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Langmuir 2001, 17, 1950-1955
Determination of the Adsorption Isotherms of Hydrogen on Activated Carbons above the Critical Temperature of the Adsorbate over Wide Temperature and Pressure Ranges P. Be´nard* and R. Chahine Institut de recherche sur l’hydroge` ne, Universite´ du Que´ bec a` Trois-Rivieres, 3351 Boulevard des Forges C.P. 500, Trois-Rivie` res, Que´ bec, Canada G9A 5H7 Received October 2, 2000. In Final Form: December 20, 2000 We present a detailed comparison between experimental adsorption measurements and the excess adsorption predicted by the Ono-Kondo lattice model. The study was performed for high-pressure adsorption of hydrogen above the critical point on microporous adsorbents. The experimental adsorption isotherms of hydrogen on the activated carbon AX-21 as well as others are compared with those of theory over the temperature range of 77-273 K and for pressures of up to 6 MPa.
1. Introduction The recent advent of activated single-walled carbon nanotubes (SWNT) has put adsorption in the forefront of research in hydrogen (H2) storage technology.1 Physical adsorption offers a promising avenue for substantially lowering the storage pressure of compressed gas fuels such as natural gas and hydrogen.2 Adsorption is also widely used for hydrogen purification using pressure-swing adsorption and in cryopumps. For storage of hydrogen on activated carbons, a low operating temperature is required to maximize the gain over compression.3 Experiments show that a system’s storage density of 26 kg/m3 and 5.8% (by weight) at 35 bar and 77 K could be achieved, compared to 14 kg/m3 and 3.1% (weight) for compressed H2 at 250 bar and 293 K. The design of adsorption-based processes thus requires the characterization of the adsorption isotherms over a wide range of pressures and temperatures above the critical point of the adsorbate. At present, there are few experimental data and no specific model available to describe hydrogen adsorption over a wide range of pressures and temperatures. Modeling is of key importance because predicting the adsorption properties over a wide range of temperatures and pressures would reduce the number of lengthy experiments required for performance evaluation. Once a suitable model is available, only a small number of experiments are required to parametrize it. All other isotherms within the range of validity of the model can be predicted. Simulations can then be performed to determine the best operating parameters and help select appropriate adsorbents by optimizing such variables as the operating temperature and pressure ranges, the density of the adsorbent, and its specific surface. A better understanding of hydrogen adsorption on activated carbon may also contribute to our comprehension of hydrogen adsorption on nanotubes, because of the similarities between the systems. The most basic approach to adsorption is the Langmuir model, which depends only on the adsorbate/adsorbent (1) Dillon, A. C.; et al. Nature 1997, 386, 377. (2) Chahine, R.; Bose, T. K. In Hydrogen Energy Progress XI; Veziroglu, T. N., et al., Eds.; Scho¨n & Wetzel GmbH: Frankfurt am Main, Germany, 1996; p 1259. (3) Chahine, R.; Be´nard, P. Advances in Cryogenic Engineering; Plenum Press: New York, 1998; Vol. 43B, p 1257.
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 covering but makes no reference to the structural properties of the adsorbent. The BrunauerEmmett-Teller (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 DubininRadushkevitch model6 and its variants, which can offer an estimate of the structural and energetic parameters of the adsorption process. The Dubinin-Astakhov approach requires the introduction of a saturation pressure, which becomes ill-defined above the critical temperature of the adsorbent. The Dubinin approach is still used in the literature along with various modifications, but the physical interpretation of the effective saturation pressure is purely empirical. 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. Because the experimentally accessible isotherms are obtained through excess adsorption measurements, the experimental isotherm will exhibit a maximum when the absolute adsorbed density approaches saturation and the bulk density starts to rise faster as a function of pressure.5 The isotherm then decreases as a function of pressure until the bulk density reaches the maximum filling density of the pores. A proper model for high-pressure adsorption (4) Dubinin, M. M.; Radushkevich, L. V. Proc. Acad. Sci. USSR 1947, 55, 331. (5) Menon, P. G. Chem. Rev. 1968, 68, 277. (6) Gregg, S. G.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982.
10.1021/la001381x CCC: $20.00 © 2001 American Chemical Society Published on Web 02/13/2001
Adsorption Isotherms of Hydrogen on Activated Carbons
Figure 1. Adsorption isotherms of hydrogen on activated carbon at 77 K and covering the range 93-273 K separated by 20 K intervals.
above the critical temperature of the adsorbate must therefore include these corrections. High-pressure excess adsorption isotherms have been studied using density functional theory. In such an approach, the excess adsorption isotherms can be obtain through a selfconsistent calculation of the adsorbed density. The objectives of this paper are to extract the physical adsorption parameters in the supercritical regime over large temperature (77-273 K) and pressure (0-6 MPa) ranges from hydrogen adsorption experiments performed at our institute and to compare them to a parametrization obtained from Ono-Kondo isotherms, extended to supercritical isotherms by Aranovich et al., which can represent accurately the adsorption isotherms of hydrogen over wide temperature (77-273 K) and pressure (0-6 MPa) ranges, particularly in the high-pressure range relevant to storage applications. This paper is divided as follows. In the first section, we present experimental results of hydrogen adsorption on activated carbon over the temperature range 77-273 K and pressure range 0-6 MPa. We also discuss the isosteric heat of adsorption and the energy of adsorption, which are important parameters in system performance modeling, which we determine from the adsorption isosteres and the second virial coefficient, respectively. In the second section, we study the Langmuir model, which is appropriate for weakly adsorbing systems, and discuss its limitations in describing the isotherms over the temperature and pressure ranges of interest. We then fit the experimental data to the Ono-Kondo isotherm. 2. Experimental Results The adsorption isotherms were obtained using a standard volumetric approach described elsewhere. The adsorbent used in the experiments was the superactivated carbon AX-21 with a specific surface area of 2800 m2/g and a bulk density of 0.3 g/cm3. The pore-volume distribution was determined using density functional theory and showed that it was sharply peaked around 12.5 Å with a half-width of about 1 Å. The adsorbate used in the experiments was 99.99% pure hydrogen. About 20 g of the adsorbent was used to determine the adsorption isotherm. The temperature was lowered from 273 to 93 K by 20 K intervals. Measurements were also made at 77 K (see Figure 1). The temperature fluctuations were maintained below 0.5 K. The pressure was varied from the maximum seen on the 77 K isotherm. This feature is characteristic of excess adsorption isotherms,5 which measures the amount of gas adsorbed in excess of the bulk gas which would normally occupy the dead volume of the adsorbent at that temperature and pressure. The maximum occurs when the bulk density increases at a higher rate than the adsorbed density. To extract with minimal model adsorption energy parameters from the experimental data we can consider the isosteric heat of adsorption and the second virial coefficient for excess adsorption isotherms.
Langmuir, Vol. 17, No. 6, 2001 1951
Figure 2. Excess adsorption isosteres for n ) 0.5, 1.0, 1.75, 3.0, 5.0, 8.0, 10.0, and 12.0 mmol/g. The isosteric pressure increases with increasing n.
Figure 3. Isosteric heat of adsorption as a function of the adsorbed density.
Figure 4. Pressure parameter p0 as a function of the adsorbed density. 2.1. Adsorption Isosteres. The adsorption isosteres are curves of the pressure as a function of temperature at a constant value of the adsorbed density n. They are shown in Figure 2 for several values of the density. The isosteric heat of adsorption q can be obtained6 from the slope of isosteres in Figure 2. A temperature-independent q leads to the following isostere:
ln(p/p0) ) q/RT
(1)
where p0 is a characteristic pressure. Therefore, the linear dependence observed in Figure 2 indicates a weakly temperaturedependent isosteric heat of adsorption. Both q and p0 are functions of the adsorbed density and can be deduced from a linear regression of the data (see Figures 3 and 4, respectively). Figure 3 shows the isosteric heat of adsorption as a function of the adsorbed density. It decreases from 6.4 to 5.0 kJ/mol over the range 0-12 mmol/g. The dependence of q on the adsorbed density is usually discussed in terms of coverage. The coverage is defined as the ratio of the adsorbed density n to the adsorption
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Langmuir, Vol. 17, No. 6, 2001
Be´ nard and Chahine BAS ) Aσ
∫ [e ∞
(-1/kT)V(y)
0
- 1] dy
and V(y) ) 2s′
[52(1y) - (1y) ] (4) 10
4
which lead to a value of 4.66 kJ/mol (560 K) for the surface-gas interaction parameter. A better representation of a nanoporous carbon such as AX-21 is obtained by assuming that the pores are formed from two adjacent parallel graphite planes.9 The overall surface-gas interaction is then obtained from superposition of two LennardJones planar potentials describing two parallel graphite planes separated by a distance which should be representative of the pore distribution. The second virial coefficient is then given by
Figure 5. Plot of the normalized virial coefficient BAS as a function of reduced inverse temperature. The open circles refer to hydrogen with s ) 5.540 kJ/mol and Rz0 ) 373 mm3/g. The solid circles refer to methane.
BAS )2 Aσ
∫
d*/2
0
[e(-1/kT)[V(y+d*/2)+V(y-d*/2)] - 1] dy
(5)
where V(y) is in (4) and capacity of a layer nm ) A/(amNA), where A is the specific surface area, am is a typical molecular area for hydrogen, and NA is Avogadro’s number. A surface area of 2800 m2/g and an effective molecular area of 13.1 Å2 (obtained from the liquid density) lead to nm ) 32.4 mmol/g. A maximum value of nm ) 49.5 mmol/g is found, assuming a hexagonal arrangement of the hydrogen molecules located at their Lennard-Jones minimum. Use of the first value of nm leads to a coverage of 0.36 for the maximum value of the adsorbed density n (12 mmol/g). The overall behavior of the isosteric heat of adsorption is similar to the one observed for nitrogen adsorption on carbon black6 for a similar range of coverage. 2.2. Virial Expansion. An important parameter in describing the adsorption isotherm is the energy of adsorption. This parameter is determined by the surface-gas interactions. Perhaps the most direct method to quantify the energy of adsorption and the other interaction parameters is through the virial coefficients. The virial series expansion of the excess adsorption isotherm is defined in the zero-pressure limit as7
n ) BAS(p/kT) + CAAS(p/kT)2 + DAAS(p/kT)3 + ...
(2)
The first nonzero coefficient (BAS) is fully determined by the interaction between a single molecule and the surface. A proper model for the adsorbent-adsorbate interactions can, therefore, allow a direct estimate of the energy of adsorption. Use of a molecule-adsorbent potential for Lennard-Jones molecules leads to the following expression for BAS:
BAS ) Rz0
∫ [exp[-x(23x3(y ∞
0
-9
)] ]
- y-3) - 1 dy
(3)
where x ) s/kT, s is the Lennard-Jones planar gas-solid interaction parameter, and R is a characteristic molecular adsorption area. The virial coefficient BAS can be obtained from the experiments by finding the intercept of a plot of n/p as a function of p. Figure 5 shows the ratio BAS/Rz0 as a function of s/kT as determined from the experimental data with adsorption data of CH4 on the activated carbon CNS-201. Fitting the data to (3) yields a Lennard-Jones surface-gas energy s of 5.54 kJ/ mol, which is close to the midrange of the values of the isosteric heat of adsorption determined above. Our result for s is close to the value of 6.0 kJ/mol reported by Steele et al.9 for hydrogen adsorption on a Saran charcoal of 2080 m2/g. We also performed a fit using a Lennard-Jones 10-4 potential, (7) Clark, A. Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (8) Pierotti, R. A.; Thomas, H. E. Physical Adsorption: The Interaction of Gases with Solids in Surface and Colloid Science; Matijevic, E., Ed.; John Wiley & Sons: New York, 1971; Vol. 4. (9) Matranga, K. R.; Myers, A. L.; Glandt, E. D. Chem. Eng. Sci. 1992, 47, 1569.
s′ ) πθσ2
(6)
If the standard Lennard-Jones intermolecular hydrogen-carbon parameter values of 30.5 K and 3.19 Å are used10 and a surface density of 0.38 atoms/Å2 is assumed for the planar density of carbon atoms,11 a good fit can be obtained using the interlayer distance and the specific surface as adjustable parameters. The best values we obtain were 8.1 Å for the interlayer distance and 2900 m2/Å for the specific surface. A good fit of the data with a potential using a single graphite plane with a 10-4 potential did not succeed if the values of the Lennard-Jones intermolecular hydrogen-carbon were used to determine the Lennard-Jones planar gas-surface interaction s′. As mentioned earlier, density functional theory analysis of AX-21 shows a strongly peaked pore distribution centered around a pore size of 12.5 Å, which within one to two molecular diameters of the fit value we obtained for the interlayer distance. However, because of the continuum approximation used during the derivation of the Lennard-Jones planar potential, the value we obtained for the interlayer distance is difficult to compare with structural parameters of the carbon on a scale of a few molecular diameters. On the other hand, the best fit value of the specific surface is very close to the one obtained from BET surface measurements. We also attempted a fit by setting s′ with the Lennard-Jones molecular values but could not obtain reasonable agreement with the experimental data.
3. Modeling of Adsorption Isotherms A practical model for the excess adsorption isotherms of microporous adsorbents must rely on a minimum number of parameters, which should have a clear interpretation in terms of the physical properties of the system. It must also predict the high-pressure maximum observed under supercritical conditions. Despite its shortcomings, the Langmuir model is relevant to weakly interacting, supercritical adsorbate because it provides a simple description of the filling of a monolayer. The Ono-Kondo equations,12 recently extended to supercritical adsorption on activated carbon,13 improve on the Langmuir approach by providing directly the excess adsorption isotherms and by including the interactions between neighboring adsorbate molecules. 3.1. Langmuir Model. Although the Langmuir model can be expected to describe with some accuracy the adsorption of hydrogen on activated carbon close to room (10) Rzepka, M.; Lamp, P.; de la Casa Lillo, M. A. J. Phys. Chem. B 1999, 102, 10894. (11) Stan, G.; Cole, M. W. Surf. Sci. 1998, 395, 280. (12) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids; Springer-Verlag: Berlin, 1960. (13) Aranovich, L.; Donohue, M. D. Carbon 1995, 33, 1369. Aranovitch, G. L. J. Colloid. Interface Sci. 1990, 141, 30.
Adsorption Isotherms of Hydrogen on Activated Carbons
Langmuir, Vol. 17, No. 6, 2001 1953
which is even lower, and a value of B0 equal to 3.876 × 10-4 kPa-1. It is interesting to point out that a fit of the Langmuir saturation density nm as a function of temperature yields the expression
nm ) 100/(1 + 0.035T) mmol/g
Figure 6. Adsorption isotherms of Figure 1 expressed as Langmuir plots. The arrow indicates the direction of increasing temperature.
for the temperature range 77-273 K. The maximum value nm ) 49.5 mmol/g would correspond to a temperature of 32.5 K, which is very close to the critical temperature of hydrogen, below which the Langmuir model fails and multilayer adsorption can occur. 3.2. Ono-Kondo Theory. The Ono-Kondo theory describes the density profile of interacting adsorbed molecules on a hexagonal lattice representing the graphite planes from the point of view of localized adsorption. In lattice approaches such as the one we will use 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 Ono-Kondo model could be adapted to describe other adsorbents with a similar carbon skeleton, such as graphite nanofibers and perhaps even carbon nanotubes. The model has been successfully applied to adsorption of methane on activated carbon over a wide range of temperatures and densities. 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
Figure 7. Langmuir plots of the three lowest temperature isotherms (77, 93, and 113 K).
temperature, it cannot predict the characteristic maximum of excess adsorption isotherms discussed earlier. The Langmuir isotherm is a monotonically increasing function of pressure, which saturates at a value of nm. It can be put into the following form:
p/n ) 1/Bnm + p/nm
B ) 1.32 × 10-5 exp(519.3/T)
(8)
from which an activation energy EA ) 4.32 kJ/mol is obtained. This value is close to the energy parameter determined from the second virial expansion coefficient with the 10-4 potential described in (4). Kinetic theory predicts the following form for B:
B ) B0 exp(EA/T)/xT
xi ) Fi/Fmc
(9)
A fit to this expression yields EA ) 3.43 kJ/mol (412.4 K),
(11)
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 N
N)C
(7)
The saturation density nm and the coefficient B can be obtained from a simple linear fit of the data to (4). Figure 6 shows that excellent fits of our data to the Langmuir isotherms are obtained at high temperatures (T > 133 K). Figure 7 shows the Langmuir plots for the three lowest temperatures with the best linear fits over the whole pressure range. Deviations from linearity can be seen. The change in slope is due to the presence of the excess adsorption maximum. The Langmuir model is valid at low pressures, where the slope is highest (and B is smaller). The change in slope leads to erroneous estimates of B and nm. A fit of the experimentally determined values of B with temperature yields
(10)
(xi - xb) ∑ i)1
(12)
The density xb is the bulk molar density of the adsorbate Fb divided by the density at maximum capacity Fcm. M is the maximum number of layers that can fit in a typical microporous slit of the activated carbon. The prefactor C in (12) takes into account the density of the active pores of the adsorbent and other structural properties of the adsorbent. Note that the model fails when xb reaches 1, i.e., when the bulk density of the adsorbate nears the saturation density of the adsorbent. The Ono-Kondo equations12-14 are a set of coupled selfconsistent nonlinear equations describing the density profile of successive layers of adsorbed molecules. The model has been recently extended to adsorption of gaseous adsorbates on activated carbon by Aranovich and Donohue and successfully applied to adsorption of methane on activated carbon over a wide range of temperatures and densities.14 The Ono-Kondo equations are
ln
(
)
xk(1 - xb)
xb(1 - xk)
+
z0E z2E - 2xk + (xk - xb) + (x kT kT k+1 xk-1) ) 0 (13)
subject to the boundary conditions (14) Be´nard, P.; Chahine, R. Langmuir 1997, 13, 808.
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Langmuir, Vol. 17, No. 6, 2001
x1 ) xN
Be´ nard and Chahine
(14)
and
ln
(
)
x1(1 - xb)
xb(1 - x1)
+
EA E (z x + z2x2 - z0xb) + ) 0 (15) kT l l kT
In (13) and (15), 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 the equations describes the interactions between adsorbate molecules, which are limited to nearest-neighbor sites of the lattice. This intermolecular interaction can be viewed as an average interaction energy between co-planar and interplanar neighboring adsorbate molecules. The on-site adsorption potential EA in the equation parametrizes the interaction between the adsorbate particles and the adsorbent surface. The number of interaction parameters of the adsorbateadsorbent system are kept to a minimum and limited to on-site and to nearest-neighbor interactions between adjacent cells. (13)-(15) form a set of nonlinear equations which must be solved self-consistently. A fit to the Ono-Kondo equations leads to the following energy parameters: EA ≈ 4 kJ/mol and E ≈ 0.4 kJ/mol. The gas-solid energy EA lies between the energy parameter obtained from the Langmuir isotherm (4.32 kJ/mol) and the one obtained from the second virial coefficient using a 10-4 potential (4.66 kJ/mol). The prefactor C of the isotherms was found to be temperature-dependent and could be simply fitted by a third-order polynomial over the temperature range 77-298 K (Figure 8):
C ) C0 + C1T + C2T2 + C3T3
(16)
with C0 ) 21.77 mmol/g, C1 ) -6.73 × 10-2 mmol/(g K), C2 ) 2.159 × 10-5 mmol/(g K2), and C3 ) 2.189 × 10-7 mmol/(g K3). The best value of the maximum density Fs we obtained was 65 mol/L (eq 11). Figure 9 shows our fit of the OnoKondo equations and the experimental data. An overall good agreement between theory and experiments was achieved, especially in the low-temperature region. We also compared the fit for the 2800 m2/g adsorbent to adsorption data of hydrogen on a pelletized carbon with 2000 m2/g with a density of 0.7 g/cm3 (see Figure 10). Comparing the adsorption data of the two carbons shows that their ratio is constant and equal to 1.47 over the whole pressure range. This value is within error bars of the ratio of the BET surface areas. The difference in the adsorption properties is fully accounted for by the changes in the BET surface area. This suggests that the OnoKondo parametrization we obtained can be applied to other carbons with similar porous structure once the prefactor is corrected by the ratio of the specific surfaces. 4. Conclusions We have presented experimental adsorption data for the temperature range 77-273 K and the pressure range 0-6 MPa. From the adsorption isostere, we found an isosteric heat of adsorption varying from 6.4 to 5 kJ/mol for adsorbed densities ranging from 0 to 12 mmol/g. Using a 10-4 planar potential and the expression for the second virial coefficient, we obtained an estimate of 4.66 kJ/mol for the adosrption energy. We also obtained a good fit to the second virial coefficient of a slit-pore potential by using the hydrogen-carbon molecular Lennard-Jones parameters to estimate the energy of adsorption and by using
Figure 8. Temperature dependence of the Ono-Kondo prefactor as a function of temperature. The points show the result of the fits for each isotherm. The line shows a third-order fit of the parameters.
Figure 9. Ono-Kondo fit of the adsorption isotherms (lines) to the experimental data (points). The adsorption density is shown as a function of bulk gas density.
Figure 10. Comparison of the low-temperature adsorption isotherms of hydrogen at 77 K for two different carbons (AX-21 powder and pellets).
an interlayer distance of 8.1 Å. A good estimate of the specific surface (2900 m2/g) was also obtained. We also attempted a fit to a single planar potential by setting s′ at the onset with the Lennard-Jones molecular values but could not obtain reasonable agreement with the experimental data. The experimental adsorption isotherms of hydrogen on activated carbon were compared with those of the Langmuir model and the Ono-Kondo approach. The Langmuir fit yielded an adsorption energy EA ) 4.32 kJ/mol. Good agreement was obtained between theory and experiments at high temperatures; however, the Langmuir model could not describe the low-temperature adsorption isotherms of interest to storage applications because of the presence of the excess adsorption maximum. We also obtained a good fit of the data from
Adsorption Isotherms of Hydrogen on Activated Carbons
the Ono-Kondo model over the whole temperature and pressure range, if the prefactor is allowed to be temperature-dependent. The same fit can be used to predict the isotherm of other microporous carbons, provided the change in specific areas is taken into account. Although it is more complex than the Langmuir model, the OnoKondo model is more useful in the pressure and temperature ranges of interest to storage applications and could
Langmuir, Vol. 17, No. 6, 2001 1955
be used as a basis to model other carbon-based structures such as carbon nanofibers. Acknowledgment. We thank E Ä ric Me´lanc¸ on, who performed some calculations for us for this paper. We acknowledge support from the Ministe`re des ressources naturelles du Que´bec and NSERC of Canada. LA001381X
