pubs.acs.org/Langmuir © 2010 American Chemical Society
Net Adsorption: A Thermodynamic Framework for Supercritical Gas Adsorption and Storage in Porous Solids Sasidhar Gumma† and Orhan Talu*,‡ †
Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India, and ‡ Department of Chemical and Biomedical Engineering, Cleveland State University, Cleveland, Ohio 44115, United States Received May 31, 2010. Revised Manuscript Received August 30, 2010
The thermodynamic treatment of adsorption phenomena is based on the Gibbs dividing surface, which is conceptually clear for a flat surface. On a flat surface, the primary extensive property is the area of the solid. As applications became more significant, necessitating microporous solids, early researchers such as McBain and Coolidge implemented the Gibbs definition by invoking a reference state for microporous solids. The mass of solid is used as a primary extensive property because surface area loses its physical meaning for microporous solids. A reference state is used to fix the hypothetical hyperdividing surface typically using helium as a probe molecule, resulting in the commonly used excess adsorption; experimentalists measure this reference state for each new sample. Molecular simulations, however, provide absolute adsorption. Theoreticians perform helium simulations to convert absolute to excess adsorption, mimicking experiments for comparison. This current structure of adsorption thermodynamics is rigorous (if the conditions for reference state helium measurements are completely disclosed) but laborious. In addition, many studies show that helium, or any other probe molecule for that matter, does adsorb, albeit to a small extent. We propose a novel thermodynamic framework, net adsorption, which completely circumvents the use of probe molecules to fix the reference state for each microporous sample. Using net adsorption, experimentalists calibrate their apparatus only once without any sample in the system. Theoreticians can directly calculate net adsorption; no additional simulations with a probe gas are necessary. Net adsorption also provides a direct indication of the density enhancement achieved (by using an adsorbent) over simple compression for gas (e.g., hydrogen) storage applications.
Introduction An adsorbed phase cannot exist autonomously; it exists only at the interface between two bulk phases, a solid phase and a fluid phase. Therefore, any thermodynamic property (amount adsorbed, enthalpy, entropy, etc.) of an adsorbed phase is measured as its value for the two-phase system relative to its value in some reference state. The “correct” reference state is not obvious. The use of different reference states by different laboratories makes comparisons complicated and time-consuming. The differences are especially significant at high pressure. International agreement on a standard reference state is needed so that experimental and/or theoretical results can be compared under all conditions.1 Adsorption in microporous solids is distinctly different from adsorption on flat surfaces,2 although the primary molecular interactions are the same. The structural differences between micropores and flat surfaces necessitate different reference states. The thermodynamic framework proposed here brings clarity to these differences, which in general have been overlooked in the past. Any application of adsorption requires that the adsorbed amount be quantified accurately and unambiguously. In this work, we present a rigorous thermodynamic framework for quantifying the adsorption equilibrium. The proposed framework has the following advantages for micropore adsorption: (a) The reference state is fixed unambiguously, a requirement for any rigorous thermodynamic framework. (b) The proposed reference state is a system property and thus needs to be measured only once for a
given experimental apparatus; other reference states in common use are sample properties and need to be measured for each new sample. (c) The proposed reference state offers a direct comparison between experiments and simulation without any need for extra information such as helium adsorption. (d) Measurement of adsorption by the proposed framework readily indicates the density enhancement achieved via adsorption as opposed to simple gas-phase compression. This enhancement factor is a critical indicator of efficiency in gas storage applications. The thermodynamic framework detailed here is based on a long-forgotten definition of an adsorption reference state. The choice of a reference state in thermodynamics is somewhat arbitrary as long as it is completely defined. A complete description and disclosure of the reference states are necessary in order to compare data from different sources. The choice of reference state is often dictated by the utility of the resulting formulations for specific applications. Our approach here is particularly suitable for describing the high-pressure adsorption of supercritical gases in microporous adsorbents. The most important contemporary application is gas storage by physical adsorption, particularly natural gas and hydrogen storage. Hydrogen storage is one of the major challenges for migration to a hydrogen economy.3,4 Any practically useful storage mechanism requires a significant enhancement of hydrogen density at nearambient temperatures. In recent years, adsorption has been widely explored as a potential candidate for hydrogen storage. The aim is
*Corresponding author. Tel: þ1 216 687 3539. Fax: þ1 216 687 9220. E-mail:
[email protected].
(3) Felderhoff, M.; Weidenthaler, C.; Helmolt, R.v.; Eberle, U. Phys. Chem. Chem. Phys. 2007, 9, 2643–2653. (4) Satyapal, S.; Petrovic, J.; Read, C.; Thomas, G.; Ordaza, G. Catal. Today 2007, 120, 246–256.
(1) Gumma, S.; Talu, O. Adsorption 2003, 9, 17–28. (2) Myers, A. L.; Monson, P. A. Langmuir 2002, 18, 10261–10273.
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to achieve storage densities comparable to that of liquid hydrogen under supercritical conditions (high pressure but low to moderate temperatures) through adsorption. Because adsorption is a surface phenomenon, a good adsorbent typically has a high unit surface area per volume, thus microporous adsorbents are deployed. It is now generally accepted that conventional adsorbents with moderate BET surface areas (500-2000 m2 g-1) may not be suitable for hydrogen storage. Substantial time and effort are being devoted to identifying and developing nanoporous adsorbents (with BET surface areas as high as 4000-5000 m2 g-1 in some cases) such as carbon nanotubes (CNT)5-14 and metal organic frameworks (MOFs).15-28 Comparing different adsorbents by experiments and simulations is complicated by the ambiguity in reference states commonly used to quantify supercritical micropore adsorption. A short review on the need for a reference state in adsorption is presented next. The source and reasons for the ambiguity in using the common reference states viz. excess and absolute adsorption are also discussed. We then introduce the net adsorption and provide the framework of thermodynamic relations. Subsequent sections highlight the implications of the net adsorption framework for experiments and simulations and for the practice of supercritical adsorption.
Gibbs Dividing Surface for Flat Surfaces A historical perspective is helpful in understanding why confusion exists in the literature with regard to the thermodynamics of adsorption in and on microporous solids. The simplest case of adsorption on flat surfaces is first examined before elaborating on micropore adsorption. In this work, the word “flat” stands for surfaces where the radius of curvature is much larger than the molecular diameter of the adsorbate. In general, adsorption refers to the perturbations in the density profile of a gas (or fluid) near a solid surface. Usually, there is an enhancement in the density of the gas due to the potential field imposed by the solid. Figure 1a shows the density profile of a gas (5) Schlapbach, L.; Zuttel, A. Nature 2005, 414, 353–358. (6) Chahine, R.; Bose, T. K. Int. J. Hydrogen Energy 1994, 19, 161–164. (7) Nechaev, Y. S.; Alekseeva, O. K. Russ. Chem. Rev. 2004, 73, 1211–1238. (8) Takagi, H.; Hatori, H.; Soneda, Y.; Yoshizawa, N.; Yamada, Y. Mater. Sci. Eng., B 2004, 108, 143–147. (9) Haas, M. K.; Zielinski, J. M.; Dantsin, G.; Coe, C. G.; Pez, G. P.; Cooper, A. C. J. Mater. Res. 2005, 20, 3214–3223. (10) Li, Y.; Yang, R. T. J. Phys. Chem. C 2007, 111, 11086–11094. (11) Lachawiec, A. J., Jr.; Qi, G.; Yang, R. T. Langmuir 2005, 21, 11418–11424. (12) Benard, P.; Chahine, R. Scr. Mater. 2007, 56, 803–808. (13) Hirscher, M.; Panella, B. Scr. Mater. 2007, 56, 809–812. (14) Murray, L. J.; Dinca, M.; Long, J. R. Chem. Soc. Rev. 2009, 38, 1294–1314. (15) Li, Y.; Yang, R. T. J. Am. Chem. Soc. 2006, 128, 726–727. (16) Li, Y.; Yang, R. T. J. Am. Chem. Soc. 2006, 128, 8136–8137. (17) Rosi, N. L.; Eckert, J.; Eddaoudi, M.; Vodak, D. T.; Kim, J.; O’Keefe, M.; Yaghi, O. Science 2003, 300, 1127–1129. (18) Duren, T.; Sarkisov., L.; Yaghi, O.; Snurr, R. AIChE J. 2004, 20, 2683– 2689. (19) Eddaoudi, M.; Kim, J.; Rosi, N. L.; Vodak, D.; Wachter, J.; O’Keefe, M.; Yaghi, O. Science 2002, 295, 469–472. (20) Pan, L.; Sander, M. B.; Huang, X.; Li, J.; Smith, M.; Bittner, E.; Bockrath, B.; Johnson, J. K. J. Am. Chem. Soc. 2004, 126, 1308–1309. (21) Sagara, T.; Klassen, J.; Ganz, E. J. Chem. Phys. 2004, 121, 12543–12547. (22) Rowsell, J. L. C.; Millward, A. R.; Park, K. S.; Yaghi, O. M. J. Am. Chem. Soc. 2004, 126, 5666–5667. (23) Garberoglio, G.; Skoulidas, A. I.; Johnson, J. K. J. Phys. Chem. B 2005, 109, 13094–13103. (24) Han, S. S.; Deng, W. Q.; Goddard, W. A., III. Angew. Chem., Int. Ed. 2007, 46, 6289–6292. (25) Poirier, E.; Chahine, R.; Benard, P.; Lafi, L.; Dorval-Douville, G.; Chandonia, P. A. Langmuir 2006, 22, 8784–8789. (26) Gerard Ferey, G.; Latroche, M.; Serre, C.; Millange, F.; Loiseau, T.; Percheron-Guegan, A. Chem. Commun. 2003, 2976–2977. (27) Li, Y.; Yang, R. T. Langmuir 2007, 23, 12937–12944. (28) Wong-Foy, A. G.; Matzger, A. J.; Yaghi, O. M. J. Am. Chem. Soc. 2006, 128, 3494–3495. (29) Myers, A. L. AIChE J. 2002, 48, 145–160.
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Figure 1. Illustration of density profiles and the Gibbs dividing surface near a flat surface.29
near a solid surface.29 Although perturbations do not change monotonically as shown in the Figure and layering/oscillations are noted in many simulations, this simple illustration is sufficient for demonstration purposes. Three regions can be identified in the Figure. From left to right, (1) the density is zero in the solid region (the illustration is for adsorption rather than absorption), (2) the density is substantially higher in the interfacial region because of the potential field of the solid, and (3) the density decays to the bulk gas density at some far away distance from the surface. The total amount of gas in the interfacial region (Γtot) per unit area (A) perpendicular to the page can be obtained from the density profile by Z Γtot ¼
Ffxg dx
ð1Þ
interfacial region
Although this definition is simple and straightforward, the limits of the interfacial region are ill-defined. The lower integration limit can be placed anywhere between x = 0 to x1, where F {x}=0. The location of x1, which may be taken as half of the molecular diameter, clearly depends on the size of the gas Langmuir 2010, 26(22), 17013–17023
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molecule. Thus, the lower integration limit in this fundamental definition of adsorption depends on the type of gas! The upper integration limit is even more problematic because the location of x2 depends on thermodynamic properties such as temperature (T ) and pressure (P). This ambiguity is clearly understood by realizing that x2 approaches x1 with decreasing gas density. The interfacial region is ill-defined because of the ambiguity in determining the integration limits in eq 1. Gibbs was the first to recognize this complication.30 To overcome the ambiguity, Gibbs partitioned the ill-defined interfacial region (with changing properties) into two distinct phases (with uniform properties). The Gibbs definition of adsorption identifies a dividing surface that separates the fluid phase and impenetrable solid phase as depicted in Figure 1b. All fluid properties are uniform on either side of the dividing surface (i.e., the densities are Fg and zero). The actual changes in the properties are attributed to a 2D adsorbed phase. In fact, the Gibbs approach is purely mathematical and applicable regardless of where the dividing surface is placed as long as it is not a function of thermodynamic properties such as T and P. With the Gibbs definition of adsorption, eq 1 is transformed to provide the surface excess amount adsorbed as Z Γex ¼
¥ low limit
ðFfxg - Fg Þ dx
ð2Þ
The integrand is the difference between the local density and the bulk gas density Fg. With this definition, the ill-defined upper limit extends to infinity where the integrand vanishes; therefore, it becomes independent of the thermodynamic properties. Note that the low limit of integration is yet to be defined. The value of the low limit corresponds to where the Gibbs dividing surface is placed. Gibbs does not provide any guidance on where to place the dividing surface in his original work, but in general the dividing surface is placed (naturally?) on the surface of the solid (x0 in the Figure). The placement of the low limit actually defines the reference state for adsorption thermodynamics. To complete the thermodynamic definition of the adsorbed phase, an additional extensive property is also necessary to fix the extent of the system. Volume, the natural extensive property for bulk phases, cannot be used in adsorption because the Gibbs definition applies to a 2D surface phase that by definition does not have a volume. Because adsorption is a surface phenomenon, the area of the solid is the natural extensive property for flat surfaces (or for interfaces with a surface curvature that is much larger than the molecular size).
Dividing Surface for Microporous Adsorbents Most practical applications of adsorption require a large surface area-to-volume ratio, hence industrial adsorbents are typically nanoporous or microporous materials. Adsorption on real adsorbents such as activated carbons, zeolites, silica gel, and MOFs bears little resemblance to the previous discussion of flat surfaces. The radius of curvature of a micropore is of the order of a few angstroms, so the surface area between guest molecules and the solid cannot be defined or measured. The surface area loses its physical significance for microporous solids; therefore, the mass of adsorbent is commonly used as the extensive property. In addition, the bulk fluid exists outside the adsorbent, so the idealized picture of an adsorbed phase between a solid and a bulk fluid phase does not apply to microporous materials. Never(30) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans and Green: New York, 1928.
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theless, since the 1930s, scientists have been treating adsorption in microporous solids using the concepts developed for flat surfaces such as Langmuir31 and BET32 formulations. The extension of these flat-surface formulations to microporous solids fails because the concept of surface area does not apply. This failure is best exemplified by the experimental observations of monolayer coverage varying with temperature in microporous solids, a clear indication of a temperature-dependent surface area that should be invariant according to the basic assumptions of these models. Irrespective of the physical insignificance of the surface area for porous materials, the concept of the Gibbs dividing surface can be extended to adsorption on these materials as long as a reference state is used to locate the dividing surface. The reference state must be clearly defined, and it must be independent of system properties such as T and P. The earliest researchers to adopt a reference state approach to adsorption in microporous solids were McBain33 and Coolidge.34,35 They abandoned the unphysical extension of flat surface formulations to microporous solids but retained the mathematical features of the Gibbs formulation. In their approach, the Gibbs dividing surface is no longer flat and no longer physically on the surface of the solid. The dividing surface is a mathematical/hypothetical hypersurface that does not have a volume. Its extent depends only on the amount of solid present and is independent of system properties such as T and P. A readily measurable reference state is therefore used to fix the location of the Gibbs dividing surface. Reference states are common in thermodynamics. Deviations of any property from that of the reference state also follow the thermodynamic relations of their counterparts (such as departure functions of nonideal gases, for example). The reference state is usually chosen for experimental and application convenience. Appropriate corrections are applied to measurements to report the results on the basis of the chosen reference state.
Reference States for Microporous Adsorption Consider a nanoporous solid adsorbent in equilibrium with a bulk gas of density Fg at some T and P as shown in Figure 2a. The container represented by the outer box in the Figure encloses the gas in the bulk phase along with the solid adsorbent and adsorbed gas molecules. The total volume of the container is V. The solid adsorbent is represented by small shaded squares (with a total volume of Vs for all squares); the pores are represented as channels between squares (with a total volume of Vp). The gas molecules are indicated by black dots. The density is higher in the channels (more black dots) than in the bulk phase away from the solid. Away from the solid, the gas occupies a volume of Vg. Although such a schematic simplifies the geometry of the nanoporous adsorbent, it correctly depicts the essential features. The known container volume is conceptually partitioned into three regions. V ¼ Vs þ Vp þ Vg
ð3Þ
Only the container volume, V, can be measured without any ambiguity. The reference state choice fixing the Gibbs dividing surface determines the partitioning among the three regions (i.e., Vs, Vp, and Vg). (31) (32) 319. (33) (34) (35)
Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361–1403. Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309– McBain, J. W.; Britton, G. T. J. Am. Chem. Soc. 1930, 52, 2198–2222. Coolidge, A. S. J. Am. Chem. Soc. 1934, 56, 554–561. Coolidge, A. S.; Fornwalt, H. J. J. Am. Chem. Soc. 1934, 56, 561–568.
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Figure 2. Schematic illustrating various reference states for the definition of adsorption. (a) A typical scenario at equilibrium between a gas and a porous solid. (b-d) Reference states for absolute, excess, and net adsorption, respectively.
The following three definitions for reference states, which essentially differ in their definitions of the bulk phase volume, provide meaningful descriptions of the adsorption process.34 These definitions are depicted in Figure 2b-d. The regions shaded black in this Figure refer to the volume that is not available to the bulk gas (impenetrable solid volume) and are considered to be on the “solid” side of the Gibbs dividing surface with F{x} = 0. The gas occupies whatever is left over, the region defined as the bulk phase volume, at a density equivalent to its bulk density at the given T and P; F{x} = Fg by the Gibbs definition. The region shaded black in Figure 2b is the volume occupied by the adsorbent, including its pores. The difference of the amount of gas present in parts a and b of Figure 2 is equal to the amount of fluid present within the pores of the adsorbent as a result of adsorption, the so-called absolute adsorption. This definition (reference state A) fixes the Gibbs dividing surface at the “outer” surface of the adsorbent; the bulk phase extends only until this surface whereas the solid phase includes all of the pores. The reference volume for the gas to be experimentally measured for absolute adsorption is Vg ¼ V - Vs - Vp
ð4Þ
The region shaded black in Figure 2c is the so-called impenetrable solid volume of the adsorbent. The difference in the amount 17016 DOI: 10.1021/la102186q
of gas present in parts a and c of Figure 2 is equal to the so-called Gibbs surface excess adsorption in the literature. Note that the channels in Figure 2c are occupied by gas molecules at a density equivalent to the bulk gas density, which in reality is not an achievable condition in experiments. This definition (reference state B) fixes the dividing surface at the “inner” surface of the adsorbent, but pores are included in the bulk phase. The reference volume for the gas to be experimentally measured for excess adsorption is Vg ¼ V - Vs
ð5Þ
A third definition (reference state C) of the Gibbs dividing surface extends the bulk phase all the way into the impenetrable solid as indicated in Figure 2d. This definition of the reference state involves the amount of gas present in the empty container (i.e., with no adsorbent present in the container). We call this the net adsorption. The net amount adsorbed is equal to the difference in the amount of gas present in parts a and d of Figure 2. The reference volume for net adsorption does not need any additional experimental measurement; it is simply the volume of the container. Vg ¼ V
ð6Þ
From a thermodynamic perspective, any reference state is acceptable as long as it is easily measurable without ambiguity. It is possible to convert and compare data based on different states Langmuir 2010, 26(22), 17013–17023
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through rigorous thermodynamic relations. Coolidge’s work34 only identifies and highlights the differences in using various reference states and concludes that the choice of a reference state may be based on the “purpose at hand”. Later developments in this area were biased toward reference state B, excess adsorption. This development in history may be due to the proximity of the excess adsorption to adsorption on flat solid surfaces. Adsorption in micropores based on excess adsorption can readily be used with models (i.e., Langmuir and BET) developed for flat surfaces. However, some works advocate the use of absolute adsorption36 on the basis of reference state A because it can be directly compared to simulation results. To the best of our knowledge, reference state C defining net adsorption was not investigated after the Coolidge era. Before elaborating on net adsorption, the following section highlights difficulties in experimentally and theoretically determining the reference states for absolute and excess adsorption (reference states A and B), which require ambiguous measurements of the pore volume and the impenetrable solid volume. At low pressure, specific properties such as the amounts adsorbed according to the three definitions are essentially equivalent whereas differential properties such as the Henry constants differ substantially. At high pressure, the differences in all properties are profound. The absolute adsorption increases monotonically with pressure whereas excess and net adsorption exhibit maxima. The maximum in the amount adsorbed occurs at a pressure where the rates of change in density (with pressure) for both the adsorbed and the gas phases are equal.
Experimental Measurement of Solid and Pore Volumes During initial development (including attempts by Mcbain33 and Fernbacher and Wenzel37), an approximate value of the structural density of the solid was used to fix the impenetrable solid volume for reference state B. In the case of vapor adsorption, where the bulk fluid density is small relative to the interfacial density, the uncertainty introduced is small. However, for gas adsorption at high pressure the estimate of structural density has a significant impact on the amount adsorbed. More realistic approaches adopted later include the measurement of the available volume using liquids such as ether and gases such as nitrogen and hydrogen. In reality, defining a bulk-phase volume for reference state B essentially requires one to distinguish between the penetrable and impenetrable volume of the porous solid. Clearly, the penetrable volume of a microporous adsorbent depends on the size of the probe molecule. Moreover, once a probe molecule is introduced into the solid, it would interact with the surface, resulting in adsorption, whereas the formulations and calculations assume that no adsorption occurs during reference state measurements. Because of its small size and low polarizability, helium is now a commonly accepted de facto standard for fixing the impenetrable solid volume of porous adsorbents for gas adsorption. Helium adsorption isotherms in silicalite were measured by Gumma and Talu1 to highlight the adsorption occurring during reference state measurements. Their results for the apparent impenetrable solid volume determined by the slope of the isotherms at zero pressure (density) as a function of temperature are shown in Figure 3. The solid volume is commonly taken at 300 K assuming no helium adsorption, which is a defacto reference state for excess adsorption in the literature. However, as can be seen in the Figure, Vs varies substantially with temperature, especially at low temperature where helium adsorption is more pronounced. The temperature (36) Salem, M. M. K.; Braeuer, P.; Szombathely, M.v.; Heuchel, M.; Harting, P.; Quitzsch, K. Langmuir 1998, 14, 3376–3389. (37) Fernbacher, J. M.; Wenzel, L. A. I & EC Fund. 1972, 11, 457–465.
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Figure 3. Helium adsorption in silicalite; data is taken from Gumma and Talu.1 Points are experimental data; the line is drawn as a guide to the eye.
dependency of apparent Vs is persistent even at temperatures much higher than ambient temperature. The effect is orders of magnitude higher than what might be caused by the thermal expansion of the solid (which was neglected in the calculation). Furthermore, this data is for helium in the inert, nonpolar pores of silicalite; the trends would be higher for polar surfaces commonly used in industry such as zeolites A and X or for high-porosity materials such as activated carbon. Surely there must be a better way to fix the reference state for adsorption other than basing all results on the ambiguity introduced by helium adsorption. We should note here that many other studies also unequivocally show or prove helium adsorption and discuss the shortcomings of excess adsorption.37-47 The measurement of the bulk phase volume for absolute adsorption (reference state A) is even more challenging; one needs to fix the pore volume in addition to the impenetrable solid volume. The pore volume is usually estimated by saturating the solid with substances such as benzene35 and nitrogen.48 Here the assumption is that the density in the pores is equal to that of liquid at same temperature. There is no reason to expect that the packing factor in the confined space of a pore is equivalent to that in a liquid. Molecules in a micropore are more often next to the solid wall than next to the other molecules (as in the case of a liquid phase). An additional complication with this approach is that sometimes substantial adsorption occurs on the outer surface and/or in the macropores of the solid particles. External adsorption is later corrected by theoretical models such as the R-plot method,48 introducing more assumptions. In another approach mercury is used to estimate the total volume of the solid (sum of impenetrable solid and pore volume) because it does not penetrate (38) Kaneko, K.; Setoyama, N.; Suzuki, T. In Proc.COPS III Symp.; Roquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; pp 593-602. (39) Maggs, F. A. P.; Schwabe, P. H.; Williams, J. H. Nature 1960, 186, 956–958. (40) Malbrunot, P.; Vidal, D.; Vermesse, J.; Chahine, R.; Bose, T. Langmuir 1997, 13, 539–544. (41) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148–5160. (42) Sircar, S. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527–1540. (43) Sircar, S. AIChE J. 2001, 47, 1169–1176. (44) Sircar, S. In Proc. FOA 7; Kaneko, K, Kanoh, H., Hanzawa, Y, Eds.; IK International: Chiba-City, Japan, 2001; pp 656-663. (45) Springer, C; Major, C. J.; Kammermeyer, K. J. Chem. Eng. Data 1969, 14, 78–82. (46) Staudt, R.; Bohn, S.; Dreisbach, F.; Keller, J. U. In Proc. COPS IV Symp.; McEnaney, B., Mays, T. J., Roquerol, J., Rodriguez-Resinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Royal Society of Chemistry: Cambridge, U.K., 1997; pp 261-266. (47) Suzuki, I.; Kakimato, K.; Oki, S. Rev. Sci. Instrum. 1987, 58, 1226–1230. (48) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscon, L; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603–619.
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the micropores. However, mercury porosimetry49 results become questionable for amorphous solids with a wide pore size distribution. With either experimental approach, vapor saturation or mercury porosimetry, the pore volume cannot be determined unequivocally without many other accompanying assumptions. Although thermodynamics requires that the reference state be identified unambiguously, no consensus exists in the scientific community on a protocol for fixing the reference states for absolute and excess adsorption (states A and B). Furthermore, most adsorption data do not include information on how the reference states are actually measured. Reference states A and B require additional experimental and/or simulation work as described above with questionable assumptions. Despite these shortcomings of reference states A and B, to the best of our knowledge, no serious attempts have been made by the scientific community at large since Coolidge34,35 to exploit the unambiguous nature of net adsorption (reference state C). A detailed analysis and benefits of this reference state in experiments and simulations are yet to be explored. In the remainder of this article, we provide the thermodynamic framework for net adsorption and highlight its benefits for experiments, simulations, and practical applications.
Unambiguous Nature of Net Adsorption Net adsorption is the total amount of gas present in the container with the adsorbent minus the amount that would be present in the empty container (without the adsorbent) at the same temperature and pressure. There is no ambiguity introduced by concepts such as pore volume and impenetrable solid volume. There is no need for another gas to probe the solid. No additional experiments are required. The net adsorption amount is always lower than excess and absolute adsorption. Furthermore, a negative net adsorption indicates that the overall density in the container with the solid is lower than that in an empty container. Net adsorption is thus a direct measure of the density enhancement for storage applications. Negative values should not be treated as a hindrance to using the net adsorption framework; several thermodynamic properties (including departure functions) are negative but are still rigorous and useful tools for analysis. Thermodynamics requires only that the dividing surface be invariant. The common practice has been constrained by the notion that adsorption has to be positive. Choosing helium (the smallest and most inert gas) as a non-adsorbing reference gas assures that other compounds show a positive excess adsorption. However, helium adsorbs even in the very inert pores of silicalite as shown earlier. Why does adsorption have to be positive? Thermodynamically, the adsorbed-phase properties in a microporous solid are completely defined as differences from a reference state. There is no overwhelming reason that these differences must always be positive. In fact, insisting that the adsorption always be positive (and be zero for helium at some T and P) results in inconsistencies and confusion. This practice should be abandoned for adsorption in microporous solids. With microporous solids, the impenetrable solid volume is directly proportional to the mass of the sample. The extent of the adsorbed phase is also directly proportional to the sample mass. Therefore, as required for a rigorous reference state, their ratio per unit mass of solid is invariant. Note that this is not true for adsorption on flat surfaces. The extent of an adsorbed phase in (49) Drake, L. C.; Ritter, H. L. Ind. Eng. Chem. Anal. Ed. 1945, 17, 787–791.
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relation to the sample weight can be varied by changing the geometric aspect ratio of the solid sample. We emphasize here that net adsorption is not useful for adsorption on flat surfaces. The solid volume loses physical significance for flat surfaces just as the surface area loses physical significance for micropores. Although primary molecular interactions are the same, adsorption on flat surfaces and adsorption in micropores are different physical manifestations of the same phenomenon. As such, their thermodynamic treatment requires different approaches: excess adsorption for flat surfaces and net adsorption for micropores. Mesoporous solids present a special case in between the two extremes. The net adsorption formalism, in general, can be used for supercritical adsorption in mesoporous solids as long as the physical nature (e.g., pore structure) does not change. However, the excess adsorption formalism may be more suitable for external adsorption on fine powders. It should be noted once again that results for all materials with different reference states can be interconverted if the material properties do not change and the reference state measurements are clearly defined. Without intending to add to the confusion, we believe that net adsorption is best suited by defining adsorption in micropores as the difference between the total amount of gas present in the container minus the amount that would be present if the adsorbent were completely absent. Because it references an empty container, net adsorption is somewhat similar in essence to the column isotherm often used in the design of adsorbers,50 which refers to the total volume of gas (interstitial gas and adsorbed) in the column per total column volume rather than void column volume (which would be similar to excess adsorption).
Thermodynamic Framework for Net Adsorption In this section, we provide the general thermodynamic framework using the net adsorption reference state for the adsorption of pure gas on a porous solid. Relationships among important thermodynamic properties based on absolute, excess, and net frameworks are also presented. Consider a solid adsorbent of true mass ms in vacuo. This solid is placed in an adsorption column of empty volume V and is allowed to equilibrate with bulk gas at a temperature T and pressure P. The total gas present in the column includes moles contained in both the adsorbed and bulk phases. Let Z be any property for the total contents present in the adsorption column, including the adsorbed and bulk gas phases. The net adsorption property Znet is defined on the basis of reference state properties Zgnet of an empty column (without solid) containing the bulk gas at the same temperature and pressure and Zs* of the clean solid in vacuo. Thus,
Z net ¼ Z - ½Zgnet fT, P, Vg þ Zs fT, P ¼ 0g
ð7Þ
The properties shown inside the curly braces are the functionalities. They are shown in some equations for clarity as needed. Because the excess and absolute properties differ in the assignment of the bulk-phase volume, they can be readily defined by replacing V in eq 7 Z ex ¼ Z - ½Zgex fT, P, V - Vs g þ Zs fT, Pg Z abs ¼ Z - ½Zgabs fT, P, V - Vs - Vp g þ Zs fT, Pg
ð8Þ
(50) Ruthven, D. M.; Farooq, S.; Knaebel, K. S. Pressure Swing Adsorption; VCH: New York, 1994; p 102.
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where Vs and Vp refer to the impenetrable solid volume and pore volume, respectively. Note that a slight difference also exists for the reference state of the solid; in absolute and excess adsorption, it is considered to be at the system pressure P (as opposed to vacuum for net adsorption). Because Vs and Vp are not unique and depend on the protocol used for their determination, ambiguity exists in definitions of excess and absolute adsorption as discussed. However, V is a property of the system alone (independent of the solid adsorbent) and can be measured unambiguously. The fundamental property relation for the total internal energy U of the column contents is dU ¼ T dS - P dV þ μg dN þ μs dms
ð9Þ
where S is the total entropy of the column contents, N is the total amount of gas present in the column (in both bulk and surface phases), μg is the chemical potential of the bulk gas (which is the same as that of the adsorbed phase), and μs is the chemical potential of the solid adsorbent when it is at equilibrium with the given bulk phase (at T and P). The corresponding property changes defined for reference states of gas and solid for net adsorption are dUgnet ¼ T dSgnet - P dV þ μg dNgnet
dUs ¼ T dSs þ μs dms
ð10Þ ð11Þ
Note that μ*s is the chemical potential of a clean solid in vacuo and is different from μs used earlier in eq 9. The P dV term vanishes at the reference state of the solid in eq 11 because P* = 0 and the solid is assumed to be incompressible, dVs = 0. From eqs 7 and 9-11, it follows that
dU net ¼ T dSnet þ μg dN net þ ðμs - μs Þ dms
ð13Þ
where Ngnet is the amount of gas present under bulk-phase conditions in a volume V. It may be obtained from an appropriate equation of state. Ngnet ¼ VFg fT, Pg
ð14Þ
Fg{T, P} is the density of the bulk gas at T and P. As in the original Gibbs formulation, the changes in the properties of the solid and the gas in the immediate vicinity of the solid are attributed to net adsorption. The extensive properties (i.e., Unet, Snet, and Nnet) are directly proportional to the mass of the adsorbent, ms. Using Euler’s theorem for homogeneous first-order systems, one can readily obtain the relation between specific properties per unit mass of the solid
unet ¼ Tsnet þ μg nnet þ ðμs - μs Þ
ð15Þ
The lowercase variables represent specific adsorption properties (i.e., properties per unit of adsorbent mass ms). The last term in eq 15 is defined as the net surface potential, which is the difference between the chemical potential of the solid Langmuir 2010, 26(22), 17013–17023
Φnet ¼ μs fT, Pg - μs fT, P ¼ 0g
ð16Þ
However, the surface potential for excess adsorption is defined as29
Φex ¼ μs fT, Pg - μs fT, P ¼ 0g - Pvs
ð17Þ
In either case, when the corresponding surface potential is zero, there would be no net (or excess) adsorption. With the definition of the net surface potential, the fundamental property relation becomes dunet ¼ T dsnet þ μg dnnet þ dΦnet
ð18Þ
We define the net Gibbs free energy as gnet unet þ Pvnet - Tsnet
ð19Þ
Because vnet = 0, usual Legendre transforms yield dgnet ¼ - snet dT þ μg dnnet þ dΦnet
ð20Þ
As is the case with excess adsorption, the net surface potential cannot be directly obtained from experiments. However, it can be readily calculated via the Gibbs-Duhem equation for the surface phase at constant temperature dΦnet ¼ - RTnnet dðln fg Þ ðconst TÞ
ð21Þ
where fg is the fugacity of the bulk gas. The integral of this equation along an appropriate path yields the surface potential. For an ideal gas phase, eq 21 reduces to a well-known form that is similar to the excess surface potential
ð12Þ
All net quantities in eq 12 are defined as the differences given in eq 7. In particular, Nnet, the net amount adsorbed, is given by N net ¼ N - Ngnet
in equilibrium with adsorbed phase and that of the clean solid in vacuo.
dΦnet ¼ - RT
nnet dP ðconst TÞ P
ð22Þ
At high pressure, where the gas nonideality is significant, the importance of using eq 21 instead of eq 22 is discussed by Talu et al.51
Property Changes on Immersion Practical applications of adsorption require energy balances because adsorption is an exothermic process. Many heats of adsorption are defined in the literature for this purpose. For excess adsorption, the isosteric heat of adsorption is commonly calculated from isotherms at different temperatures. The nature of the gas-solid pair determines the isosteric heat, which displays a wide range of dependencies on the amount adsorbed (e.g., ref 52). Heat is a process variable that is dependent on the path of change; it is not a system property independent of the path. It is desirable to use system properties in energy balances over pathdependent energy terms. The isosteric heat is same as the differential enthalpy of adsorption, as clearly discussed by Myers.29 The relevant adsorption energy terms for net adsorption analogous to differential enthalpy are linked to immersion properties. A property change on immersion is defined as the property of an adsorbed phase with reference to a compressed gas and an (51) Talu, O.; Zhang, S. Y.; Hayhurst, D. T. J. Phys. Chem. 1993, 97, 12894– 12898. (52) Talu, O.; Kabel, R. L. AIChE J. 1987, 33, 510–514.
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evacuated clean solid. The property changes are typically expressed as extensive variables per unit adsorbent mass, or as massspecific properties. In particular, following Myers,29 the Gibbs free energy of immersion for excess adsorption is Δgimm, ex ¼ Φex þ Pvs
ð23Þ
Making use of eqs 16 and 17, the immersion Gibbs free energy for net adsorption is simply Δgimm, net ¼ Φnet
ð24Þ
The physical significance of the net surface potential is now evident; it is directly the Gibbs free energy change upon immersion. When an evacuated clean adsorbent is placed in contact with a bulk gas at constant T and P, adsorption occurs. Although there would be no change in the free energy of the gas as it adsorbs (because the chemical potential for gas and adsorbed phases are the same), the change in the surface free energy of the solid manifests itself as the free energy of immersion. This relation is in contrast to that of excess adsorption, where an unnecessary additional Pvs term appears to account for a change in the chemical potential of the clean solid as it is compressed from vacuum to system pressure.29 Other immersion variables can be obtained through the differentiation of eq 24. In particular, the enthalpy of immersion that is useful for energy balances is given by Δh
imm
¼
! DðΦnet =TÞ Dð1=TÞ
ð25Þ P
The use of net adsorption clarifies the energy balances for columns also. Consider an adsorption column of volume V containing a clean adsorbent of mass ms charged with an amount of gas Ncharge. The total enthalpy of the column contents is " H col ¼ N charge hg þ ms
DðΦ =TÞ Dð1=TÞ net
#
þ ms hs
ð26Þ
P
The three terms on the RHS represent the enthalpy of the gas (adsorbed and otherwise), the net enthalpy change due to adsorption in the solid (enthalpy of immersion), and the enthalpy of a clean solid at temperature T. The energy balances on adsorption columns can be performed directly using the net adsorption framework without invoking the need for excess or absolute adsorption; reference state properties such as the pore volume and the impenetrable solid volume are also not necessary. In this section, we have presented a general thermodynamic framework for net adsorption. It was demonstrated that rigorous thermodynamic relations can be obtained for variables of interest similar to those based on excess adsorption. Conversion between various thermodynamic properties in different frameworks is also possible, provided that data on reference state properties (vs for example) is available. Subsequent sections detail the benefits of using net adsorption for experiments and simulations and in practice.
Experimental Benefits of Using Net Adsorption We concentrate on two of the most common methods used for measuring adsorption, viz., the volumetric and gravimetric techniques.53 (53) Talu, O. Adv. Colloid Interface Sci. 1998, 76-77, 227–269.
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Figure 4. Common isotherm measurement systems.
A simple volumetric setup for the measurement of pure-gas adsorption equilibrium is shown in Figure 4a. Initially, the solid adsorbent contained in the adsorption column is activated (to remove previously adsorbed species) under appropriate conditions, and the column is sealed under vacuum. Both the column and reservoir are maintained at the desired temperature T. The reservoir is then charged with gas to a predetermined pressure P0. The valve between the reservoir and column is opened, and adsorption equilibrium is established between the solid and the gas; the final equilibrium pressure Pg is noted. The contents of the column (in both adsorbed and bulk phases) at equilibrium are equal to the moles exchanged between the reservoir and the column, ΔN, ΔN ¼ V0 ðF0 - Fg Þ
ð27Þ
where V0 is the volume of the charge reservoir and F0 and Fg are the molar densities in the charge reservoir before and after the valve is opened. In general, the amount adsorbed nads per unit adsorbent mass ms is given by nads ¼
ΔN - V ref Fg ms
ð28Þ
where Vref is the volume assigned to the bulk phase on the basis of the chosen reference state. Apart from Vref, an experimental determination of adsorption requires independent knowledge of the reservoir volume V0 and sample mass ms, irrespective of the chosen reference state. As explained earlier, Vref for net, excess, and absolute adsorption are Vcol, Vcol - Vs, and Vcol - Vs - Vp, respectively. In addition to other advantages of net adsorption outlined earlier, from a purely experimental perspective one needs to determine, Vref (= Vcol) only once because it is fixed for a given experimental apparatus. Subsequent changes in the sample do not necessitate additional measurements. In contrast, reference states for both absolute and excess adsorption involve the solid adsorbent, and Vref needs to be measured every time the adsorbent sample is changed (using helium as the probe gas). A simple gravimetric setup is shown in Figure 4b. Initially, the adsorbent is placed in a bucket, activated under appropriate conditions, and sealed. The signal from the microbalance under vacuum m0 is due to the mass of the sample (ms) and the bucket (mbucket). The sample is then allowed to equilibrate with the gas of interest at equilibrium pressure Pg and temperature T (at a gas Langmuir 2010, 26(22), 17013–17023
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molar density of Fg). The signal from the microbalance mf is recorded under equilibrium conditions. The change in the microbalance signal Δm is a result of adsorption occurring on the solid surface and the total buoyancy force. In general, the adsorbed amount per unit solid mass is given by ! 1 Δm ads ref þ V Fg ð29Þ n ¼ ms MWg where MWg is the molar mass of the gas and Vref is the volume experiencing the buoyancy force. To determine the adsorbed amount in addition to Vref, a knowledge of the mass of the clean adsorbent in vacuo (i.e., ms) is necessary. The buoyancy volume Vref is fixed on the basis of the reference state. In the case of net adsorption, the buoyancy correction is needed only for forces acting on the bucket. For excess adsorption, the buoyancy correction for the impenetrable solid volume is also needed. Finally, for absolute adsorption, yet another correction for the buoyancy acting on the pore volume is also necessary. Thus, V ref is Vbucket, Vbucket þ Vs, and Vbucket þ Vs þ Vp for net, excess, and absolute adsorption, respectively. The reference state correction for net adsorption depends only on the bucket volume Vbucket. It needs to be determined only once for the given experimental system, similar to the determination for the volumetric apparatus. However, the reference states for both excess and absolute adsorption involve the sample; separate measurements for each sample are necessary. As an example of adsorption experiments, the differences among isotherms calculated using each of these three reference states for the adsorption of methane on Norit R1 Extra are shown in Figure 5 for data taken from Herbst and Hartig.54 As a unique feature in this article, the net adsorption is directly reported as the so-called reduced mass, Ω, following the protocol first suggested by Staudt et al.55 It is the experimental data corrected only for the buoyancy of the volume of the bucket. An excess isotherm calculated with additional information from helium experiments at 298 K is also reported in the article; the impenetrable volume, vs, was determined to be 0.462 cm3 g-1. Here, the absolute adsorption is also included for completeness. A further correction corresponding to a pore volume of 0.516 cm3 g-1 was used by Ustinov et al.56 to obtain the absolute adsorption. Obviously, the differences in adsorption based on the three reference states increase with bulk gas pressure and the contribution of the correction term increases with density. Net adsorption becomes negative at around 21.9 MPa, but both absolute and excess adsorption remain positive throughout the experimental range. The implication of this observation is provided later.
Comparison of Net Adsorption with Simulation Results Typical adsorption simulations are performed in the grand canonical ensemble (GCMC) where the simulation volume Vbox (containing the solid and pore space), temperature, and chemical potential are fixed. Simulations yield the average number of molecules present in the system, ÆNæ. The absolute amount adsorbed is directly obtained from nabs ¼
1 ÆNæ ms NA
ð30Þ
(54) Herbst, A.; Harting, P. Adsorption 2002, 8, 111–123. (55) Staudt, R.; Saller, G.; Keller, J. U. Ber. Bunsen-Ges. Phys. Chem. B 1993, 97, 98–105. (56) Ustinov, E. A.; Do, D. D.; Herbst, A.; Staudt, R.; Harting, P. J. Colloid Interface Sci. 2002, 250, 49–62.
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where ms is the mass of solid in the simulation and NA is Avogadro’s number. The excess amount adsorbed from experiments can be calculated via the pore volume 1 ÆNæ - Vp Fg ð31Þ nex ¼ ms NA The bulk fluid density, Fg, can be obtained from an appropriate equation of state. Some of the early work in molecular simulations attempts to compare the simulation results and experiments directly without attempting to correct for the difference in reference states.57,58 In later years, attempts were made to correct for these differences from the structure of the adsorbent59-61 to determine Vp. More recently, helium simulations were performed to determine the correction mimicking experimental measurements of excess adsorption.62,63 In essence, extra effort is spent to fix the location of the dividing surface as in the experiments for excess adsorption. Fortunately, molecular simulations can be performed at absolute zero pressure (by direct Monte Carlo integration), circumventing the problem of helium itself adsorbing at finite pressures. The conversion is based on the slopes of the helium isotherm at zero pressure as described by Talu and Myers.64,65 The accessible volume in the simulation is calculated from the configurational integral for helium from Z 1 e - φ=kT dV ð32Þ Vp ¼ ms Vbox where φ is the helium-solid interaction potential and k is the Boltzmann constant. The volume calculated by this equation depends on the molecular size of helium (collision diameter) used in φ. A new approach proposed by Do et al.66 for simulations is worth mentioning here. They essentially suggest using a zeropotential hypersurface for the adsorbing molecule to define the dividing surface for excess calculations. Such a definition renders excess adsorption always positive (absolute adsorption is always positive). Similar calculations were performed by Talu and Myers65 to show clearly that the zero-potential surface depends on the molecular size (as was noted by Liu et al.61 and many others also). These researchers never suggested the use of a zeropotential surface as a reference state because it clearly depends on the molecular size. In addition, this reference state becomes a property of the gas-solid pair, whereas excess adsorption is a property of the helium-solid pair. Furthermore, a zero-potential surface approach cannot be duplicated in experiments because the interaction potential cannot be turned off in an experiment. The simulation results based on such an approach, although theoretically elegant, are practically inconsequential because they cannot be compared to real data. In contrast, the net adsorption reference state does not depend on either the solid or the gas. (57) Goodbody, S. J.; Watanabe, K.; MacGowan, D.; Walton, J. P. R. B.; Quirke, N. J. Chem. Soc., Faraday Trans. 1991, 87, 1951–1958. (58) Smit, B. J. Phys. Chem. 1995, 99, 5597–5603. (59) Myers, A. L.; Calles, J.; Calleja, G. Adsorption 1997, 3, 107–115. (60) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148–5160. (61) Liu, J.; Culp, J. T.; Natesakhawat, S.; Bockrath, B. C.; Zande, B.; Sankar, S. G.; Garberoglio, G.; Johnson, J. K. J. Phys. Chem. C 2007, 111, 9305–9313. (62) Talu, O.; Myers, A. L. In Fundamentals of Adsorption: Proceedings of the Fifth International Conference on Fundamentals of Adsorption; Le Van, D. M., Ed.; Kluwer Academic Publishers: Boston, 1996; pp 945-952. (63) Talu, O.; Myers, A. L. AIChE J. 2001, 47, 1160–1168. (64) Talu, O.; Myers, A. L. In Fundamentals of Adsorption: Proceedings of the Sixth International Conference on Fundamentals of Adsorption; Meunier, F., Ed.; Elsevier: New York, 1998; pp 861-866. (65) Talu, O.; Myers, A. L. Colloids Surf., A 2001, 187-188, 83–93. (66) Do, D. D.; Do, H. D.; Nicholson, D. J. Phys. Chem. B 2009, 113, 1030– 1040.
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Figure 5. Differences among absolute (9), excess (b), and net (2) adsorption isotherms for methane on Norit R1 Extra at 298 K; lines are drawn as a guide to the eye. The experimental data and reference state for the excess isotherm (vs = 0.462 cm3/g) are taken from the work of Herbst and Harting.54 Absolute isotherms are calculated on the basis of the pore volume (vp = 0.516 cm3/g) as suggested by Ustinov et al.56
Figure 6. Comparison of simulation results for hydrogen adsorption at 298 K on HKUST-1, MOF. The lines represent nabs ( 3 3 3 ), nex (---), and nnet (-). The reference state corrections used are vp = 0.83 and vs = 0.31 cm3/g. All results are recalculated from the work of Liu et al.61
Using net adsorption, a comparison between experiments and simulation results is straightforward; one is no longer required to fix the location of the dividing surface at an arbitrary interface to delineate the solid and bulk phases. Simulation results can be converted to net adsorption for comparison with experiment using the volume of the simulation box. No other simulation is necessary to calculate the net adsorption from simulations. It is given as net
n
1 ÆNæ ¼ - Vbox Fg ms NA
ð33Þ
As an example of the simulation results, the data of Liu et al.61 for the adsorption of hydrogen on metal-organic framework HKUST-1 is shown in Figure 6. Net adsorption results follow eq 33. HKUST-1 is a metal-organic framework with a unit cell volume of 1.14 cm3 g-1.28 The pore volume for this material obtained from N2 adsorption experiments at 77 K is 0.83 cm3 g-1.61 If adsorption on outer surface is negligible, then all of the pore volume may be assigned to the micropores and the impenetrable solid volume will be 0.31 cm3 g-1. This value represents the lower limit for estimating the impenetrable solid volume necessary for excess adsorption used here. Because hydrogen does not adsorb substantially, the differences among net, excess, and absolute adsorption in Figure 6 are less dramatic than the differences for methane shown in Figure 5. However, this case shows the significant differences in the Henry’s constants (slope of the isotherm at the origin) for the three definitions. The differences in the Henry’s constants for heavily adsorbing methane are not visible in Figure 5.
Practical Benefits for Gas Storage Because net adsorption is measured with a reference based on an empty container (with no solid), it is a direct measure of the additional amount that is stored in the container as a result of adsorption under identical conditions. This is in contrast to excess adsorption, which measures only the additional amount present in the void space of the container without any accounting for the space occupied by the solid. A negative net adsorption value indicates that the amount stored in the container is actually lower 17022 DOI: 10.1021/la102186q
Figure 7. Contributions of adsorption to hydrogen storage on MOF-177 at 298 K, calculated using the experimental data of Li and Yang.67 Total storage capacity (9) and net adsorption (b); the difference between the two quantities indicates the amount that would be stored in the same volume without the adsorbent when the bulk gas is compressed under identical conditions (O). Lines are drawn as a guide to the eye. The reference state conditions used are vp = 1.58 and vs = 0.76 cm3/g.
than what could be stored in the same container when there is no adsorbent at the same T and P. This does not indicate the absence of adsorption but simply indicates that the additional amount present in the container due to adsorption is less than the amount of gas that would be in the space occupied by the solid. For example, net adsorption goes to zero at about 21.9 MPa in Figure 5, indicating that no advantage exists in using Norit R1 for natural gas storage above this pressure; above this pressure, storing the compressed gas in its bulk phase (without the adsorbent in the container) results in a larger storage capacity. The same conclusion cannot be readily reached from excess or absolute isotherms without performing additional calculations to account for the ambigious pore and/or impenetrable solid volume. It is worthwhile to explore this advantage for hydrogen storage on MOFs. As an example, MOF-177, which is claimed to have (67) Li, Y.; Yang, R. T. Langmuir 2007, 23, 12937–12944.
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one of the highest hydrogen storage capacities among MOFs,28,67 was analyzed using the net adsorption framework. This MOF has a unit cell volume of vuc = 2.34 cm3 g-1 and a pore volume of 1.58 cm3 g-1. The corresponding volume of the impenetrable solid is 0.76 cm3 g-1. Experimental data for excess adsorption at 298 K from Li and Yang67 was used to calculate the net adsorption (Figure 7). Although the excess adsorption was about 3.1 mmol g-1 at the highest pressure, there is very little net adsorption (about 0.22 mmol g-1). The presence of an adsorbent offers little advantage in increasing the storage capacity for hydrogen; the highest enhancement of about 12% above simple gas compression is achieved at about 20 bar. The enhancement is only about 2.6% at 97 bar. Practically, this implies that most of the storage is achieved as a result of bulk gas compression, and hydrogen adsorption on the solid has a negligible effect on increasing the storage capacity. High values of excess adsorption must be carefully weighed with the bulk-phase volume that is lost because of the presence of a solid; net adsorption requires no such correction and is a direct measure of the storage enhancement.
Conclusions The measurement of excess adsorption at low pressure by either a volumetric or a gravimetric technique is now common practice throughout the world. The terminology “excess” refers to the amount of gas adsorbed relative to helium gas used for the reference state measurements. The small amount of uncertainty introduced by helium adsorption becomes significant at high pressure because excess adsorption is the difference between two numbers having nearly the same value. At high pressure, the density of the gas phase approaches and finally exceeds the density of the adsorbed phase so that the determination of excess adsorption becomes very sensitive to the adsorption of helium. This has led in recent years to a flurry of discussion about the proper way to measure helium adsorption. Should helium adsorption be measured under specific reference conditions set by international agreement, or should helium adsorption be measured as a function of temperature and pressure? The comparison of experimental adsorption data with theory has been hampered by the fact that molecular simulation yields absolute adsorption. Therefore, the molecular simulation of adsorption requires an additional simulation step with helium if the results are to be compared with excess adsorption data.
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Practical applications such as the design of adsorption columns and/or quantifying the capacity of an adsorptive gas storage container (e.g., hydrogen storage container) necessitate many approximations for the interparticle and intraparticle void spaces in addition to the excess adsorption isotherms. The intraparticle void space in particular is problematic to determine because it can be measured only by using a probe gas, thus the complications of helium adsorption are introduced once again. All of these experimental, theoretical, and practical problems vanish in the net adsorption framework. Experiments are simplified because helium measurements need not be performed for each sample. Molecular simulations provide net adsorption directly without any need for helium simulations. In practical applications, net adsorption directly shows the amount of material in the system (in both the adsorbed and gas phase) without the need for any additional approximations. We realize that any attempt to change fundamental techniques for quantifying adsorption will be an uphill battle. Almost all previous experimental data in the literature has been reported as excess adsorption, usually without any clarification about how the helium reference state is determined. However, most of the experimental data in the literature is at low pressure and pertains to heavily adsorbed vapors. The differences among absolute, excess, and net adsorption capacities are significant only at high pressure for supercritical gases. Interest in the adsorptive storage of light gases such as methane and hydrogen has recently stimulated experimental and theoretical studies at high pressure. Most of this data can be converted to net adsorption simply through the impenetrable solid volume for excess adsorption and the additional pore volume for absolute adsorption. Migration to the net adsorption framework will eliminate the confusion and controversy that exist in the measurement of high-pressure adsorption data under supercritical conditions. The confusion is partially due to the adaption of concepts for adsorption on flat surfaces to adsorption in microporous adsorbents. Net adsorption is the natural theoretical framework for quantifying micropore adsorption, which is separate from and different than excess adsorption, the natural framework for adsorption on flat surfaces. Acknowledgment. We gratefully acknowledge discussions with Professor Alan Myers over the years that greatly contributed to the development of concepts presented in this article.
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