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Langmuir 2003, 19, 314-320
Modeling of High-Pressure Adsorption Using the Bender Equation of State Alexander M. Puziy,*,† Alexander Herbst,‡ Olga I. Poddubnaya,† Joachim Germanus,‡ and Peter Harting‡ Institute for Sorption and Problems of Endoecology, Naumov St. 13, Kiev, Ukraine, and Institut fu¨ r Nichtklassische Chemie an der Universita¨ t Leipzig, Permoserstr. 15, Leipzig, Germany Received June 12, 2002. In Final Form: November 5, 2002 A model describing high-pressure adsorption based on the simplified local density model and the Bender equation of state is proposed. The Bender equation of state was used for the description of both bulk and adsorbed fluid. The model is capable of characterizing adsorbent material in terms of structural parameters such as the skeletal density, the micropore volume, and the energy of adsorption. The model was successfully applied to high-pressure adsorption isotherms of nitrogen and methane up to 50 MPa at 40 °C on a series of synthetic carbons and at 25-70 °C on the commercial carbon Norit R1. The structural parameters of the investigated carbons obtained by the proposed method are in good agreement with those obtained by independent methods based on standard and advanced characterization from nitrogen adsorption at 77 K.
1. Introduction Physical adsorption at high pressures is of great interest in the fields of energy storage,1-5 gas separation,3,6,7 adsorbent regeneration,8 and supercritical fluid chromatography.9 Understanding the thermodynamics and structure of the gas-solid interface is essential to the design of adsorption-based processes.10 Description of the adsorption isotherms over a wide range of pressures requires an adsorption model based on a minimal set of parameters with clear physical interpretation. Such a model should be able not only to describe and predict adsorption but also to characterize the adsorbent in terms of structural characteristics. Theoretical approaches to understanding and predicting high-pressure supercritical adsorption range from simple empirical fits (Dubinin-Astakhov, Freundlich, Toth isotherms)11 to theoretically sound methods such as statistic thermodynamic theory and computer simulation.12 Computer simulations such as grand canonical ensemble Monte Carlo semiquantitatively predict the cusplike behavior near the critical point.13 However, such methods are computationally intensive. Simulations are difficult near * Corresponding author. E-mail:
[email protected]. Fax: (38-044) 452-9327. † Institute for Sorption and Problems of Endoecology. ‡ Institut fu ¨ r Nichtklassische Chemie an der Universita¨t Leipzig. (1) MacDonald, J. A. F.; Quinn, D. F. Carbon 1996, 34, 1103. (2) Malbrunot, P.; Vidal, D.; Vermesse, J. Appl. Therm. Eng. 1996, 16, 375. (3) Sircar, S.; Golden, T. C.; Rao, M. B. Carbon 1996, 34, 1. (4) MacDonald, J. A. F.; Quinn, D. F. Fuel 1998, 77, 61. (5) Cheng, H. M.; Yang, Q. H.; Liu, C. Carbon 2001, 39, 1447. (6) Yang, R. T. Gas separation by adsorption processes; Butterworth: London, 1987. (7) Ruthven, D. M.; Shamasuzzaman, F.; Knaebel, K. S. Pressure swing adsorption; VCH: New York, 1994. (8) Recasens, F.; McCoy, B. J.; Smith, J. M. AIChE J. 1989, 35, 951. (9) Parcher, J. F.; Strubinger, J. R. J. Chromatogr. 1989, 479, 251. (10) Salem, M. M. K.; Braeuer, P.; Szombathely, M. v.; Heuchel, M.; Harting, P.; Quitzsch, K.; Jaroniec, M. Langmuir 1998, 14, 3376. (11) Amankwah, K. A. G.; Schwarz, J. A. Carbon 1995, 33, 1313. (12) Peterson, B. K.; Gubbins, K. E.; Heffelfinger, G. S.; Marconi, U. M. B.; Swol, F. V. J. Chem. Phys. 1988, 88, 6487. (13) Van Megan, W.; Snook, I. K. Mol. Phys. 1982, 45, 6.
the critical point due to fluctuations and require a large number of molecules and consequently significant amounts of supercomputer time. On the other hand, traditional empirical and semiempirical methods that are computationally undemanding are unable to account for the wide variety of shapes of adsorption isotherms observed near the critical region. Another disadvantage is that semiempirical methods do not characterize adsorbents in terms of structural or energetic parameters, quantities that are needed to discriminate between adsorbents for practical purposes. Recently the simplified local density model14 (SLD) has been developed to correlate adsorption isotherms of a pure component confined in pore space. The model is based on the idea of the equality of chemical potential at any point of an equilibrium system. The SLD model with the van der Waals equation of state gives a qualitatively correct description of adsorption at intermediate pressures (up to 10 MPa), exhibiting the cusplike behavior for subcritical fluid, the isotherm crossover, and the correct temperature dependence. Further, the quality of adsorption modeling was greatly improved by the use of more accurate equations of state such as the Peng-Robinson15,16 and Elliot-Suresh-Donohue17 equations. However, our attempts to apply the SLD model with the Elliot-SureshDonohue equation of state to experimental data in a wide pressure range (up to 50 MPa) have led to unsatisfactory results even when the heterogeneity of the adsorbent was taken into account: the model underpredicts adsorption at lower pressures and overpredicts at higher pressures.18 The purpose of the present paper is the application of the SLD model with the Bender equation of state for (14) Rangarajan, B.; Lira, C. T.; Subramanian, R. AIChE J. 1995, 41, 838. (15) Chen, J. H.; Wong, D. S. H.; Tan, C. S.; Subramanian, R.; Lira, C. T.; Orth, M. Ind. Eng. Chem. Res. 1997, 36, 2808. (16) Subramanian, R.; Lira, C. T. In Fundamentals of Adsorption; LeVan, M. E., Ed.; Kluwer: Boston, 1996; p 873. (17) Soule, A. D.; Smith, C. A.; Lira, C. T. Adsorption modeling with the ESD equation of state. Langmuir, submitted February 2000. http:// www.egr.msu.edu/∼lira/esdpaper.pdf. (18) Puziy, A. M.; Harting, P.; Herbst, A.; Poddubnaya, O. I.; Germanus, J. Eurocarbon 2000, 9-13 July 2000, Berlin; pp 659-660.
10.1021/la026062f CCC: $25.00 © 2003 American Chemical Society Published on Web 12/12/2002
Modeling of High-Pressure Adsorption
description of nitrogen and methane adsorption over a wide range of pressures (up to 50 MPa) on carbons with varying porous structure and surface chemistry. The Bender equation of state is a modification of the BenedictWebb-Rubin equation19 and is one of the most sophisticated equations of state that gives a good description of thermodynamic properties of fluids over a wide range of temperatures and pressures including the vapor and the liquid state.20 About 50 pure fluid parameter sets of the Bender equation have been reported in the literature.21-27 2. Experimental Section 2.1. Adsorbents. The adsorbents used in present study were synthetic carbons that differ in porosity and surface chemistry. C Series. These carbons were obtained from styrene-divinylbenzene copolymers by carbonization up to 800 °C in an argon atmosphere followed by steam activation at the same temperature for different times. The last index in the carbon abbreviation means activation time in hours. SP Series. These carbons were obtained from the same styrene-divinylbenzene copolymers. Before carbonization, the copolymer was impregnated with H3PO4. The impregnation ratio was 0.98. Further carbonization and activation procedures were the same as in the C series. SCS Series. These carbons were obtained from the same styrene-divinylbenzene copolymer by carbonization and activation as the C series. The sample was then divided in three parts. One part was left unchanged. This sample is abbreviated as SCS3. The second part was oxidized with 20% HNO3 for 5 h, followed by washing with 1-3% NH3 and then with hot water in a Soxhlet reactor until colorless wash water was obtained. The sample was then rinsed with 1-3% HCl and then with water until neutral pH was reached. This sample is abbreviated as SCS-3-Ox. The third part was treated with CCl4 at 400 °C for 30 min. This sample is abbreviated as SCS-3-CCl4. The commercially available carbon Norit R1 (Norit Deutschland GmbH, Du¨sseldorf, Germany) was also used for highpressure adsorption experiments. 2.2. Characterization. The porous structure of the adsorbents was characterized by low-temperature nitrogen adsorption. The nitrogen adsorption isotherms were measured at 77 K using an ASAP 2000 apparatus (Micromeritics, Norcross, GA). The nitrogen adsorption isotherms were analyzed by conventional methods: the Brunauer-Emmett-Teller (BET) method, the Rs-method, and the two-term Dubinin-Radushkevich equation. In addition to conventional methods, the carbons were characterized by pore size distribution (PSD) obtained using the recently proposed BET-Kelvin method.28,29 The PSD was obtained by solving the adsorption integral equation using the CONTIN method30 and the BET-Kelvin equation as the local isotherm. The True (pycnometric) density of the adsorbents was measured by an He pycnometer AccuPyc 1330 (Micromeritics, Norcross, GA). (19) Benedict, M.; Webb, G. B.; Rubin, L. C. J. Chem. Phys. 1940, 8, 334. (20) Bender, E. Der Berechnung von Phasengleichgewichten mit einer thermishen Zustandsgleihung - dargestellt an der reinen Fluiden Argon, Stickstoff, Sauerstoff und an ihren Gemischen. Habilitationsschrift, Ruhr-Universita¨t Bochum, Bochum, Germany, 1971. (21) Bender, E. Ka¨ ltetechnik - Klimatisierung 1971, 23, 258. (22) Bender, E. Cryogenics 1975, 15, 667. (23) Bender, E. VDI-Forschungsh. 1981, 609, 15. (24) Bu¨hner, K.; Maurer, G.; Bender, E. Cryogenics 1981, 21, 157. (25) Polt, A. W. Zur Beschreibung der thermodynamischen Eigenschaften reiner Fluide mit “Erweiterten BWR-Gleihungen”. Ph.D. Dissertation, Universita¨t Kaiserslautern, Kaiserslautern, Germany, 1987. (26) Platzer, B.; Polt, A.; Maurer, G. Thermophysical Properties of Refrigerants; Springer-Verlag: Berlin, 1990. (27) Polt, A. W.; Maurer, G. Fluid Phase Equilib. 1992, 73, 27. (28) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608. (29) Puziy, A. M.; Poddubnaya, O. I.; Martı´nez-Alonso, A.; Sua´rezGarcı´a, F.; Tasco´n, J. M. D. Appl. Surf. Sci. 2002, 200, 196. (30) Puziy, A. M.; Matynia, T.; Gawdzik, B.; Poddubnaya, O. I. Langmuir 1999, 15, 6016.
Langmuir, Vol. 19, No. 2, 2003 315 The surface chemistry of carbons was estimated from cation exchange capacity (CEC) measurements. A modified method based on Boehm’s31 was used to measure the cation exchange capacity of carbons. A weighed amount of adsorbent (0.1 ( 0.0001 g) was placed into an Erlenmeyer flask. A volume of 20 mL of 0.1 M NaOH solution was added. To attain equilibrium, the flasks were shaken for 24 h. After equilibration, the NaOH concentration was measured by titration with HCl. The consumed quantity of NaOH was converted to CEC and expressed in mmol/g. 2.3. High-Pressure Adsorption. High-pressure adsorption measurements were performed on a magnetic suspension balance (Rubotherm, Bochum, Germany) that can be operated up to 50 MPa. The balance has the advantage of noncontact weighing of samples. The unit consists of a conventional high-pressure stainless steel sample cell, which is connected to a gas reservoir via an air-driven gas booster (Haskel, Burbank, CA). Typically, a sample basket was loaded with 1-5 g of dried (110 °C, 2 h) carbon and the system was evacuated overnight at 2.6 × 10-3 Torr. This procedure ensured removal of preadsorbed gases. Equilibrium weights were achieved in 10-30 min. The equilibrium pressure was recorded by a pressure transducer (model 4091B, Kistler, Wintertur, Switzerland) and a digital pressure indicator (Rubotherm, Bochum, Germany). The weight measurements were carried out by microbalances (MettlerToledo, Geifensee, Switzerland). Each isotherm was comprised of 20 data points up to 50 MPa. Selected pressures were tested for reversibility of adsorption. In all cases the isotherms were reversible, indicating that adsorption equilibrium was achieved. Selected replicate runs were also carried out; the reproducibility of the adsorption isotherms was within 1-2%. The gases used in high-pressure adsorption experiments were ultrahigh purity grade (99.999%) nitrogen and methane supplied by Air Liquid. 2.4. SLD Adsorption Model. The excess adsorption is the quantity obtained directly from the experiment. According to the Gibbs definition, the excess adsorption is determined by the difference between the densities of bulk and adsorbed fluids in the adsorbed phase:
Γ ) W(Fa - Fb)
(1)
where Γ is excess adsorption, Fa and Fb are the densities of adsorbed fluid and bulk fluid, respectively, and W is the volume of the adsorbed phase. The density of a nonideal bulk fluid is determined by temperature and pressure through equation of state Fb ) f(P,T). The density of adsorbed fluid can also be calculated provided the equation of state for the adsorbed fluid and the pressure in the adsorbed phase are known. In the present study, the Bender equation of state (EoS) is used for the calculation of both the bulk fluid and adsorbed fluid densities. An equation to account for the influence of adsorption potential on the density of adsorbed fluid can be deduced from the condition of thermodynamic equilibrium. At equilibrium, the chemical potential of any component is equal in the different adsorption phases. For the system with a spatial heterogeneous external field (adsorption potential), the chemical potential at equilibrium is
µa ) µb + U
(2)
where µb is the chemical potential of the bulk fluid, µa is that of a fluid in a field of adsorption potential, and U is the adsorption potential. The chemical potential of a nonideal bulk fluid is defined in terms of fugacity as
µb ) µ0 + RT ln
() fb
f0
(3)
where µ0 is the standard-state chemical potential, and fb and f0 are the fugacity of the bulk fluid and the standard state, respectively. The chemical potential of the adsorbed fluid is defined using the same standard state as for the bulk fluid: (31) Boehm, H. P. Carbon 1994, 32, 759.
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()
µf ) µ0 + RT ln
ff
0
f
Puziy et al.
From eqs 2-4, the relation between the fugacity of adsorbed fluid and the adsorption potential can be developed:
(
fa ) fb exp -
U RT
)
(5)
Thus in eq 1 the density of bulk fluid Fb is obtained from the Bender EoS using the bulk pressure (see eq A1 of the Appendix) whereas the density of adsorbed fluid is calculated from the same Bender EoS using the fugacity of adsorbed fluid from eq 5 (see eq A2 of the Appendix). Unknown parameters for the calculation of the excess adsorption from eq 1 are the volume of the adsorbed phase W and the adsorption potential U, which affects the density of the adsorbed fluid Fa through eq 5 and the Bender EoS. These parameters can be obtained by fitting to the experimental data. 2.5. Data Treatment. The weight recorded by the balance during an adsorption experiment is comprised of (i) the mass of sample holder, (ii) the mass of the sample, and (iii) the mass of adsorbed fluid, all corrected for buoyancy in bulk fluid:
(
) ( ) ( )
Fb Fb Fb w ) mw 1 + ms 1 + ma 1 Fw Fs Fa
(6)
where mw, ms, and ma are the mass of the sample holder, the mass of the sample, and the mass of the adsorbed fluid; and Fb, Fw, Fs, and Fa are the densities of bulk fluid, sample holder, sample, and adsorbed fluid, respectively. Buoyancy results from the displacement of the bulk fluid by the sample holder, the sample, and the adsorbed phase. Equation 6 may be rearranged as follows:
(
w - mw 1 -
) ( )
Fb Fb ma - ms 1 ) (F - Fb) Fw Fs Fa a
(7)
In eq 7, ma/Fa ) W is the volume of the adsorbed phase or volume of pores occupied by the adsorbed fluid and Fa - Fb is the difference between the densities of the adsorbed phase and the gas phase. Thus the left-hand side (LHS) term of eq 7 is equal to Gibbs’ excess adsorption (compare with eq 1):
(
w - mw 1 -
) ( )
Fb Fb - ms 1 ) W(Fa - Fb) ≡ Γ Fw Fs
Table 1. Physical Characteristics of Investigated Carbons
(4)
(8)
In the LHS of eq 8, the weight recorded by the balance w, the mass of sample holder mw, and the mass of sample ms are obtained from the experiment, while the density of the bulk fluid Fb is calculated from the Bender EoS (eq A1 of the Appendix). In principle, the density of the adsorbent Fs may be measured by independent methods (e.g., using a He pycnometer). However, the helium density differs from the density of the adsorbent, Fs, used in high-pressure adsorption experiments. The discrepancy stems from the difference in size of helium atoms and adsorbate molecules.32 The adsorption of the helium at room temperature during density measurements33 must also be taken into account, which will cause the density to increase. Instead of independent measurements of the density of the adsorbent, Fs, we assume it as a fitting parameter obtained from highpressure experiments. Thus the LHS of eq 8 is the experimentally measured Gibbs excess adsorption, Γexp, with an unknown parameter Fs, the density of the adsorbent, whereas the right-hand side (RHS) of eq 8 is equal to the calculated Gibbs excess adsorption, Γcalc (eq 1), with unknown parameters W, the volume of the adsorbed phase, and U, the adsorption potential, related to the density of the adsorbed phase Fa through eq 5 and the Bender EoS. Thus, the calculated excess adsorption, Γcalc, may be fitted to the experimental excess adsorption, Γexp, by adjusting the unknown (32) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148. (33) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. K. Langmuir 1997, 13, 539.
carbon
burnoff [%]a
packing density [cm3/g]
FHe [cm3/g]
CECb [mmol/g]
C-A2 C-A4 C-A6 SP-A2 SP-A3 SCS-3 SCS-3-Ox SCS-3-Cl4 Norit R1
27 46 65 57 72 50 NA NA NA
0.53 0.45 0.39 0.36 0.30 0.41 0.37 0.44 0.39
2.05 2.02 2.08 2.00 2.05 2.04 2.08 2.21 2.08
0.26 0.18 0.34 0.18 0.10 0.58 3.16 1.30 0.20
a
NA ) not available. b CEC ) cation exchange capacity.
parameters Fs, W, and U using a nonlinear least-squares fitting algorithm. The functional to be minimized is given by
(Γexp - Γcalc)2 ) min
(9)
3. Results and Discussion 3.1. Characterization of Adsorbents. Table 1 presents the physical characteristics of several series of synthetic carbons together with the commercial carbon Norit R1 used in the present study. The carbon C and SP series with progressively increased burnoff were chosen to investigate the impact of porous structure on adsorption properties at high pressures. The carbon SP series was prepared using the same conditions as for the C series but with carbonization in the presence of phosphoric acid. The carbon SCS series was chosen to reflect the influence of surface chemistry on adsorption behavior at high pressure. With the increase of burnoff, the packing density of the carbon C and SP series increases, indicating the development of porosity (Table 1). The helium density of the carbons slightly increases upon activation though the increase is small, practically at the level of experimental error. For the carbon SCS-3 series, a slight increase of helium density upon oxidation with nitric acid and a significant increase upon treatment with carbon tetrachloride were observed. It is apparent that such increase of helium density is due to incorporation of heavier chlorine atoms in the carbon structure. The surface chemistry of the carbons was estimated by cation exchange capacity. All investigated carbons are essentially basic, except oxidized carbon SCS-3-Ox and carbon treated with carbon tetrachloride SCS-3-CCl4 which show considerable consumption of base (see CEC, Table 1). The porous structure of the adsorbents has been characterized by low-temperature nitrogen adsorption (see Figures S1-S4 of the Supporting Information). All isotherms belong to type IV of the IUPAC classification.34,35 A significant part of all isotherms at low relative pressures is attributed to micropore filling. A steep rise at high relative pressures for synthetic carbons is characteristic of capillary condensation in mesopores. Table 2 lists various textural parameters of the investigated carbons calculated from nitrogen adsorption isotherms at 77 K using standard (BET, Rs, and two-term Dubinin-Radushkevich) and advanced (BET-Kelvin) methods. All samples show a developed microporous structure with high surface area. Activation of synthetic carbons (C and SP series) progressively increases the (34) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. (35) Rouquerol, J.; Avnir, D.; Fairbridge, C. W.; Everett, D. H.; Haynes, J. H.; Pernicone, N.; Ramsay, J. D. F.; Sing, K. S. W.; Unger, K. K. Pure Appl. Chem. 1994, 66, 1739.
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Table 2. Porous Structure Parameters of Investigated Carbons
carbon C-A2 C-A4 C-A6 SP-A2 SP-A3 SCS-3 SCS-3-Ox SCS-3-Cl4 Norit R1
Rs-method Dubinin-Radushkevich method BET-Kelvin method BET Sme Vmi W01 E01 E02 W02 V(5-10 Å) V(11-31 Å) V(
