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Equilibrium and Kinetics of Water Adsorption in Carbon Molecular Sieve: Theory and Experiment S. W. Rutherford* and J. E. Coons Los Alamos National Laboratory, Engineering Sciences and Applications Division, MS C930, Los Alamos, New Mexico 87545 Received March 15, 2004. In Final Form: July 1, 2004 Measurements of water adsorption equilibrium and kinetics in Takeda carbon molecular sieve (CMS) were undertaken in an effort to characterize fundamental mechanisms of adsorption and transport. Adsorption equilibrium revealed a type III isotherm that was characterized by cooperative multimolecular sorption theory. Water adsorption was found to be reversible and did not display hysteresis upon desorption over the conditions studied. Adsorption kinetics measurements revealed that a Fickian diffusion mechanism governed the uptake of water and that the rate of adsorption decreased with increasing relative pressure. Previous investigations have attributed the observed decreasing trend in the rate of adsorption to blocking of micropores. Here, it is proposed that the decrease is attributed to the thermodynamic correction to Fick’s law which is formulated on the basis of the chemical potential as the driving force for transport. The thermodynamically corrected formulation accounted for observations of transport of water and other molecules in CMS.
1. Introduction The adsorption of water molecules on carbon surfaces is a phenomenon that is relevant to a wide variety of commercial processes.1,2 In tribological applications, for example, the presence of water is believed to enhance the flow of lubricant on the carbon coating of magnetic storage media, thereby providing improved protection against abrasion.3,4 Carbon-supported platinum catalysts employed in fuel cells are also affected by the presence of water vapor. Adsorption of water on carbon is reported to increase platinum utilization, thereby allowing increased fuel cell efficiency.5 In adsorption applications, the presence of water is believed to enhance the storage of methane in activated carbon6 but adversely affects commercial air separation processes that employ carbon molecular sieve, by reducing breakthrough times and lowering production rates.7-9 Despite the relevance to these commercial applications, the fundamental understanding of water adsorption in carbon is still rudimentary.10,11 Current theories relate surface chemistry and structure of carbon to water adsorption affinity.10,11 These theories propose that carbon adsorbents are composed of misaligned aromatic platelets that form a nanoporous network in which oxygenated * Corresponding author: phone (505) 6676124; Fax (505) 6657836; e-mail
[email protected]. (1) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. (2) Yang, R. T. Adsorbents: Fundamentals and Applications; Wiley: New York, 2003. (3) Karis, T. E. J. Colloid Interface Sci. 2000, 225, 196-203. (4) Shukla, N.; Svedberg, E.; van de Veerdonk, R. J. M.; Ma, X. D.; Gui, J.; Gellman, A. J. Tribol. Lett. 2003, 15, 9-14. (5) Maruyama, J.; Abe, I. J. Electroanal. Chem. 2003, 545, 109-115. (6) Zhou, L.; Sun, Y.; Zhou, Y. P. AIChE J. 2002, 48, 2412-2416. (7) Harding, A. W.; Foley, N. J.; Norman, P. R.; Francis, D. C.; Thomas, K. M. Langmuir 1998, 14, 3858-3864. (8) O’Koye, I. P.; Benham, M.; Thomas, K. M. Langmuir 1997, 13, 4054-4059. (9) Foley, N. J.; Thomas, K. M.; Forshaw, P. L.; Stanton, D.; Norman, P. R. Langmuir 1997, 13, 2083-2089. (10) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982. (11) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999.
functional groups are contained.12 The oxygenated functional groups form primary sites for the adsorption of water.10 Surface chemistry and the distribution of the primary adsorption sites play an important role in the mechanism of water adsorption according to molecular simulation.13-15 At very low relative pressures, the polar nature of the water molecule allows it to bond with individual oxygenated functional groups. Henry’s law is followed at very low pressures, and the amount of water adsorbed is determined by the total number of primary adsorption sites. At higher relative pressures, hydrogen bonding between free and adsorbed water molecules occurs and the formation of clusters begins. At even higher relative pressures, water clusters grow and hydrogen bonding between clusters occurs. The nature of the resulting equilibrium isotherm is dependent not only on the total number of primary adsorption sites but also on their surface density.13-15 In carbon materials with high adsorption site densities, type II (in the BET classification scheme) equilibrium isotherms are observed.16-18 Carbons with low site density produce type III or V equilibrium isotherms, and carbons with medium site density produce hybrid equilibrium isotherms.16,19 Attempts have been made to model water-carbon interactions and to capture the type II, III, V, and hybrid behavior. Methods based on Dubinin’s theory extended to water adsorption20-23 have (12) Marsh, H., Ed.; Introduction to Carbon Science; Butterworths: London, 1989. (13) Muller, E. A.; Rull, L. F.; Vega, L. F.; Gubbins, K. E. J. Phys. Chem. 1996, 100, 1189-1196. (14) McCallum, C. L.; Bandosz, T. J.; McGrother, S. C.; Muller, E. A.; Gubbins, K. E. Langmuir 1999, 15, 533-544. (15) Brennan, J. K.; Thomsom, K. T.; Gubbins, K. E. Langmuir 2002, 18, 5438-5447. (16) Bandosz, T. J., Jagiello, J.; Schwarz, J. A.; Krzyzanowski, A. Langmuir 1996, 12, 6480-6486. (17) Salame, I. I.; Bandosz, T. J. J. Colloid Interface Sci. 1999, 210, 367-374. (18) Salame, I. I.; Bagreev, A.; Bandosz, T. J. J. Phys. Chem. B 1999, 103, 3877-3884. (19) Salame, I. I.; Bandosz, T. J. Langmuir 2000, 16, 5435-5440. (20) Dubinin, M. M.; Zaverina, E. D.; Serpinsky, V. V. J. Chem. Soc. 1955, 1760-1766. (21) Dubinin, M. M. Carbon 1980, 18, 355-364. (22) Dubinin, M. M.; Serpinski, V. V. Carbon 1981, 19, 402. (23) Stoeckli, F. Carbon 1998, 36, 363-368.
10.1021/la049330d CCC: $27.50 © 2004 American Chemical Society Published on Web 08/24/2004
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Figure 1. Schematic of experimental apparatus.
been employed but cannot account for Henry’s law behavior.24 Molecular simulations have also been employed,13-15,25 but these techniques are computationally expensive and have difficulty predicting equilibrium behavior at low relative pressure.14 Cooperative multimolecular sorption (CMMS) theory has been recently employed to account for type II, III, V, and hybrid equilibrium isotherms observed in carbon-water systems.26 In this study, we apply CMMS theory to characterize the equilibrium adsorption isotherm of water in a carbon molecular sieve (CMS). Water adsorption kinetics are also studied in an effort to characterize fundamental mechanisms of adsorption and transport. 2. Experimental Section A volumetric, batch adsorption apparatus was constructed as indicated in the schematic of Figure 1. The apparatus consists of a sample chamber connected to two dosing volumes by a series of valves. The sample and dosing chambers consist of Swagelock cylinders modified by the addition of seal-welded VCR end fittings for high-vacuum holding capability. Swagelock BG series valves capable of operating up to 315 °C under high pressures and vacuum are used to regulate vapor flow into the dosing and sample chambers. The system is connected to two pressure transducers (10 and 1000 Torr, MKS Baratron type 615A) capable of measuring pressures up to 1000 Torr. The system includes a vacuum source generated by a Varian turbo pump (V70LP) and a Varian mechanical diaphragm pump (MDP 30). The vapor source consists of a glass vessel that transitions to a Swagelock VCR face seal fitting. The vapor source is connected to the system with Teflon-lined vacuum bellows and a 60 µm frit gasket that buffers against rapid pressure changes when flowing gas to the dosing chamber. The vessel is initially filled with double distilled water, allowing a small amount of headspace, and exposed to vacuum for a period of 5 days to remove dissolved gases. The water is assumed to be free of dissolved gases when pressure in the headspace is equal to the known water vapor pressure at ambient temperature (17.5 Torr at 20 °C).27 Operation of the volumetric adsorption apparatus was initiated by degassing the whole system (not including the vapor source) to less than 10-9 Torr. Water vapor was expanded into the dosing chamber until the desired pressure was reached. The sample chamber was subsequently exposed, and the pressure drop was monitored as adsorption of water vapor into the sample occurred. By this method, the sample was incrementally loaded with water vapor. Once the maximum desired loading was reached, the pressure in the dosing chamber was reduced to less than that in the sample chamber, and the valve between the dosing and sample chambers was opened. During desorption, the system (24) Kapoor, A.; Ritter, J. A.; Yang, R. T. Langmuir 1989, 5, 11181121. (25) Striolo, A.; Chialvo, A. A.; Cummings, P. T.; Gubbins, K. E. Langmuir 2003, 19, 8583-8591. (26) Rutherford, S. W. Carbon 2003, 41, 622-625. (27) Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York, 1985.
Rutherford and Coons pressure increased as removal of adsorbed water from the sample took place. The dynamic pressure change was logged to a PC using National Instruments hardware and software. The system volume was determined by helium expansion into a previously calibrated vessel. The batch adsorption experiment was conducted differentially as outlined in Rutherford and Do.28 The increments in pressure were taken small enough such that an approximately linear relationship existed between the change in the gas and adsorbed phase concentrations. It was assumed that under these conditions the diffusivity was held constant throughout each increment. Also, small absolute changes in pressure ensured that small amounts were adsorbed and hence that the adsorbent temperature remained constant. 2.1. Materials. The adsorbent investigated in this study is a carbon molecular sieve manufactured by Takeda Chemical Co. The surface chemistry of the Takeda CMS has not been investigated, and hence the type and quantity of surface groups are unknown. However, the material has been studied in a previous investigation by application of gas permeation, gas adsorption, and mercury intrusion to characterize pore size.28 The pore volume is derived from micropores (of size around 0.5 nm) and macropores (of size around 0.27 µm), with negligible mesopore content. The micropores of the Takeda CMS are contained within a grain structure consisting of crystalline and amorphous carbon,12 and the macropores form in the voids between grains. In this investigation, disks of Takeda CMS with radius 0.09 cm and thickness 0.1 cm were studied and were initially outgassed for a period of 14 days. The sample was also outgassed for at least 60 h at less than 10-9 Torr between each of the five adsorption experiments performed. 2.2. Mass Transfer Considerations. In the single gas batch adsorption experiment, the process of mass transfer in carbon molecular sieve may be rate limited by macropore diffusion, micropore diffusion, or transport through the pore mouth barrier. In many cases the first two mechanisms control.1 However, large molecules transported in CMS may be restricted from entry into the micropore by a barrier at the pore mouth. Characteristic of this barrier is a kinetic uptake which adheres to a non-Fickian linear driving force (LDF) model.29-37 LDF kinetics were not observed in this investigation, so we disregard the influence of the pore mouth barrier in water uptake. The influence of macropore diffusion can be assessed using the procedure suggested by Ruthven1 in which the ratio of macropore to micropore diffusion resistance is calculated. For Takeda CMS, this has been calculated by Rutherford and Do,28 and it was found that the influence of macropore diffusion in the transport of carbon dioxide, oxygen, nitrogen, and argon can be disregarded. For water transport, the value of the macropore to micropore mobility ratio calculated for this investigation is around 0.08. On the basis of this calculation, consideration of macropore diffusion for water transport was not included. Similar to carbon dioxide, oxygen, nitrogen, and argon,38,39 the rate-limiting process of water transport in CMS was attributed to micropore diffusion. 2.3. Consideration of Heat Transfer and Adsorption on Chamber Walls. It is known that the exothermic adsorption (28) Rutherford, S. W.; Do, D. D. Langmuir 2000, 16, 7245-7254. (29) Koresh, J.; Soffer, A. J. Chem. Soc., Faraday Trans. 1981, 77, 3005-3018. (30) LaCava, A. I.; Koss, V. A.; Wickens, D. Gas Sep. Purif. 1989, 3, 180-186. (31) Chagger, H. K.; Ndaji, F. E.; Sykes, M. L.; Thomas, K. M. Carbon 1995, 33, 1411. (32) Liu, H., Ruthven, D. M. In Proceedings of the 5th International Conference of the Fundamentals of Adsorption; Le Van, M. D., Ed.; Kluwer Press: Boston, 1996. (33) Loughlin, K. F.; Hassan, M. M.; Fatehi, A. I.; Zahur, M. Gas Sep. Purif. 1993, 7, 264. (34) Srinivasan, R.; Auvil, S. R.; Schork, J. M. Chem. Eng. J. 1995, 57, 137-144. (35) Reid, C. R.; Thomas, K. M. Langmuir 1999, 15, 3206-3218. (36) Reid, C. R.; Thomas, K. M. J. Phys. Chem. B 2001, 105, 1061910629. (37) Rutherford, S. W.; Nguyen, C.; Coons, J. E.; Do, D. D. Langmuir 2003, 19, 8335-8342. (38) Rutherford, S. W.; Coons, J. E. Carbon 2003, 41, 405-411. (39) Ruthven, D. M. Chem. Eng. Sci. 1992, 47, 4305-4308.
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Table 1. BET Classification and Characteristics of the CMMS Isotherm and Examples of the Simplified Forms of the CMMS Isotherm BET classification
isotherm
type I, II, III, V, and hybrid types type I
CMMS
type II
extended BET
type III
Ising
type V
Ising
type II/III
extended BET
type II/V
CMMS
Langmuir
characteristics
K0
K1
Kas
low to high site density with or without side association high primary site density with no side association allowed high primary site density with side association allowed low primary site density with no side association allowed low primary site density with no side association allowed medium primary site density with side association allowed medium primary site density with side association allowed
0 to ∞
0 to ∞
0 to 1
50
50
0
10
10
0.77
0.05
1
0
0.057
1.71
0
1
1
0.88
0.27
3.5
1
process liberates heat which can interfere with mass transfer. The influence of heat transfer can be assessed using the protocol of Ruthven (1984). This has been performed by Rutherford and Do40 for Takeda CMS. We estimate a heat transfer coefficient to be on the order of 10 W/(m2 K)41 and consider the CMS to have thermal properties of coal.41 We also estimate the isosteric heat of adsorption around 15 kJ/mol, which is a value obtained by molecular simulation under similar water concentrations and pore size.25 With these considerations, we calculated that increments less than 0.2 mmol/g would yield isothermal conditions under which consideration of heat transfer would not be required. Hence, increments less than 0.2 mmol/g were taken in this investigation. It is also known that water vapor can adsorb on metal surfaces,42 and it is therefore necessary to consider the possibility of water adsorption on the walls of the dosing and sample chambers. Separate experiments have been performed that involve expansion of water from the vapor source into the dosing chamber. The pressure is monitored continuously over the time scales of water adsorption in Takeda CMS (around 1000 s). The pressure change over this period is negligible in comparison to the pressure changes during exposure of water to Takeda CMS, and we therefore ignore the influence of water adsorption on the chamber walls.
3. Water Adsorption Equilibrium Water adsorption equilibrium is inherently related to the surface chemistry of carbon.10 Oxygenated functional groups distributed within the carbon form the primary adsorption sites and govern the degree of water affinity and the nature of the equilibrium isotherm.10,11 Types II, III, and V of the BET classification scheme have been observed in carbon, and there are few theories that can account for all of these observed equilibrium types. One exception is the CMMS theory proposed by Malakhov and Volkov43 to explain the uptake of alcohols in a high free volume polymer. This theory has been subsequently applied to water-carbon systems and employed to characterize the various types of equilibrium isotherms observed.26 It was shown that CMMS theory can account for type II, III, V, and hybrid equilibrium isotherms which fall between these types. Additionally, the parameters derived from fitting the equilibrium data were shown to be correlated with the primary adsorption site density.26 CMMS theory considers a primary site triad that is composed of a central site with side units on either side of the central site. In the case of water adsorption in carbon, the central site is represented by the oxygenated functional (40) Rutherford, S. W.; Do, D. D. Carbon 2000, 38, 1339-1350. (41) Levenspiel, O. Engineering Flow and Heat Exchange; Plenum Press: New York, 1984. (42) Holloway, S.; Bennemann, K. H. Surf. Sci. 1980, 101, 327-333. (43) Malakhov, A. O.; Volkov, V. V. J. Polym. Sci., Ser. A 2000, 42, 1120-1126.
group. Secondary interactions are also incorporated by inclusion of side associates which form from the unit triad and allow for the formation of dimers, trimers, etc. On the basis of these consideration, the CMMS equilibrium isotherm equation can be derived and represented by the following analytical form:
θ)
K0a C (1a) ) Csat (1 - K a)[K a + w2(1 - K a)] as 0 as
where θ is the fractional loading, C is the concentration of sorbate in the adsorbent which is in equilibrium with the gas phase, Csat is the adsorption capacity of the primary adsorption sites, a is the relative pressure (i.e., the ratio of gas-phase pressure to saturated vapor pressure), K0 is the equilibrium constant for sorption of the central unit on the primary site, and Kas is the equilibrium constant for sorption of the side associate. The parameter w is given by
w)
(
K1a 1 + 12 1 - Kasa
x(
1-
)
K1a 1 - Kasa
2
+
)
4K0a 1 - Kasa (1b)
where K1 is the equilibrium constant for sorption of the side unit on the primary site. The versatility of the CMMS isotherm equation is demonstrated by its transformation to simpler equilibrium isotherm equations by examining limits and manipulating equilibrium constants. At the low-pressure limit (a f 0), the CMMS equation reduces to a linear isotherm representing Henry’s law. The Henry’s law limit is a thermodynamic requirement24 that is not met by other isotherm equations such as those based on Dubinin’s theory.20-23 At higher pressures, the Langmuir isotherm equation, which is representative of type I equilibrium, is obtained by setting Kas to 0 and equating K0 and K1. The extended BET isotherm,44,45 which is representative of type II equilibrium, is obtained by equating K0 and K1, and the standard BET equation is obtained by equating K0 and K1 and assigning Kas equal to 1. The Ising equation, which is representative of type III or V equilibrium, is obtained setting Kas to 0. To show the functional form for the various isotherm types accounted for by the CMMS isotherm equation, example isotherms were chosen according to Table 1 and plotted in Figure 2. Table 1 shows the general CMMS (44) Anderson, R. B. J. Am. Chem. Soc. 1946, 68, 686. (45) Brunauer, S.; Skalny, J.; Bodor, E. E. J. Colloid Interface Sci. 1969, 30, 546.
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Figure 2. Example equilibrium isotherms predicted by the CMMS isotherm equation (expressed as eqs 1a and 1b) showing the various possible isotherm types in the BET classification. The values for each example isotherm type are shown in Table 1.
Rutherford and Coons
results in an increasing isotherm slope with increasing pressure. If the micropores in which the primary sites are contained are large enough but still on the order of molecular size, large amounts of side association can occur. If the micropores are much larger than molecular dimensions, condensation can also occur at pressures approaching the saturated vapor pressure. Hysteresis in the equilibrium isotherm is also observed when condensation occurs. The micropores of Takeda 3A CMS are sized on the order of molecular dimensions, and there is negligible mesoporosity. It is therefore expected that side association would be hindered, and condensation and the associated hysteresis would not be observed. The equilibrium data points shown in Figure 3 were obtained from adsorption and desorption measurements, and no appreciable hysteresis is evident. The absence of hysteresis in the adsorption of water in CMS was also reported by O’Koye et al.8 Under conditions of minimal side association and absence of hysteresis, the Ising equation can be employed to characterize the isotherm and is therefore fitted to the data of Figure 3. The numerical fitting is based on a leastsquares minimization procedure which adjusts the parameters K0, K1, and Csat to obtain the minimum sum of squares error. It is evident that the theory can characterize the equilibrium isotherm, and the isotherm parameters obtained from the fitting are presented in Table 2. The isotherm constants derived in a previous study26 from data obtained by Bandosz et al.16 for Westvaco carbon are presented for comparison in Table 2. The values of the isotherm constants determined for both carbon materials are similar. 4. Kinetics of Water Adsorption
Figure 3. Amount of water adsorbed in Takeda CMS at 20 °C for various values of relative pressure. The solid line represents the fit of a simplified form of the CMMS isotherm equation.
equation, the isotherm types covered by the theory, and the range of acceptable values for the equation parameters. Also included are the simpler forms of the equation including the Langmuir isotherm which can be employed to characterize carbons with high primary site density where no side association is allowed. When side association is allowed, the extended BET isotherm could be employed to characterize carbons with high primary site density. Also included in the table is the Ising equation which can be applied to carbons with low primary site density when no side association is allowed. 3.1. Water Adsorption Equilibrium in CMS. The adsorption equilibrium isotherm for water in Takeda CMS at 20 °C was measured and is shown in Figure 3. According to the BET classification scheme, the isotherm is type III which corresponds to a material with low density of primary adsorption sites. At low pressures the isotherm appears to be linear, indicative of Henry’s law and the binding of the water molecules to the primary adsorption sites. At intermediate pressure, the cooperative effect is induced by the formation of side associates and hydrogen bonding of water molecules with adsorbed water. This
Kinetic uptake data reveal important information relating to transport and adsorption mechanisms and are essential for characterizing the kinetic selectivity of CMS. Previous transport studies focused on CMS propose that the two main barriers for mass transfer are (1) diffusion through the micropores34 and (2) transport through the barrier at the pore mouth.29-37 As stated in section 2.2, the rate-limiting process in the mass transfer of water in Takeda CMS is attributed to micropore diffusion. Additionally, the uptake experiments were conducted with small differential increments of pressure to negate the concentration dependence of the diffusivity throughout the uptake period. Furthermore, the total pressure change is kept at less than 10%, allowing an effectively constant boundary condition. For this case, the fractional uptake of water (F) can be expressed by Fick’s law applied to a spherical particle:1
F)1-
6
∞
∑
1
π2n)1n2
(
exp -
)
Dµn2π2t Rµ2
(2)
where Dµ is the diffusivity of the adsorbed species in the micropore and Rµ is the radius of the grain. Often the grain radius is unknown, and it is convenient to define the mobility parameter, Dµ/Rµ2, which is used for analysis. At large times (t f ∞), the fractional uptake (F) is given by1
ln(1 - F) ) ln
()
Dµπ2t 6 π2 Rµ2
(3)
If the uptake is Fickian, a plot of the left-hand side of eq 3 vs time will be linear with an intercept of ln(6/π2) and
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Table 2. Isotherm and Kinetic Parameters Obtained from Optimization adsorbate
T (°C)
adsorbent
water
20
water
25
carbon dioxide
20
Takeda CMS, type III isotherm (this investigation) Westvaco carbon, type III isotherm (taken from ref 16) Takeda CMS, type I isotherm (taken from ref 28) Takeda CMS- Linear isotherm (taken from ref 28)
oxygen a
20
Using value of Pv ) 19.2 atm.27
b
Ko
Kas
0.02
1.45
0
48
0.0016
0.072
1.55
0
40.2
N/A
)K0
0
2.45
0.0003
)K0
0
N/A
0.0013
96a 0.00076b
Csat (mmol/g)
Dµ0/Rµ2 (/s)
K1
Supercritical conditions (Pv unavailable), so K0Csat/Pv reported in units of mmol/(g Torr).
a slope proportional to the mobility parameter for micropore diffusion. 4.1. Thermodynamically Corrected Formulation. Errors arise in the employment of Fick’s law at high pressures if the assumption of constant diffusivity is rigidly maintained. This is evident when the chemical potential is used as the true driving force for diffusion, which leads to the following thermodynamic correction:46
Dµ 2
)
Rµ
Dµ0 ∂(ln f) Rµ2 ∂(ln C)
(4)
where f is the fugacity of the vapor phase in equilibrium with the adsorbed phase and Dµ0/Rµ2 represents the zero loading mobility parameter. For single-component measurements, the relationship becomes equivalent to the single-component Maxwell-Stefan expression,46,47 also known as the Darken relation.48 These expressions relate the diffusivity to the fractional loading and relative pressure as follows:
Dµ 2
Rµ
)
Dµ0 ∂(ln a) Rµ2 ∂(ln θ)
(5)
Transport of molecules that experience hindrance from the pore mouth barrier in CMS obeys a non-Fickian uptake which can be characterized by a LDF rate constant (k).29-37 The thermodynamic correction has also been applied to the LDF rate equations to characterize the dependence of the LDF rate constant upon pressure:34
∂(ln a) k ) k0 ∂(ln θ)
(6)
where k0 is the zero loading LDF rate constant. Equations 5 and 6 are referred to in this investigation as the thermodynamically corrected formulations for Fickian diffusivity and LDF rate constant, respectively. Evaluation of these formulations requires an equilibrium isotherm to relate the relative pressure to the fractional loading. By applying the CMMS equilibrium isotherm equation, the thermodynamically corrected formulations can be evaluated by substitution of eqs 1a and 1b into either of eq 5 or eq 6. However, analytical expressions cannot be obtained, and numerical methods are required for evaluation. For the simpler case of Langmuir equilibrium behavior, the following analytical equation is derived:49
Dµ k ) ) 1 + K0a Dµ0 k0
(7)
(46) Karger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992. (47) Krishna, R.; Wesselingh, J. A. Chem. Eng. Sci. 1997, 52, 861911.
Figure 4. Dependence of the Fickian diffusivity and the LDF rate constant upon relative pressure as predicted by the thermodynamically corrected formulation for each isotherm type. Plots are obtained by numerical solution of eq 5 and eqs 1a and 1b for the example isotherms shown in Table 1.
Equation 7 predicts a constant diffusivity or LDF rate constant at low pressures (K0a , 1) where Henry’s law applies and linear isotherms are observed. This is a requirement that is met by employing the CMMS isotherm equation but would not be met by employing isotherms based on Dubinin’s theory extended to water adsorption20-23 because they cannot account for Henry’s law behavior.24 Equation 7 also predicts a linear increase in the diffusivity or LDF rate constant with relative pressure. This is shown in Figure 4, which plots the predicted dependence of the Fickian diffusivity and LDF rate constant for the example equilibrium relationships shown in Figure 2. For type I equilibrium, the diffusivity and rate constant are predicted to increase linearly with relative pressure. This linear behavior has been observed in the Fickian diffusivity46 and in the LDF rate constant by Srinivasan et al.34 for nitrogen in CMS. For type II equilibrium, the observed diffusivity and rate constant are predicted to increase and then decrease with relative pressure. This has been observed in the Fickian diffusivity for gases in zeolites46 and in the LDF rate constant for gases in CMS.35,36 For type III equilibrium, which has been observed for water in CMS,7 the diffusivity and LDF rate constant are predicted to decrease with increasing relative pressure up to high relative pressures. This is consistent with measurements of the LDF rate constant by Harding et al.7 for water in CMS (bax950 carbon). For type V equilibrium, the diffusivity and rate constant are predicted to decrease until a relative pressure of around 0.6 is reached, at which point it begins to increase. This (48) Darken, L. Trans. AIME 1948, 74, 184. (49) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1978.
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Rutherford and Coons
Figure 6. Mobility parameter (Dµ/Rµ2) for adsorption and desorption of water in Takeda CMS at 20 °C for various values of relative pressure. The solid line represents the prediction of eq 5.
Figure 5. (a) Adsorption experiment at 20 °C and 5.06 Torr. Circles represent data points, and the solid line represents the fit of the large time Fickian diffusion equation expressed as eq 3. (b) Desorption experiment at 20 °C and 5.03 Torr. Circles represent data points, and the solid line represents the fit of the large time Fickian diffusion equation expressed as eq 3.
behavior in the LDF rate constant has been observed for water in CMS by Harding et al.7 and O’Koye et al.,8 who have attributed the observation to blockage of pores by clusters of water molecules. In this investigation, we apply the thermodynamically corrected formulation to characterize this behavior and assess the formulation by comparison with measurements of mobility parameter at a range of pressures. 4.2. Kinetics of Water Adsorption in CMS. The uptake of water in Takeda CMS at 20 °C was measured over a range of pressures, and the kinetic uptake curves for adsorption and desorption appear in parts a and b of Figure 5, respectively. The data were fit to eq 2 and were found to be in agreement as evidenced by conformity with the large time asymptote also included in the figures. The resulting mobility parameter, obtained from both adsorption and desorption experiments, is plotted in Figure 6 against the relative pressure. Similar values were obtained for both adsorption and desorption experiments, which is indicative of a Fickian diffusion process such as micropore diffusion.46 It also appears that the mobility parameter decreases with relative pressure. The solid line in Figure 6 represents the mobility parameter resulting from the least-squares fit of eq 5 coupled with eqs 1a and 1b which have fixed equilibrium constant values shown in Table 2. Only the zero loading mobility parameter was adjusted
Figure 7. Adsorption equilibrium isotherms for water (obtained in this investigation), oxygen, and carbon dioxide (taken from Rutherford and Do28) in Takeda CMS at 20 °C for various values of relative pressure. The solid lines represent the prediction of simple forms of the CMMS isotherm for each species. Oxygen data have been fitted with a linear isotherm, carbon dioxide data with a Langmuir isotherm, and water data fitted with the Ising equation.
in the fitting procedure, and the optimum value was determined to be 0.0016 per second. It is evident from Figure 6 that the thermodynamically corrected formulation accurately captures the decreasing trend in the mobility parameter for water adsorption. 5. Comparison of Water with Other Species The mechanism of adsorption of nonpolar species in carbon solids differs from the mechanism for polar species such as water molecules. Polar molecules may interact with the oxygenated functional groups, and nonpolar molecules interact with the aromatic platelets that form the graphitic network of carbon solids.50 As there is a greater amount of graphitic carbon than there are functional groups, there are a greater number of adsorption sites for nonpolar species than polar species. As a result, the adsorption site density is higher for nonpolar species, and therefore type I or II isotherms are commonly (50) Rao, M. B.; Jenkins, R. G.; Steele, W. A. Langmuir 1985, 1, 137-141.
Water Adsorption in Carbon Molecular Sieve
Figure 8. Mobility parameter (Dµ/Rµ2) for adsorption of water (obtained in this investigation), oxygen, and carbon dioxide (taken from Rutherford and Do28) in Takeda CMS at 20 °C for various values of relative pressure. The solid lines represent the prediction of eq 5 for each species.
observed. In Takeda 3A CMS, carbon dioxide has been shown to display type I equilibrium,28 as is shown in Figure 7. A characterizing isotherm of type I behavior is the Langmuir equation which has been fitted to these data and the parameters displayed in Table 2. Also shown in Figure 7 is the isotherm data for oxygen. These data have been shown to be linear in the range of pressures shown,28 and the Henry’s law constant for oxygen is indicated in Table 2. The comparison of all three adsorbates indicates that although the number of sites for polar water molecules to adsorb is fewer, the cooperative nature of the water adsorption allows many molecules to bond with the primary adsorption sites, and hence a comparable amount of water can be adsorbed. The comparison of adsorbates in Figure 7 also shows that CMMS theory and the simpler isotherms derived from this theory can account for the adsorption of polar and nonpolar species in carbon. The uptake of nonpolar oxygen and carbon dioxide molecules in Takeda CMS displays characteristics of micropore diffusion according to previous studies.28 The thermodynamically corrected formulation has been shown to characterize the pressure dependence of the diffusivity across a wide range of pressures.28 This is shown in Figure
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8, which indicates a constant mobility parameter for oxygen due to the Henry’s law equilibrium within the range shown. Carbon dioxide displays a type I isotherm which is characterized by the Langmuir equation and a mobility parameter that increases with pressure. The thermodynamically corrected formulation predicts a linear increase in diffusivity with pressure, and this formulation appears to accurately characterize the data as shown in Figure 8. Figure 8 also shows the mobility parameter for water vapor and highlights the unique decreasing trend with increasing pressure. According to the thermodynamically corrected formulation, this unique trend is a result of the type III equilibrium. By comparison of all three adsorbates in Figure 8, it is clear that the thermodynamically corrected formulation coupled with the CMMS equation can account for transport of polar and nonpolar species. 6. Conclusion In this investigation, a quantitative method for characterizing water adsorption equilibrium and kinetics in carbon adsorbents is proposed and applied to measurements of water adsorption and desorption in Takeda CMS. The equilibrium data indicate a reversible type III isotherm, which was characterized by the CMMS equilibrium isotherm equation. The kinetics of water adsorption and desorption were found to be Fickian in the range of measurement, and the resulting mobility parameters decreased with increasing pressure. The decreasing trend in the values for the mobility parameter was accurately predicted by the thermodynamically corrected formulation that was developed using the CMMS isotherm equation. Agreement with the type III adsorption behavior demonstrates the versatility of CMMS theory, which can also be applied to characterize type I isotherms observed in CMS. In the case of water, oxygen, and carbon dioxide sorbates in CMS, the thermodynamically corrected formulation was shown to correctly represent uptake not only at low sorbate concentrations where Henry’s law applies and the mobility parameter becomes independent of pressure but also at higher pressures where more complex behavior is observed. LA049330D
