9.71
35.99 48.18 64.96 98.79 121.43
TABLE 3: Ar Encapsulation Capacity of Sodalite at 534 K C/g of zeolite press. density (A-3) ee (STP vol in cm3) (atm) 6.42 X 1.23 X 2.29 X 4.02 X 5.40 X 6.56 X 7.56 X
10-4 lk3 lk3 lO-’ lk3 lk3 lk3
0.097 0.184 0.335 0.558 0.706 0.804 0.870
600
800
1000
Pressure ( a h )
1 The subscripts a and 6 stand for the a-and &cages, respectively. At 623 K and 4300 atm 8 . 4 of the zeolite A is 1. In all cases Ng = 1.
49.5 98.7 197.4 394.8 592.2 789.6 987.0
400
200
TABLE 2 Change of Kr Encapsulation Capacity of 40% Potassium-ExchangedZeolite A with varying R at 623 K and 4300 atma C/g of zeolite R a (A) Nu Nu -tNg (STP vol in cm3) 1.88 2.85 4.19 6.78 8.71
.
10
Figure 3. Encapsulation model.
3.78 3.95 4.14 4.50 4.87
0
0
v
4.34 8.25 15.02 25.07 31.37 35.99 38.94
like to mention that the kinetic diameter of hydrogen is less than the radius of the @-cagecavity. Therefore, in our previously studied hydrogen encapsulations in Cs-substituted zeolite A cases4s5this restriction was not used. Regardless, R, of the cy-cage was determined through the calculation of Kr encapsulation capacities of 40% potassiumexchanged zeolite A at 623 K and 4300 atm by varying the cavity free radius by using the above encapsulationmodel. The variation of encapsulation capacity with the free radius is shown in Table 2. Since the Kr-40% potassium-exchanged zeolite A encapsulation system has been known to be one of the most stable gas encapsulation systems, we determined R by comparing our data with Breck‘s’ Kr encapsulation data measured at the same temperature and pressure. At R = 4.408 A to give R, = 0.63 A, theyshowedthebestagreement(Le., C= 90.5cm3/gofzeolite). Since sodalite consists of only &cages, we used eq 8 for the calculation of Ar encapsulation capacities of the sodalite. On the other hand, since 40%potassium-exchangedzeoliteA consists of both cy- and &cages, both eqs 1 and 8 were used for the calculation of CH4, C2H4, Ar, and Kr encapsulation capacities of the zeolite A.
Figure 4. Ar encapsulation isotherm curve of sodalite at 534 K. Experimental data points are from Barrer and Vaughan in ref 2.
a
TABLE 4 Encapsulation Capacity of 40% Potassium-Exchanged Zeolite A at 623 K press. density C/g of zeolite (A-3) 8, Nu 8, (STPvolinc”) (atm) 8.45 42.75 86.29 177.11 384.73 653.45 1024.35 1399.77 1550.91 1893.08 2088.14 2300.80 2532.50 2784.67
0.0001 O.OOO5 0.0010 0.0020 0.0040 0.0060 0.0080 0.0095 0.0100 0.0110 0.0115 0.0120 0.0125 0.0130
0.075 0.323 0.546 0.802 0.970 0.997 1.OOO 1.000 1.OOO 1.000 1.OOO 1.OOO 1.000 1.000
1.018 1.090 1.183 1.379 1.822 2.345 3.055 3.498 3.692 4.102 4.320 4.547 4.782 5.026
0.015 0.075 0.146 0.276 0.505 0.697 0.846 0.922 0.941 0.968 0.978 0.985 0.990 0.994
1.14 5.34 9.91 17.30 28.44 37.98 48.82 55.32 57.98 63.45 66.30 69.23 72.23 75.34
TABLE 5 C f i Encapsulatim Capacity of 40% Potassium-Exchanged Zeolite A at 523 K (atm)
(A-9
8.
Nu
66
C/g of zeolite (STPvolincm3)
7.07 34.56 67.35 130.00 193.97 360.42 661.80 774.00 824.33 906.27
0.0001 O.OOO5 0.0010 0.0020 0.0030 0.0050 0.0070 0.0075 0.0077 0.0080
0.074 0.314 0.520 0.757 0.879 0.980 0.999 1.000 1.000 1.OOO
1.008 1.042 1.085 1.178 1.283 1.535 1.865 1.963 2.004 2.067
0.015 0.072 0.136 0.246 0.344 0.543 0.763 0.814 0.834 0.861
1.12 4.98 8.73 14.18 18.34 25.53 32.74 34.62 35.38 36.50
press.
density
The Ar encapsulation capacities of sodalite in the pressure range 0 to 1000 atm at the isotherm of 534 K are shown in Table 3 and plotted together with Barrer and Vaughan’s experimental data in Figure 4. The calculated encapsulation capacities of CH4 up to 2700 atm at 623 K, CzH4 up to 900 atm at 523 K, Ar up to 2900 atm at 623 K, and Kr up to 5000 atm at 623 K of 40% potassiumexchanged zeolite A are shown in Tables 4-1 and plotted in Figure 5 . The comparison of the calculated data with with Breck’s experimental data at several selected data points is shown in Table 8.
The Journal of Physical Chemistry, Vol. 98, No. 12, 1994 3205
Encapsulation of Simple Gases in Zeolites
TABLE 6 Ar Encapsulation Capacity of 40% Potassium-Exchanged Zeolite A at 623 K press. density C/g of zeolite (A-3) 0, N, 0.4 (STPvol incm’) (atm) ~~
~~
8.50 42.75 86.20 175.80 480.49 877.31 1224.21 1664.44 2226.57 2563.56 2943.90
~
O.OOO1 0.0005 0.0010 0.0020 0.0050 0.0080 0.0100 0.0120 0.0140 0.0150 0.0160
0.075 0.323 0.545 0.799 0.988 1.000 1.000 1.000 1.000 1.000 1.000
1.025 1.123 1.248 1.505 2.339 3.285 3.984 4.740 5.552 5.978 6.738
0.015 0.075 0.146 0.275 0.584 0.798 0.893 0.952 0.983 0.991 0.995
80
1.15 5.49 10.35 18.52 36.28 51.16 61.11 71.32 8 1.89 87.33 91.90
8
o.Ooo1 0.0005 0.0010 0.0020 0.0050 0.0080 0.0100 0.0120 0.0130 0.0140 0.0150 0.0152 0.0160
0.075 0.322 0.541 0.794 0.988 1.000 1.ooo 1.000 1.000 1.000 1.000 1.Ooo 1.Ooo
1.172 1.087 1.177 1.368 2.043 2.921 3.651 4.519 5.010 5.543 6.118 6.239 6.738
0.015 0.075 0.144 0.271 0.588 0.834 0.937 0.985 0.994 0.998 1.ooo 1.000 1.000
1.14 5.31 9.77 16.98 32.60 46.98 57.40 68.86 75.12 81.83 89.06 90.56 96.81
Throughout this work all the integrals involving rdf were computed by the trapezoidal rule with the integral interval of 0.01 D.
Discussion As was done before,4ss the one-component 3D lattice gas theory and the theoretical gas rdf were used to calculate the gas encapsulation isotherm curves of sodalite and 40% potassiumexchanged zeolite A. Using the 3D lattice gas theory and the gas rdf theory, we were able to calculate the gas encapsulation probabilities of a zeolite crystal and the maximum number of molecules that can be encapsulated in the zeolite cavity, respectively. The use of the 3D lattice gas theory and the gas rdf theory in our encapsulation study becomes possible by the assumption that gas encapsulations occur mainly via molecular diffusion. In fact, Cohen De Lara16 and the collaborators have shown that diffusion is the dominant mechanism for a gas encapsulation by interpreting their neutron scattering data on H2 and CH4 encapsulation in zeolite A with jump models. Throughout this work the gases examined were treated as LJ molecules. According to the theories of gases and liquids, the LJ molecular potential is known to be one of the most commonly used realistic intermolecular potentials, enough to yield various equilibrium physical properties such as pressure, free energy, and rdf of a highly compressed gas or liquid system consisting of spherical molecules such as CH4, Ar, or Kr.11J2 Since encapsulation arises within a few-molecular-diameteredcavities where the intermolecular interactions between the gas molecules are the greatest, it is emphasized to use such a realistic intermolecular potential to obtain the gas rdf s. So far, as can be seen in Figure 4, the calculated Ar encapsulation capacities of sodalite at 534 K show a good agreement with the experimental data at all the pressures examined. Moreover, if one expands the calculated Ar encapsulation isotherm curve (up to =2000 atm), one may find that the curve approaches the encapsulation saturation limit of 45 cm3/g of zeolite predicted by BarrerZ and Vaughan. This suggests
00
0
I)
0 : 014, 623 K A : cW4. 123 K 0 : &, 62.3 K 0 : Kr. 825 K
TABLE 7: Kr Encapsulation Capaci Potassium-Exchanged Zeolite A at 62 KOf 40% press. density C/g of zeolite (A-3) 0, N, (STPvolincm3) (atm) 8.49 42.46 85.18 172.77 486.00 982.59 1511.89 2293.08 2809.29 3428.25 4165.53 4300.00 5037.35
1
0 0
*O
t?
t
lo
1000
0
2000
3000
4000
5000
Pressure ( a h )
Figure 5. Gas encapsulation isotherm curves of zeolite A.
TABLE 8 Encapsulation of Gases in 40% Potassium-Exchanged Zeolite A C/g of zeolite (STP vol in cm3) gas temp (K) press. (atm) theory exptb 73.7 98-105 CH4 623 2650 523 830 35.5 7681 C2H4 Ar Kr a
623 623
2650 4300
89.4 90.5
77-109 90.5
Calculated by letting R = 4.408 A. From Breck’s data in ref 1.
that the Ar-sodalite system is a truly existing one component 3D lattice gas model proposed by Hill? In light of Hill’s 3D lattice model, one may interpret the restriction that no more than one Ar molecule/fl-cage is allowed in the sodalite that each 8-cage in the sodalite is empty or occupied by one Ar molecule. Since only one Ar molecule/cage is allowed in the Ar-sodalite system, the gas encapsulation capacity is independent of Ar rdf. Unlike sodalite, in 40% potassium-exchanged zeolite A the bigger a-cages, which are capable of encapsulating more than one gas molecule (i.e., g(r) > 0), now participate in the gas encapsulation in addition to the @-cages. Therefore, the gas encapsulation isotherm curves of the 40% potassium-exchanged zeolite A is dependent not only on the fractional coverage but also on the gas rdf. The fractional coverage dependence and the rdf dependence are well presented in Busvalues and in Nuvalues, listed in Tables 4-7. In all cases, the increase of 0 with increasing pressure is significant in the low-to-medium pressure range, but as the pressure increases further the increment becomes less and less and finally it becomes essentially constant (e 1). In the Kr encapsulation case, OU and 0, become constant at -500 and =3500 atm, respectively. Therefore, the increase of Nu (thus, the rdf) becomes increasinglyimportant with increasing pressure. Because both 8 and gas rdf contribute to the increase of gasencapsulation capacities of the zeolite A with increasing pressure, the gas-encapsulation isothermcurves from the zeolite A increase more rapidly with increasing pressure than does the Ar encapsulation isotherm curve from the sodalite, as can be seen in Figure 5 . On the other hand, we would like to mention that because of the lackof experimentaldata to compare withour data, thevalidity of the encapsulation isotherm curves from the zeolite A was not examined fully. However, it is very much encouraging to find that there is a good agreement between our data and Breck‘s’ experimental data at the data points examined, as can be seen in Table 8.
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3206 The Journal of Physical Chemistry, Vol. 98, No. 12, 1994
The significant underestimationof CzH4 encapsulationcapacity by the 3D lattice gas approach may be originated from the anisotropy of C2H4 molecule. For strongly anisotropicmolecules, reduction of the reference system to an equivalent hard-sphere system is unlikely to yield a reliable CzH4 rdf, which plays a central role in calculating the gas-encapsulation capacities. In the C2H4case, theanisotropic reference potential should be related to an equivalent hard-core potential that itself is noncentral. Apart from the C2H4 rdf problem, the anisotropy leads to a permanent quadrupole moment in C2H4 molecule. Many adsorption studies have shown that quadrupolar molecules such as CzH4 and C02 have a tendency to be adsorbed more in a zeolite particularly by the anisotropic interactions between the quadrupole moment of adsorbates and the metal cations in the host zeolite.17 Since in our model such multipolar interactions between gas encapsulates and host zeolites is not included, it is thought that the neglect of thequadrupolar interactionsis responsiblefor the underestimation of the C2H4 encapsulation capacities, as well.” One more interestingphenomenonrelated to the rdf dependence is that the increase of the first rdf maximum with increasing pressure can open a way to store more gas molecules per unit volume in the zeolite A than in a normal bulky gas container under the same pressure and temperature condition, at least theoretically. In fact the first rdf maximum of 2.7 of the gas at the highest pressure in Figure 2 means that the gas density at the separation at which the first rdf maximum appears is 2.7 times greater than the gas density in bulk. As far as the penetration radius is concerned, if one could estimate the anisotropicintermolecularinteractionswith accuracy between the encapsulated gas molecules and the a-cavity-wallconstituting molecules, a more rigorous way would be provided in determining the penetration radius R,. Also, with such a method one would estimate that attractive intermolecular interactions may exist between the encapsulated molecules and the cavity constituting molecules. It is believed that such attractive intermolecular interactions can increase the calculated gas encapsulation capacities of the zeolites further. In summary, by applying Hill’s 3D lattice gas theory together with the BH perturbation theory of gas rdf, an expression to
Yoon and Huh calculate a gas encapsulation capacity of a 3D lattice cavity applicableto a very high pressure (density) was deduced. Analyses from the calculated Ar encapsulation capacities of sodalite at 534 K and C h , Ar, and Kr encapsulation capacities of 40% potassium-modified zeolite A at temperatures up to 623 K and pressures up to 5000 atm by the deduced expression shows that this approach seems to be very much promising to predict a very high pressure gas encapsulation capacity of a gas-zeolite system.
Acknowledgment. The authors acknowledge the financial support from the Institute of Industrial Technologyin Kyungpook Sanup University. References and Notes (1) Breck, D. W. J. Chem. Educ. 1964,41,678. Nicol, J. M.; Eckert, J.; Howard, J. J. Phys. Chem. 1988, 92, 7117. (2) Barrer, R. M.; Vaughan, D. E. W. J. Phys. Chem. Solids. 1971,32, 731. (3) Harper, R. J.; Steifel, G. R.; Anderson, R. B. Can. J. Chem. 1969, 47, 4661. (4) Yoon, J.-H.; Heo, N. H. J . Phyk Chem. 1992, 96,4997. ( 5 ) Yoon, J.-H. J . Phys. Chem. 1993, 97, 6066. (6) Loughlin, K.F.; Ruthven, D. N. J . Chem. Phys. Solids. 1971, 32, 2541. (7) Fraenkel, D.; Shabtai, J. J. Am. Chem. Soc. 1977,99,7074. ( 8 ) Breck, D. W. Zeolite Molecular Sieues; John Wiley & Sons: New York, 1974; Chapter 2. (9) Hill, T. L. Statisticul Mechanics;McGraw-HillBookCo.: New York, 1956; p 402. (10) For example: Hill, T. L. InrroductionroSrarisrical Thermodynamics; Addison-Wesley Publishing Co. Inc.: London, 1960; p 130. (1 1) McQuarrie,D. A. SratisricalMechanics;Harper & Row: New York, 1976; Chapters 13 and 14. (12) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47,2856. (13) A listing of a FORTURAN program that gives a hard-sphere rdf is given in: McQuarrie, D. A. Srotisrical Mechanics; Harper & Row: New York, 1976; p 600. (14) Yoon, J.-H.; Hacura, A.; Baglin, F. G. J. Chem. Phys. 1989, 91, 5230. (15) Smith, W. R.; Henderson, D. Mol. Phys. 1970,19, 411. (16) Kahn, R.; Cohen De Lara, E.; Viennet, E.J . Chem. Phys. 1989,91, 5097. Cohen De Lara, E.; Kahn, R. Zeolites 1992, 12, 256. (17) Happer, R. J.; Stifeld, G. R.; Anderson, R. B. Can. J . Chem. 1969, 47, 4661 and references therein. (18) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1967.
