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Langmuir 1997, 13, 808-813
Modeling of High-Pressure Adsorption Isotherms above the Critical Temperature on Microporous Adsorbents: Application to Methane P. Be´nard and R. Chahine* Institut de recherche sur l’hydroge` ne, Universite´ du Que´ bec a` Trois-Rivie` res, 3351 Boulevard des Forges, C.P. 500, Trois-Rivie` res, Que´ bec G9A 5H7, Canada Received August 27, 1996. In Final Form: November 22, 1996X We present a detailed comparison between experimental adsorption measurements and the excess adsorption predicted by the Ono-Kondo equations. The study was done for high-pressure adsorption of methane above the critical point on microporous adsorbents. The experimental adsorption isotherms of CH4 on the activated carbon CNS-201 as well as others are compared with theory over the temperature range 243-333 K and for pressures up to 16 MPa. Extension of the model to zeolites is also discussed.
I. Introduction Physical adsorption on activated carbons and zeolites is widely used for the separation and purification of gas mixtures.1 It also offers a very promising avenue for lowering the storage pressure of compressed gas fuels such as natural gas and hydrogen.1,2 The design of adsorptionbased processes usually requires the characterization of the behavior of the adsorption isotherms over a wide range of pressures and temperatures above the critical point of the adsorbate. One of the difficulties in this area has been to develop a simple model for the adsorption isotherms under such conditions for adsorbents with small pores. Such a model must yield an isotherm based on a minimal set of parameters, which must have a clear physical interpretation in terms of the thermodynamic properties of the gas-solid system. The most widely used models of adsorption were developed for absolute adsorption. As such, they predict a monotonically increasing adsorption isotherm, which generally saturates at the maximum adsorption capacity in accordance with thermodynamical requirements. The adsorption experiments however measure excess adsorption, which is defined as the excess gas present in the pores and on the surface of the adsorbent, over and above that corresponding to the density of the gas in the bulk phase at that temperature and pressure. Although excess adsorption and absolute adsorption differ little at low pressures, the high-pressure excess adsorption isotherms of microporous adsorbents in the above critical region often go through a maximum followed by a steady decline as a function of adsorbate density. In this paper, we apply a modified version of the Ono-Kondo equations to the study of experimental adsorption isotherm curves of methane on activated carbons and zeolites along the lines recently proposed by Aranovich and Donohue for microporous carbon adsorbents.3,4 In our approach however, the adsorption process is directly mapped onto the sublattices of the hexagonal graphite planes, taking into account the Lennard-Jones repulsion between nearest-neighboring adsorbed molecules on the graphite layers. In section II we present a brief review of adsorption models for microporous adsorbents, followed by a discus* Corresponding author. X Abstract published in Advance ACS Abstracts, January 15, 1997. (1) Sircar, S.; Golden, T. C.; Rao, M. B. Carbon 1996, 34, 1. (2) Chahine, R.; Bose, T. K. Int. J. Hydrogen Energy 1994, 19, 161. (3) Aranovich, G. L.; Donohue, M. D. Carbon 1995, 33, 1369. Aranovich, G. L. J. Colloid Interface Sci. 1990, 141, 30. (4) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1990, 94, 6061.
S0743-7463(96)00843-8 CCC: $14.00
sion in section III of the Ono-Kondo approach to adsorption. In section IV we compare our experimental adsorption isotherms of methane on the activated carbon CNS201 with the Ono-Kondo model over the temperature range 243-333 K. The model is also compared to the experimental data of Payne et al.5 In section V the model is extended to adsorption on zeolites. Finally in section VI we discuss our results and present our conclusions. II. Review of Adsorption Models for Microporous Adsorbents The most basic approach to adsorption is the Langmuir model, which depends only on the adsorbate/adsorbent interactions, completely neglecting the interactions between adsorbate molecules. As such this model is mostly useful at low pressures and high temperatures. The virial model, which expresses the adsorption isotherm as a density expansion, can yield the isosteric heat of adsorption in the limit of zero coverage but makes no reference to the structural properties of the adsorbent. The BET approach improves on the Langmuir model by taking into account adsorbate/adsorbate interactions and thus allows for multilayer adsorption. However this model is mostly useful for mesoporous adsorbents, where multilayer adsorption can occur with little contribution from the structure of the adsorbent. In contrast the structure of microporous adsorbents can be expected to influence the adsorption isotherm, because of the close proximity of the solid surfaces. For microporous adsorbents such as activated carbons, the model most frequently used has traditionally been the Dubinin-Radushkevitch model6 and its variants,7 which can offer an estimate of the structural and energetic parameters of the adsorption process. The DubininAstakhov approach predicts the well-known adsorption isotherm for subcritical adsorbate gases:
Γ/Γ0 ) exp(-(RT ln(P/Ps)/ED)2)
(1)
where Ps is the saturation pressure and ED is a characteristic energy of adsorption. When the adsorbate is above the critical conditions, as is relevant to many industrial processes, the saturation (5) Payne, H. K.; Sturdevant, G. A.; Leland, T. W. IEEC Fundam. 1968, 363. (6) Dubinin, M. M.; Radushkevich, L. V. Proc. Acad. Sci. USSR 1947, 55, 331. (7) Gregg, S. G.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982.
© 1997 American Chemical Society
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pressure is no longer defined. The Dubinin approach is still used in the literature along with various modifications,6,7 but such models become more phenomenological, since they require the introduction of an effective saturation pressure, defined on a purely empirical basis. In general, all of the approaches mentioned above eventually fail for a wide variety of gases at high pressures above the critical conditions because they can only be applied over a small temperature range and because they were designed for the absolute adsorption isotherm or the low-pressure regime. Since the experimentally accessible isotherm is obtained through excess adsorption measurements, the experimental isotherm will exhibit a maximum when the bulk density starts to rise faster than the adsorbed density as a function of pressure. A proper model for high-pressure adsorption above the critical temperature of the adsorbate must therefore include these corrections. High-pressure excess adsorption isotherms have been studied using density functional theory (see ref 4 and references therein). In such an approach, the excess adsorption isotherms can be obtained through a selfconsistent calculation of the adsorbed density. III. Ono-Kondo Approach to Adsorption The slit model of activated carbons8 describes the microporous structure of activated carbons as two parallel planes of graphite spaced by a few molecular diameters. The planes are usually considered large enough to neglect finite-size effects. The graphite planes form an hexagonal matrix for the adsorption process. In lattice approaches such as the one used here, adsorption occurs on a discrete lattice, whose symmetry reflects the periodic arrangement of the adsorption sites on the adsorbent. At most one particle can be adsorbed on an adsorption site. The lattice used in the model will thus be formed of layers of hexagonal planes separated by a distance of one adsorbate molecule. The molecular fraction per adsorption site xi on the ith layer, is related to the molar density Fi by the expression
xi ) Fi/Fmc
(2)
where Fmc is the density at maximum capacity, i.e. the molar density corresponding to a completely filled adsorption layer. The excess adsorption, which is the experimentally relevant physical observable, is given by M
(xi - xb) ∑ i)1
N)C
(3)
The density xb is the bulk molar density of the adsorbate Fb divided by the density at maximum capacity Fmc. M is the maximum number of layers that can fit in a typical microporous slit of the activated carbon. The prefactor C in eq 3 takes into account the density of the active pores of the adsorbent and other structural properties of the adsorbent. The Ono-Kondo equations9 are a set of coupled selfconsistent nonlinear equations describing the density profile of successive layers of adsorbed molecules. They are obtained from using a local density assumption on the three-dimensional Ising (or lattice gas) model with a polarizing field on one of its surfaces. This field models the action of the attractive interactions between the adsorbent surface and the adsorbate molecules. The model (8) Matranga, K. R.; Myers, A. L.; Glandt, E. D. Chem. Eng. Sci. 1992, 47, 1569. (9) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids; Springer-Verlag: Berlin, 1960.
has been previously used in the context of adsorption of solutes in liquid solutions and recently extended to adsorption of gaseous adsorbates on activated carbon by Aranovich and Donohue.3 The Ono-Kondo equations can thus be written as
ln
(
)
xi(1 - xb)
xb(1 - xi)
+
z0E (x - xb) + kT i z2E (x - 2xi + xi-1) ) 0 (4) kT i+1
subject to the following boundary conditions for eq 4:
x1 ) xM
(5)
and
(
ln
)
x1(1 - xb)
xb(1 - x1)
+
EA E (z1x1 + z2x2 - z0xb) + ) 0 (6) kT kT
In eqs 4 and 6, z2 ) 1 is the interplane coordination number, z1 ) 6 is the in-plane coordination number, and z0 ) z1 + 2z2 is the full lattice coordination number of the Ising system. The parameter E in eqs 4 and 6 describes the interactions between adsorbate molecules, which are limited to nearest-neighbor sites of the lattice. The onsite adsorption potential EA in eq 6 parametrizes the interaction between the adsorbate particles and the adsorbent surface. The number of interaction parameters of the adsorbate-adsorbent system is kept to a minimum and limited to onsite and to nearest-neighbor interactions between adjacent cells. Equations 4-6 form a set of nonlinear equations which must be solved self-consistently. The parameter z0 only comes into play in the equations as a scaling factor for the intermolecular interaction parameter E. However the results of the self-consistent equations will depend on the ratio z1/z0. Changing this ratio is similar to modifying the interlayer interaction energy with respect to the in-plane interactions. A more formal approach to adsorption would allow two values of E to represent the interplane interactions and the inplane interactions (E⊥ and E|). This would introduce however another fitting parameter. Distinguishing between E⊥ and E| would be crucial in order to properly describe the behavior of CH4 in the supercritical regime in large micropores or mesopores, where the interplane molecular interactions can lead to multilayer adsorption even above the critical temperature, as discussed by Aranovich and Donohue.3,10 However because we are interested in a temperature regime well above the supercritical region, we have considered the interaction E to model the average interaction between adsorbed molecules in order to limit the number of fitting parameters. As we will see later, this simplification does not hinder the model. When the interactions between the adsorbed molecules can be neglected (E ) 0), eqs 4-6 can be easily solved, yielding the following expression:
xb(1 - xb)(1 - exp(EA/kT))
N0 ) 2C
xb + (1 - xb) exp(EA/kT)
(7a)
N0 corresponds to an interactionless excess adsorption isotherm. Dividing both sides of eq 7a by xb and taking the inverse of both sides of the resulting equation leads to (10) Aranovich, G. L.; Donohue, M. D. J. Colloidal Interface Sci. 1996, 180, 537.
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xb ) N0
1+
Be´ nard and Chahine
(1 - xb) exp(EA/kT) xb
(1 - xb) 2C (1 - exp(EA/kT)) xb xb
)
(1 - xb)
+ exp(EA/kT)
2C(1 - exp(EA/kT))
(7b)
This isotherm depends on only three parameters, EA, C, and Fmc, which can be found through a fit of the linear equation
Y ) aX + b
(8)
where the excess adsorbtion density N and the bulk density F are expressed in terms of the reduced variables Y and X defined as
Y)
F F and X ) N Fmc - F
a)
2C(1 - exp(EA/kBT))
and
b)
Fmc exp(EA/kT) 2C(1 - exp(EA/kT))
Table 1. Values of the Fit Parameters for the Ono-Kondo Excess Adsorption Isotherm of Methane on Activated Carbons and Zeolites
(9)
The fit parameters a and b are related to the physical parameters of the isotherm through the expressions
Fmc
Figure 1. Fit of the experimental excess adsorption isotherms of methane on the activated carbon CNS-201 to the Ono-Kondo theory.
(10)
The density at maximum capacity Fmc is first determined by optimizing the correlation coefficient of the linear fit of eq 8. The activation energy EA and the prefactor C can be found using eqs 10. IV. Adsorption of Methane on Activated Carbon In this section we present a detailed comparison between our own experimental results on the excess adsorption of methane on activated carbon and the Ono-Kondo theory presented in section III. We also compare the theory with other experimental work. The case of zeolites will be discussed in section V. IV.1. Adsorption Measurements. We carried out excess adsorption measurements in the 0-10 MPa pressure range and 243-333 K temperature range using a dielectric method described in detail in ref 11.11,12 The volumes of the adsorption cell (96 mL) and of the expansion cell (151 mL) were determined with an accuracy better than 1% by weighing techniques. Both cells were immersed in a circulation bath with a temperature control of (0.1 K. A piezoelectric digital manometer (HEISE 620, scale 0-2000 PSIA) with an accuracy of 0.1% at full scale was used for pressure measurements. Helium gas was used for the determination of the dead volume. Activated CNS-201 carbon from AC-carbon Canada was used as the adsorbent. It has a BET surface area of 1150 m2/g, a micropore volume of 0.45 mL/g, as determined from the DR method, and an average micropore size distribution of 10.8 Å with very few mesopores. Degassing of the adsorbent was done under high vacuum at 473 K for 3 h. The adsorbate was ultrahigh purity grade methane supplied by Matheson. The GERGref virial equation was (11) Bose, T. K.; Chahine, R.; Marchildon, L.; St-Arnaud, J.-M. Rev. Sci. Instrum. 1987, 58, 2279. (12) Chahine, R.; Bose, T. K. International Symposium on Measurement Properties and Utilisation of Natural Gas; John Libbey & Company Ltd: Paris, 1988; p 669. (13) Jaeschke, M.; et al. High accuracy compressibility factor calculation for natural gases and similar mixtures by use of a truncated virial equation. GERG Technical Monograph TM2, 1988.
isotherm parameter M E (K) EA (K) C (mmol/g) Fmc (mol/L)
activated carbons CNS-201 and CGG4
zeolite Linde 5A
excess adsorption isotherm
excess adsorption adsorption M M (xi - xb) N ) C∑i)1 xi N ) C∑i)1
2 50 -1332 9.47(1 - T/59K) 26.5
0 -1524 3.31 40
0 -1524 2.30 30
used to derive gas densities from PVT data. Its expected accuracy for methane13 is better than 0.06%. The methane adsorption isotherms were measured up to 16 MPa over the temperature range 243-333 K in steps of 10 K (see ref 12). IV.2. Fit to the Ono-Kondo Equations. We have fitted the experimental excess adsorption isotherms of methane on CNS-201 to the Ono-Kondo self-consistent equations. The results are shown in Figure 1. The adsorption isotherms are plotted as a function of the bulk methane density at the corresponding pressure and temperature. The bulk density was determined using the GERGref virial equation of state. For the purpose of clarity, only the methane adsorption isotherms at extreme and intermediate temperatures are shown. The deviations are well within the experimental uncertainties. The fit parameters are given in the second column of Table 1. The number of micropore layers M was set to 2 because the pore size distribution in CNS-201, as determined by density functional theory, shows that 90% of the slits of the carbon skeleton have a width smaller than 13 Å. The spacing between adjacent graphite planes (10.98 Å centerto-center between the planes8) in CNS-201 is thus very narrow and close to the optimum value of two methane molecules. The activation energy obtained by fitting the OnoKondo equations to the experimental isotherms was EA ) -1332 K. This is in excellent agreement with the value EA ) -1325 K found by Blu¨mel et al.14 by comparing the adsorption isotherms of krypton with those of methane. It is also in good agreement with the well depth 1s ) -1208 K of the Lennard-Jones 10-4 potential describing (14) Blu¨mel, S.; Ko¨ster, F.; Findenegg, G. H. J. Chem. Soc., Faraday Trans. 2 1982, 78, 1753. (15) Dubinin, M. M.; Neimark, A. V.; Serpinsky, V. V. Carbon 1993, 31, 1015. (16) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Unpublished. (17) Menon, P. G. Chem. Rev. 1968, 68, 277. (18) Hill, T. L. An Introduction to statistical thermodynamics; Dover: New York, 1986. (19) Plischke, M.; Bergersen, B. Equilibrium Statistical Physics; Prentice-Hall: Englewood Cliffs, NJ, 1989.
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Figure 2. Temperature dependence of the isotherm prefactor C obtained from the fits to the Ono-Kondo isotherms of Figure 1 over a temperature range of 90 K.
Figure 3. Comparison between the experimental excess adsorption isotherms of methane on Columbia Grade G (CGG) from Payne et al.5 (filled triangles) and CNS-201 (open circles) and the Ono-Kondo equations (lines).
the interactions between the graphite layers and the methane molecules.8 The prefactor C of the adsorption isotherm was found to vary linearly with temperature (Figure 2). A fit yields the following expression for C:
(
C ) 9.47 1 -
)
T mmol/g with T0 ) 591 K T0
(11)
The variation of C with T we have found is very similar to the variation of the maximum capacity volume Vs found by Payne5 et al. in their fit of adsorption experiments of methane on Columbia Grade G (CGG) on their Eyring adsorption isotherm. In their case T0 ) 583 K. The largest deviation between the linear fit (eq 11) and the parameters determined from the experiments is 1.76% at T ) 333 K. This temperature dependence of the prefactor C is not predicted by the Ono-Kondo approximation. A temperature dependence such as this has been predicted by Dubinin et al. for subcritical systems.15 Their calculation yields a small but measurable temperature-dependent correction due to compressibility effects (which are directly related to density fluctuations), even if the adsorbed phase is more or less in a liquid state. Such effects can be expected to be greater for adsorbates in the above critical temperature range, because of the greater compressibility. We have found that the best fit of experimental adsorption isotherms of methane on activated carbon over a 90 K temperature range occurs for a positive value of E (E ) 50 K). Nul or negative values of E could not yield good and consistent fits over the range of temperature for both the low- and high-density regions of the isotherms. Positive values of the parameter E influence mostly the high-density region of the isotherm close to the maximum and have little effect at low values of the bulk densities. As discussed in the next section, positive values of E are not surprising because of the very strong Lennard-Jones repulsion between adsorbed molecules on nearestneighbor graphite adsorption sites. Furthermore the density at maximum capacity Fmc is somewhat above the liquid density of the adsorbate. This also suggests that the repulsive part of the adsorbate-adsorbate interactions is the dominant contribution to the average intermolecular interaction (parametrized by E) at high densities. The whole temperature range of isotherms could be fitted with a constant density at maximum capacity Fmc ) 26.5 mol/L. This value is slightly above the liquid density of methane. The very high pressure experiments of Malbrunot et al.16 show that the excess adsorption isotherm of methane goes through a maximum followed by a steady, almost linear decrease. The rate of decrease slows somewhat before the bulk density reaches the density at maximum capacity Fmc ) 26.5 mol/L, and a minimum is observed close to 30 mol/L. A linear ex-
Figure 4. Excess adsorption as a function of temperature and density for methane on activated carbon, using the Ono-Kondo equations and the parameters from Table 1.
trapolation of their data leads to a value of Fmc very close to the value we obtained. The results of the fits show little or no dependence of Fmc on temperature over a range of 90 K. The same behavior can be seen from the excess adsorption experiments of nitrogen on activated coal and alumina of Menon et al.,17 which suggest that the extrapolated value of Fmc should vary little with temperature. Experimental points at higher adsorbate bulk densities would be required to observe a significant deviation from the constant value we obtained. The fit parameters determined from CNS-201 (second column of Table 1) are also in excellent agreement with the data of Payne et al.5 Their measurement of the experimental adsorption isotherms of methane was done on a Columbia G grade (CGG) activated carbon very similar to CNS-201. The fits to the experimental excess adsorption isotherms on both carbons at 283 and 323 K are shown in Figure 3. Agreement is excellent overall, especially for values of the bulk densities up to the maximum of the adsorption isotherm. For higher densities, a weak but systematic deviation can be observed. The deviation can be accounted for by allowing a weak temperature dependence of the density at maximum capacity (Fmc ) 25 mol/L for T ) 283 K and Fmc ) 23.5 mol/L for T ) 323 K). A three-dimensional plot of the Ono-Kondo excess adsorption surface as a function of density (up to 6 mol/L) and temperature (from 243 to 333 K) is shown in Figure 4, using the parameters determined by fitting the experimental data on CNS-201 (second column of table 1). The M ) 2 Ono-Kondo excess adsorption surface for the whole density and temperature range is characterized by a restricted set of five experimentally determined parameters, including the characteristic temperature scale and the amplitude of the linear fit to the temperature dependence of the prefactor C.
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Be´ nard and Chahine
adsorption sites of the graphite plane are occupied by one methane molecule:
Fhex )
Figure 5. Ideal graphite planes. The carbon atoms of the graphite planes form hexagonal bidimensional layers which can be divided into three distinct hexagonal sublattices. The minimum of the Lennard-Jones interaction between two methane molecules is at 4.28 Å, very slightly beyond the distance between second-nearest-neighbor sites on the graphite planes.
IV.3. Physical Interpretation. The interaction between CH4 molecules and the carbon atoms of the adsorbate leads to the existence of adsorption sites at the center of the hexagonal cells of graphite. In the OnoKondo approach, the Lennard-Jones (LJ) interaction between adsorbed CH4 molecules is modeled by an Isinglike interaction representing the effects of the in-plane and interplane interactions. Figure 5 shows the graphite planes on which the adsorption process occurs. The minimum of the LJ potential lies at 4.285 Å for methane,18 and the onset of its repulsion core is at 3.82 Å. Because the distance between nearest-neighboring adsorption sites on activated carbon is 2.46 Å, the interaction between adsorbed molecules will be strongly repulsive, as discussed by Plischke19 for adsorbed helium atoms on activated carbon. The effects of in-plane intermolecular repulsion between nearest-neighbor adsorbed CH4 molecules should thus be even greater because the minimum of the LJ potential between methane molecules is located at a distance slightly larger than the position of the secondnearest-neighbor on the hexagonal graphite plane. In fact the repulsion deduced from the LJ potential between nearest-neighbor CH4 molecules is 80 times stronger than the adsorbate-adsorbent interaction parameter EA found in Table 1. This suggests that adsorbed methane molecules will tend to avoid configurations with nearestneighbor adsorbed molecules. In the case of helium, this situation leads to modeling of the adsorption of helium atoms on graphite onto the frustrated antiferromagnetic Ising model.19 The mapping to an antiferromagnetic Ising model stems from the repulsive interaction between molecules adsorbed on nearest-neighboring sites. The model is said to be frustrated because it is impossible to separate the hexagonal lattice of graphite into two sublattices. This leads to a frustrated ground state, where some pairs of spins in the ground state remain ferromagnetically aligned even at T ) 0. Adsorption will first proceed by occupying all available second-nearest-neighbor sites on the graphite lattice. Note that adsorbed methane molecules are only expected to crystallize on the surface of the carbon layers at temperatures below 55 K.20 Therefore no long-range order for the methane molecules at the temperatures we investigated can be expected to be observed. Note that longrange order is not established in the two-dimensional antiferromagnetic Ising model on an hexagonal lattice at finite temperatures because of the strong degeneracy of the minimum energy state (i.e. the presence of three possible sublattices on which adsorption can occur).19 The density at maximum capacity of the graphite layer for methane corresponds to the density when all the (20) Kim, H.-Y.; Steele, W. A. Phys. Rev. B 1992, 45, 6226.
2 ) 83.24 mol/L 3a2dCH4 cot(30)NA
(12)
where a ) 1.418 Å is the distance between two carbon atoms on the graphite lattice, dCH4 ) 3.82 Å is the diameter of a methane molecule, and NA is Avogadro’s number. Figure 5 shows that the hexagonal lattice can be divided into three hexagonal sublattices. On each of these sublattices, the adsorption sites correspond to the secondnearest neighbor sites of the parent graphite lattice. The density at maximum capacity of each of the three sublattices of the graphite plane is therefore one-third of the total, yielding the estimate Fmc ) 27.5 mol/L, in excellent agreement with the value Fmc ) 26.5 mol/L we obtained from the fits. This value is also very close to the liquid density of methane. Surprisingly, in a discrete lattice interpretation of the results, the fits are consistent with a picture where the Ono-Kondo equations primarily describe sublattice adsorption at a local level because of very strong repulsive interactions between molecules on nearest-neighbor sites of the graphite plane. When the bulk density of the gas becomes high enough that the LJ interaction between molecules in the bulk phase becomes repulsive and comparable to the interaction between adsorbed molecules, adsorption on nearest-neighbor sites of the graphite layer becomes energetically favorable and the excess adsorption isotherm will start to rise again, as observed in the high-pressure data of Menon17 and Malbrunot.16 This behavior does not show up in the OnoKondo adsorption isotherm. This approach predicts instead that the excess adsorption isotherm vanishes as xb ) Fb/Fmc approaches 1. The failure of the Ono-Kondo approach is due to the fact that it cannot describe the adsorption process on a scale smaller than a sublattice (which is what occurs when the density of the bulk exceeds the density at maximum capacity). For such strong pressures, adsorption will depend on the detailed behavior of the intermolecular interactions. V. Adsorption Isotherm of Methane on Zeolite Adsorbents We have compared the excess adsorption isotherm of methane on the zeolite Linde 5A at 297 K measured at our laboratory with the interactionless excess adsorption isotherm (eq 7). Figure 6 shows the result of a fit of the excess adsorption isotherm using the interactionless isotherm proposed in section III (eq 7). Figure 6a shows that the experimental isotherm can be plotted as a line when expressed in terms of the reduced variables defined in eqs 9 and 10, when the density at maximum capacity is set to 40 mol/L. The density at maximum capacity was obtained by maximizing the correlation coefficient of a first-order linear regression of the reduced data set. The parameters obtained from the linear regression analysis of the experimental data are shown in the third column of Table 1. The full isotherm is shown in Figure 6b (solid line). The quality of the fit to the interactionless isotherm is excellent over the whole bulk density range, suggesting that, at 300 K, the interaction between adsorbed CH4 molecules on zeolites can be neglected. These results show that the room temperature excess adsorption isotherms are well described by the interactionless isotherm of eq 7 for zeolites. The parameters relevant to the adsorption process can thus be determined through first-order linear regression and optimization of the correlation coefficient. The interactionless isotherm of eq 7 should therefore be useful for adsorption systems where the adsorbate/ adsorbate interactions can be neglected, such as zeolites
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Langmuir, Vol. 13, No. 4, 1997 813
Figure 6. (a) Fit of the interactionless excess adsorption isotherm to the experimental adsorption isotherm of methane on the Linde 5A zeolite at 297 K. The experimental data are expressed in terms of reduced variables. (b) Comparison between fits of the experimental data using the adsorption isotherm without the high-pressure corrections and the excess adsorption isotherm. The fit parameters are shown in Table 1.
at temperatures well above the critical temperature of the adsorbate. For tubular zeolites, low temperature deviations from the interactionless isotherm could be included in a simple way by using the one-dimensional Ising model, which can be solved exactly as a function of the chemical potential. In order to show the importance of the corrections due to the high-pressure behavior, we have compared the excess adsorption results with a fit to the adsorption isotherm, neglecting the high-pressure corrections. In this case, the adsorption isotherm N’ is given by the expression N
xi ∑ i)1
N′ ) C
(13)
which leads to the simple expression
xb exp(-EA/kT) N0′ ) 2C 1 - xb + xb exp(-EA/kT)
(14)
The fit of the experimental data to eq 14 is illustrated in Figure 6b as a dashed line. The parameters are given in the third column of Table 1. They were obtained by a linear fit of the inverse of the measured adsorbed density as a function of the inverse density. Note that the three parameters in eq 14 are highly correlated for a given isotherm, so the temperature dependence of the adsorption isotherm must also be studied in order to fully determine the value of the three parameters. Thus the activation energy EA obtained from the excess adsorption isotherm of eq 7 was used. Clearly, Figure 6a shows that the adsorption isotherm N’ cannot properly fit the high-density region of the experimental data. Fitting the experimental data to eq 13 underestimates the density at maximum density Fmc by 25% and the prefactor C by 30%. Neglecting
the high-density behavior observed in excess adsorption can therefore lead to important systematic errors in determining the physical parameters characterizing the adsorption isotherm. VI. Conclusion We have presented fits of experimental data for the adsorption isotherms of methane on activated carbon covering the temperature range 243-333 K. Our experimental data on CNS-201, as well as the data provided by Payne et al.5 for a similar activated carbon, have been fitted using the Ono-Kondo equations adapted for microporous adsorbents. The agreement between the experimentally measured dependence on the adsorbate bulk density of the adsorption isotherms and the results of Ono-Kondo theory is excellent. The model easily accounts for the high-pressure maximum observed in the excess adsorption isotherms. The determined value of the adsorption energy EA is in good agreement with the results of Blu¨mel et al.,14 as well as the value of the minimum of the 10-4 Lennard-Jones interaction between CH4 molecules and activated carbon. Our data could be fitted with a single value of Fmc for the whole temperature range. The density at maximum capacity was found to be in good agreement with the very high pressure results of Malbrunot et al. The density Fmc is very close to the density at maximum capacity of methane on one of the three sublattices of the hexagonal graphite planes. The Ono-Kondo approximation to the Ising model provides an excellent fit of the density dependence of the experimental adsorption isotherms of methane on activated carbon over a wide range of temperatures, provided that the adsorption coefficient C is allowed to vary linearly with temperature. The variation is similar to the one observed by Payne et al. for the adsorbed volume. Such a result is not surprising, since the Ono-Kondo solution of the frustrated antiferromagnetic Ising model neglects important density correlations. Our comparison between theory and experiments suggests that this approach can be interpreted mainly in terms of sublattice adsorption on the hexagonal graphite planes. This occurs because of the strong repulsive Lennard-Jones interactions between adsorbed CH4 molecules on nearest-neighbor sites of the graphite plane. The high-density experiments of Malbrunot16 and of Menon17 have shown that the excess adsorption curve does not vanish when the asdorbate bulk density reaches the density at maximum capacity. As soon as the bulk density of the adsorbate is such that the LJ interaction in the bulk becomes repulsive and comparable to the one experienced between two nearest-neighbor adsorbed molecules on a graphite plane, adsorption on nearestneighboring sites can begin. This gives rise to the minimum observed in the excess adsorption isotherm at very high adsorbate bulk densities, followed by an increase. For adsorption of methane on activated carbon, a consistent fit of the low- and high-density parts of the adsorption isotherm requires that intermolecular interactions are included. On the other hand, the high-temperature adsorption isotherms of zeolites can be fitted very well by an interactionless isotherm. They can thus be characterized by only three parameters for all adsorbate bulk densities investigated. This suggests that intermolecular interactions between adsorbed CH4 molecules in Linde 5A are weak. Acknowledgment. We wish to thank Daniel Cossement and P. Malbrunot for useful discussions. This research was financed by the government of Que´bec through the Synergie program of the Fonds de de´veloppement technologique. LA960843X
