Binary Isosteric Heats of Adsorption in Carbon Predicted from Density

James A. Ritter and Huanhua Pan , Perla B. Balbuena. Langmuir 2010 26 (17), 13968-13975 ... Yufeng He and Nigel A. Seaton. Langmuir 2005 21 (18), 8297...
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Binary Isosteric Heats of Adsorption in Carbon Predicted from Density Functional Theory Huanhua Pan, James A. Ritter, and Perla B. Balbuena* Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, South Carolina 29208 Received October 27, 1998. In Final Form: April 8, 1999 The nonlocal density functional theory of Kierlik and Rosinberg was used to predict isosteric heats of adsorption (qi) of three binary gas mixtures: (1) CO2-C2H4, (2) CH4-C2H6, and (3) CH4-C3H8 in homogeneous and heterogeneous carbons at 350 K and 1 bar. The qi’s of the components in the mixture were different from their pure state qi°’s and showed complex behavior for the nonideal systems (2 and 3). Adsorbent heterogeneity also played an important role in determining the behavior of the qi’s compared with the qi°’s. The differences were attributed to effects caused by the loading of the opposing component and by differences in the intermolecular forces between the adsorbate molecules.

Introduction Knowledge of the heats of adsorption of the components in a multicomponent gas mixture is critical in the design and control of gas separation processes, such as pressure swing adsorption (PSA) or thermal swing adsorption (TSA). This is especially true when bulk concentrations are present, which usually makes thermal effects more pronounced.1 However, because of the difficulties involved in measuring the heats of adsorption of the components in a mixture, there are essentially no experimental data available in the literature, except for that obtained recently by Dunne et al.2 Sircar proposed some thermodynamic expressions to calculate the isosteric heats of adsorption of the components in binary gas mixtures,3 and later used them along with an extended Toth model to predict binary isosteric heats of adsorption;1 Al-Muhtaseb and Ritter4 extended a virial-type model5 and derived expressions for predicting single component and multicomponent isosteric heats of adsorption. Both of these approaches require only pure component adsorption isotherm parameters to predict multicomponent isosteric heats of adsorption; however, the accuracy of these models is questionable because there are essentially no experimental data available for comparison. Molecular simulation has also been used for predicting isosteric heats of adsorption on different adsorbents,6-11 but only the study by Karavias and Myers7 determined the isosteric heats of adsorption of the components in several binary gas mixtures on type X zeolite. There has * To whom correspondence should be addressed. E-mail: [email protected]. (1) Sircar, S. Langmuir 1991, 7, 3065. (2) Dunne, J. A.; Rao, M.; Sircar, M.; Gorte, R. J.; Myers, A. L. Langmuir 1997, 13, 4333. (3) Sircar, S. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527. (4) Al-Muhtaseb, S. A.; Ritter, J. A. Ind. Eng. Chem. Res. 1998, 37, 684. (5) Taqvi, S. M.; LeVan, M. D. Ind. Eng. Chem. Res. 1997, 36, 2197. (6) Razmus, D. M.; Hall, C. K. AIChE J. 1991, 37, 769. (7) Karavias, F.; Myers, A. L. Langmuir 1991, 7, 3118. (8) Matranga, K. R.; Karavias, F.; Stella, A.; Segarra, E. I. In Fundamentals of Adsorption; Suzuki, M., Ed.; Elsevier: New York, 1992. (9) Bottani, E. J.; Bakaev, R.; Steele, W. Chem. Eng. Sci. 1994, 49, 2931. (10) Vuong, T.; Monson, P. A. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer Academic Publishers: Boston, 1996. (11) Myers, A. L.; Calles, J. A.; Calleja, G. Adsorption 1997, 3, 107.

also been interest in using density functional theory (DFT) for predicting single component isosteric heats of adsorption;12,13 however, none of these studies have dealt with binary gas mixtures. Therefore, the objective of this study is to demonstrate the use of DFT for predicting the isosteric heats of adsorption of the components in binary gas mixtures adsorbed on carbon materials. Nonlocal density functional theory (NDFT) has been relatively successful in describing simple fluids confined in narrow pores,14-16 in deriving pore size distributions of adsorbents,17 in predicting mixed-gas adsorption on carbon materials,18-21 and, as stated above, in predicting single component isosteric heats of adsorption.12,13 There are several different versions of NDFT,22,23 but only the Kierlik and Rosinberg22,24 version is readily applicable to multicomponent fluids, because its weighting functions are purely geometric and characteristic of each type of molecule. Therefore, the NDFT of Kierlik and Rosinberg (hereafter referred to as KRDFT) is used to predict the adsorption isotherms and the corresponding isosteric heats of adsorption of three binary mixtures: CO2-C2H4, CH4C2H6, and CH4-C3H8 in a homogeneous carbon comprised of slit-shaped pores, and in a heterogeneous activated carbon with a multinodal pore size distribution.13 The effects of pore size and adsorbed phase loading on the isosteric heats of adsorption of the components in these binary gas mixtures are studied. (12) Balbuena, P. B.; Gubbins, K. E. Langmuir 1993, 9, 1801. (13) Pan, H.; Ritter, J. A.; Balbuena, P. Ind. Eng. Chem. Res. 1998, 37, 1159. (14) Evans, R.; Tarazona, P. Phys. Rev. Lett. 1984, 52, 557. (15) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789. (16) Evans, R.; Marini Bettolo Marconi, U. J. Chem. Phys. 1986, 84, 2376. (17) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (18) Kierlik, E.; Rosinberg, M. L. Mol. Phys. 1992, 75, 1435. (19) Jiang, Shaoyi; Gubbins, K. E.; Balbuena, P. B. J. Phys. Chem. 1994, 98, 2403. (20) Balbuena, P. B.; Gubbins, K. E.; Jiang, Shaoyi; Sowers, S. Adsorption News 1994, 4, 6. (21) Sowers, S. L.; Gubbins, K. E. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer Academic Publishers: Boston, 1996. (22) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1990, 42, 3382. (23) Davis, H. T. Statistical Mechanics of Phases, Interfaces, and Thin Films; VCH Publishers: New York, 1996. (24) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1991, 44, 5025.

10.1021/la981511q CCC: $18.00 © 1999 American Chemical Society Published on Web 05/26/1999

Binary Isosteric Heats of Adsorption in Carbon

Langmuir, Vol. 15, No. 13, 1999 4571

Theory The framework of KRDFT is similar to other versions of DFT. It is based on the idea that the grand free energy of an inhomogeneous fluid can be expressed as a functional of the density profile in the pore, where the fluid-fluid interactions are separated into repulsive and attractive contributions. The attractive part is treated in the meanfield approximation, and the repulsive part of the intrinsic Helmholtz free energy is modeled by the free-energy functional of a reference hard sphere fluid. For a multicomponent fluid under an external potential vi(r), and at a fixed temperature T and chemical potential µi, the grand potential functional Ω is22

Ω[{Fi}] ) FHS[{Fi}] +

∫∫dr dr′ Fi(r) Fj(r′)φattr ij (|r r′|) + ∫dr Fi(r)[vi(r) - µi] (1)

1 2

where FHS is the hard sphere Helmholtz free energy and Fi is the density of species i. In KRDFT, the excess contribution to FHS[{Fi}] is

Fex HS[{Fi}] )

∫dr kTΦ({nR(r)})

∑i ∫dr′ Fi(r′)

ω(R) i (r

- r′)

n1n2 n32 1 + (4) 1 - n3 24π (1 - n )2 3

∑i

FiR(R) i

(5)

and

4 (1) (2) 2 (3) 3 R(0) i ) 1, Ri ) Ri, Ri ) 4πRi , Ri ) πRi 3

(6)

where Ri is the radius of hard sphere species i. The four weighting functions ω(R) i (r) are related to the successive derivatives of the Heaviside step function θ(r) as24

ω(3) i (r) ) θ(Ri - r)

(7a)

ω(2) i (r) ) δ(Ri - r)

(7b)

ω(1) i (r) ) ω(0) i (r) ) -

/k (K)

Tc (K)

Pc (bar)

Vc (L/mol)

28.0 190 137 205 230 254 72.9 61.9 75.8 80.2 84.3 197.4 177.5 186.5

304.2 190.6 282.4 305.4 369.8

73.76 80.96 50.36 48.84 42.46

0.094 0.099 0.129 0.148 0.203

a

Viscosity data.

The fluid-fluid pair interaction uij is described by a cut and shifted Lennard-Jones (LJ) potential, 1/6 uij(r) ) -ij - uLJ ij (rc), r < 2 σij

1 δ′(Ri - r) 8π

1 1 δ′′(Ri - r) + δ′(Ri - r) 8π 2πr

(7c) (7d)

(8a)

LJ 1/6 ) uLJ ij (r) - uij (rc), 2 σij < r < rc

(8b)

) 0, rc < r

(8c)

with

[( ) ( ) ]

(3)

n0, n1, n2, and n3 are the reduced variables of the SPT, with

nR )

σ (Å) 3.4 3.996 3.822 4.232 4.418 5.061 3.698 3.611 3.816 3.909 4.231 4.114 4.120 4.442

uLJ ij (r) ) 4ij

The functional form of Φ is taken from scaled-particle theory (SPT)25 or, equivalently, from the Percus-Yevick compressibility equation of state as26,27

Φ ) -n0 ln(1 - n3) +

interaction C-C CO2-CO2 CH4-CH4 C2H4-C2H4 C2H6-C2H6 C3H8-C3H8 C-CO2 C-CH4 C-C2H4 C-C2H6 C-C3H8 CO2-C2H4 CH4-C2H6 CH4-C3H8

(2)

where kTΦ is the excess Helmholtz free energy density of a uniform hard sphere mixture expressed as a function of four weighted densities nR(r) (R ) 0, 1, 2, 3) that are given by

nR(r) )

Table 1. Lennard-Jones Parameters34 a and Critical Constants35 of the Adsorbates

σij r

12

-

σij r

6

(8d)

and cutoff radius rc ) 2.5σij. The LJ parameters are given in Table 1; Lorentz-Berthelot mixing rules are used to determine the LJ parameters that describe the interactions between different species. The solid-fluid interactions φsi are modeled using Steele’s 10-4-3 potential,28 which is given by

φsi(z) ) 2πsiFsσsi2∆

[(

) ( )

2 σsi 5 z

10

-

σsi z

4

-

σsi4

]

3∆(z + 0.61∆)3

(9)

with solid density Fs ) 0.114 Å-3 and the interlayer separation for graphite ∆ ) 3.35 Å.28 Note that although the most important intermolecular forces for adsorption are given by solid-fluid interactions, the adsorbateadsorbate interactions also play a significant role. The LJ function used here for the fluid-fluid and solid-fluid interactions are “effective” intermolecular potentials. Therefore, any electrostatic interactions (such as dipoledipole, dipole-quadrupole, etc.) are implicitly included through the value of these effective parameters, and instead of writing a Hamiltonian split into dispersion and electrostatic interactions, the LJ parameters are considered to comprise all the possible effects. For fluids confined in slit-shaped pores, the external potential νext is given by

vext(z) ) φsf(z) + φsf(H - z)

(10)

(25) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. (26) Thiele, E. J. Chem. Phys. 1963, 39, 474. (27) Lebowitz, J. L. Phys. Rev. 1964, 133, 895. (28) Steele, W. A. The interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974.

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Figure 1. Experimental,31 KRDFT, and GCMC31adsorption isotherms of CH4-C2H6 mixture (yCH4 ) 0.48) adsorbed in BPL-6 activated carbon at 308 K.

Figure 3. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CO2 (1)C2H4 (2) in homogeneous pores with H ) 20 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

where VV1, VV12, and VV2 are the van der Waals constants, which are given by

VVi ) -

[ ( ) ( )]

σi σi 32 16 π (21/6σi)3 + πiσi3 3 9 i 9 rc rc

9

(13)

i ) 1, 12, 2

Figure 2. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CO2 (1)C2H4 (2) in homogeneous pores with H ) 10 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

Note that in eq 10, H is defined as the distance between the centers of two closest carbon atoms belonging to opposite walls of a slit pore. The bulk phase pressure P and chemical potential µi are calculated from

P ) PHS +

F2 2 (y VV + 2y1y2VV12 + y22VV2) (11) 2 1 1

µ1 ) µHS 1 + F1VV1 + F2VV12 µ2 )

µHS 2

+ F2VV12 + F1VV12

(12a) (12b)

The hard sphere contributions to eqs 11 and 12 (PHS and µiHS) are calculated from SPT.29,30 From these relations, the isosteric heats of adsorption of the components in a binary gas mixture are calculated as follows. The grand potential in eq 1 is minimized with respect to Fi to solve for the equilibrium density profile, based on eqs 1-13. The mean pore fluid density Fji is calculated from

Fji )

1 H

∫0HFi(z) dz

(14)

For convenience, the slit width H and fluid density Fi are scaled with respect to the fluid-fluid molecular diameter of the first component in the mixture, as H* ) H/σ11 and Fi* ) Fiσ113. The isosteric heat of adsorption qi of each component in a binary gas mixture is calculated (29) Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J. Chem. Phys. 1965, 43, 774. (30) Rosenfeld, Y. J. Chem. Phys. 1988, 89, 4272.

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Figure 4. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CH4 (1)C2H6 (2) in homogeneous pores with H ) 10 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

Figure 5. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CH4 (1)C2H6 (2) in homogeneous pores with H ) 20 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

carbon) is calculated from from the relations given by Sircar:3

∫HH

max

{ [( ) ( ) [( ) ( ) [( ) ( ) { [( ) ( ) [( ) ( ) [( ) ( )

q1 1 ∂N1 ) RT P ∂T 1 ∂N1 y1 ∂P

p,y1

T,y1

∂N1 ∂y1

q2 1 ∂N1 ) RT P ∂T 1 ∂N1 y2 ∂P

∂N2 ∂T

T,P

p,y1

T,y1

∂N1 ∂y1

∂N2 ∂y1

∂N2 ∂T

T,P

-

p,y1

∂N2 ∂P

∂N2 ∂y1

( )( ) ] ( ) ( ) ]}/ ( ) ( ) ] ( )( ) ] ( ) ( ) ]}/ ( ) ( ) ]

∂N1 ∂y1 p,T

-

T,y1

-

p,T

-

p,y1

∂N2 ∂P

∂N1 ∂T

T,y1

p,y1

∂N2 ∂P

∂N1 ∂P

∂N1 ∂y1

∂N1 ∂T

-

p,T

∂N2 ∂T

P,y1

p,T

p,y1

∂N2 ∂P

∂N1 ∂P

P,y1

+

qi(H,P) F(H,P) dV(H)

∫HH

F(H,P) dV(H)

max

min

P,y1

(16)

where the cumulative pore volume V(H) is calculated from the known pore size distribution of an adsorbent, f(H),through

T,y1

∂N2 ∂y1

∂N2 ∂T

qi(P) )

min

(15a)

T,P

-

V(H) )

∫HH

f(H′) dH′

min

(17)

Results and Discussion

P,y1

T,y1

∂N2 ∂y1

(15b)

T,P

where Ni is the loading of species i, which maybe replaced with the mean density, since Ni ) FjiV and V cancels out when substituted into eq 15. The partial derivatives in eq 15 are evaluated numerically to find qi(H,P) in homogeneous slit-shaped pores. Once qi(H,P) of each component at different slit widths and different pressures is determined, qi(P) of each component on carbon consisting of a distribution of slit widths (i.e, a heterogeneous porous

Predicting Adsorption Isotherms of a Binary Mixture by KRDFT. As mentioned earlier, very few experimental isosteric heats of adsorption for mixtures are reported in the literature, especially for activated carbon. This made it difficult to test the accuracy of the KRDFT model for predicting isosteric heats of adsorption of mixtures. However, it was possible to test the accuracy of the KRDFT model for predicting binary adsorption equilibria, based on the recent work by Gusev and O’Brien.31 They experimentally measured the adsorption isotherms of a CH4-C2H6 mixture adsorbed in a BPL activated carbon at 308 K, and compared the results with grand canonical Monte Carlo (GCMC) molecular simulations that included the effect of the pore size distribution. (31) Gusev, V. Yu.; O’Brien, J. A. Langmuir 1998, 14, 6328.

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Figure 6. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CH4 (1)C3H8 (2) in homogeneous pores with H ) 10 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

The same system was used here, with the pore size distribution taken from ref 32. The results are presented in Figure 1. The KRDFT results agreed very well with the experimental results at low pressures, with deviations starting at about 1 bar. The KRDFT results also agreed very well with the GCMC results over the entire pressure range for the light component, but for the heavy component, the two models agreed with each other only at low pressures, again with deviations starting at about 1 bar. These deviations between KRDFT and GCMC simulations for the heavy component were interesting and most likely caused by a shape effect that was included in the GCMC simulations, but not in KRDFT. The model used for ethane in the GCMC simulations was a two-center model; in contrast, the KRDFT model considered only spherical LJ molecules. On the basis of this comparison, and since the effective diameters of propane, carbon dioxide, and ethylene do not differ appreciably from ethane, the KRDFT model should also be reasonably accurate for these molecules at pressures below about 1 bar. In the following studies, the bulk pressure was therefore fixed at 1 bar. Isosteric Heats of Adsorption in Homogeneous Carbon. Isosteric heats of adsorption of the components in three binary gas mixtures, CO2-C2H4, CH4-C2H6 and CH4-C3H8, in homogeneous slit-shaped pores of width H ) 10 and 20 Å were calculated using the procedure described above. The temperature and the total pressure of each mixture were fixed at 350 K and 1 bar, respectively. (32) Gusev, V. Yu.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815.

Pan et al.

Figure 7. (A) Binary and single component isosteric heats of adsorption and (B) corresponding mean densities of CH4 (1)C3H8 (2) in homogeneous pores with H ) 20 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

The isosteric heats of adsorption of the components in each mixture were compared with the corresponding pure component values taken at the same partial loading of the component in the mixture. The results obtained for the CO2 (1)-C2H4 (2) system are shown in Figures 2 and 3 for H ) 10 and 20 Å, respectively. The same information is shown in Figures 4 and 5, and Figures 6 and 7 for the CH4 (1)-C2H6 (2) and CH4 (1)-C3H8 (2) systems, respectively. In all of these figures, Figures 2a-7a display the isosteric heats of adsorption and Figures 2b-7b display the corresponding component densities, both as functions of the adsorbed phase composition in terms of x2. The CO2 (1)-C2H4 (2) system is usually considered to be an ideal adsorbed mixture, which can be asserted based on the similarity in the solid-fluid and fluid-fluid LJ parameters and critical constants shown in Table 1, and evidence from the literature.7 Karavias and Myers7 suggested that if binary adsorbed solutions are nearly ideal, the qi’s of the components in the mixture should increase (or decrease) monotonically from the value for the single component to the value at infinite dilution. This behavior is shown very clearly in Figures 2a and 3a, where the qi’s of CO2 and C2H4 both increased steadily and linearly with increasing x2. The densities in Figures 2b and 3b also exhibited fairly ideal behavior, varying nearly linearly over the entire composition range. These same trends were apparent for both pore sizes; however, the smaller pore size gave rise to higher densities and higher heats of adsorption because of the increased interaction of the adsorbates with the very dense carbon walls. The qi°’s of both components also increased with their respective mean densities (or loading, noting that the

Binary Isosteric Heats of Adsorption in Carbon

Langmuir, Vol. 15, No. 13, 1999 4575

Figure 8. Binary and single component isosteric heats of adsorption of C3H8 in CH4(1)-C3H8(2) in homogeneous pores with H ) 10, 10.5, 11, 12, 15, 16, 17, and 18 Å. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

partial loading of the light component increased with decreasing x2), as expected in a homogeneous carbon composed of one pore size at low densities.13 However, only the qi of the heavy component (C2H4) in the mixture increased with its respective density; the qi of the lighter component (CO2) decreased with an increase in its density for both H ) 10 and 20 Å. This behavior was caused by the overall density of the adsorbed phase increasing with an increase in x2, which gave rise to stronger adsorbateadsorbate interactions and thus increases in both qi’s with increasing x2. Both of the qi’s in the mixture were also higher than the qi°’s at the same partial loading. This behavior was caused by the presence of C2H4 in the same

amount of CO2 relative to the pure state (or CO2 in the same amount of C2H4 relative to the pure state). Thus, the effective pore size was smaller and of different character for each of the components in the mixture because of the presence of the other component. This gave rise to the higher qi’s compared to qi°’s at the same partial loading, as explained above for qi being higher in smaller pores. The CH4-C2H6 system showed behavior similar to that of the ideal CO2-C2H4 system in the two single pores, except that the qi’s did not change linearly with x2, especially for the heavy component, C2H6 (Figures 4a and 5a). The CH4-C3H8 system exhibited even more nonideal behavior (Figures 6a and 7a), which was expected based

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Pan et al.

Figure 9. Pore size distribution function of Westvaco BAX activated carbon, taken from Pan et al.13

on the differences in the solid-fluid and fluid-fluid LJ parameters and critical constants shown in Table 1. The behavior of the qi’s in the CH4-C3H8 system also changed depending on the pore size. For both pore sizes, the qi’s for the light component, CH4, behaved similarly to the light component in the other two systems, except for the maximum exhibited near x2 f 1.0 for H ) 20 Å. This maximum was also exhibited for both qi°and qi of C3H8 in the 20 Å pore. Since this maximum occurred for both the pure and mixed isosteric heats of adsorption of C3H8 at high adsorbed phase densities, this rapid drop in the heat of adsorption as x2 f 1.0 was most likely caused by a layering transition of the molecules confined in the pore.13 Supporting this assumption was the fact that the presence of CH4 had little effect on the qi compared to qi° of C3H8 for H ) 20 Å; however, this was not the case for the qi of CH4. The presence of C3H8 caused the same behavior in the qi of CH4, indicating the dominant effects of C3H8 on the qi’s of the two components in this system. It has been shown13 that when pure C3H8 is adsorbed in carbon, the first layering transition begins at a mean density of around 0.2 ()Fσff3, where σff is the diameter of the C3H8 molecule) at H ) 20 Å and T ) 353 K, which becomes 0.085 when converted to the dimensionless density based on the diameter of CH4. This was approximately in the same range of density where the corresponding q’s exhibited peaks in Figure 7a. Figure 8 shows the qi’s and qi°’s of C3H8 in different pore sizes ranging from 10 to 18 Å. The onset of the formation of a peak near x2 f 1.0 occurred when increasing the pore size from 16 to 17 Å, suggesting that layering transition occurs more easily in larger pores, in agreement with the previous study.13 Since the temperatures are supercritical, these layering transitions were of second order; therefore they did not show a discontinuity in the isotherm or heat of adsorption. The most nonideal behavior was exhibited by C3H8 at H ) 10 Å, where qi showed a minimum and also became equal to qi° at some intermediate composition denoted by x2*. This very nonideal behavior was most likely caused by the large differences in the molecular size and energetic parameters between these two species, coupled with confinement in a small pore. For x2 < x2*, the presence of CH4 in the same amount of C3H8 as in the pure state gave rise to stronger interactions between the molecules in the mixture than those that existed in the pure state, and this effect resulted in qi > qi°. However, for x2 > x2*, the presence of CH4 in an increasing amount of C3H8 in such a confined pore most likely produced smaller carbonC3H8 interactions as compared with that in pure state; this effect caused qi < qi° until x2 f 1.0. The complex

Figure 10. (A) Binary and single component isosteric heats of adsorption and (B) corresponding binary and total loadings of CO2 (1)-C2H4 (2) in activated carbon. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

interplay between like and unlike interactions coupled with enhanced geometric effects in a confined pore also gave rise to the minimum in qi for C3H8 in Figure 6, which resulted in the maximum difference between qi and qi°. For x2 > x2min, the amount of C3H8 in the pore was becoming sufficient to overcome the effect of the presence of CH4, and thus qi began to approach qi°. This nonideal behavior existed only in very small pores and was very sensitive to pore size. Figure 8 shows the transition of these nonideal features to more ideal features with very small changes in the pore size. For example, at H ) 10.5 Å, the qi’s and qi°’s of C3H8 were similar to those at H ) 10 Å, except that no minimum in qi was exhibited. At H ) 11 Å, both nonideal features disappeared and the qi°’s and qi°’s of C3H8 were similar to those exhibited by the other two more ideal systems. Isosteric Heats of Adsorption in Heterogeneous Activated Carbon. The qi’s of the components in each binary gas mixture adsorbed in a heterogeneous activated carbon were determined from the previous results (i.e., from the qi’s of the components in each mixture adsorbed in a homogeneous carbon of different slit widths) using eq 16 along with the known pore size distribution (PSD) of the activated carbon. Westvaco BAX activated was used because of its well-characterized PSD, which is shown in Figure 9.13 For each binary mixture, 20 different reduced slit widths were used that ranged between 1.6 and 20. The results obtained for the CO2 (1)-C2H4 (2), CH4 (1)C2H6 (2) and CH4 (1)-C3H8 (2) systems are respectively displayed in Figures 10, 11, and 12, and plotted similarly to the homogeneous carbon results.

Binary Isosteric Heats of Adsorption in Carbon

Langmuir, Vol. 15, No. 13, 1999 4577

Figure 11. (A) Binary and single component isosteric heats of adsorption and (B) corresponding binary and total loadings of CH4 (1)-C2H6 (2) in activated carbon. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

Figure 12. (A) Binary and single component isosteric heats of adsorption and (B) corresponding binary and total loadings of CH4 (1)-C3H8 (2) in activated carbon. The partial loadings correspond to the same single component loadings. T ) 350 K and P ) 1 bar.

There is some experimental data in the literature for the adsorption of these gas mixtures, but on a different activated carbon and at different temperatures.33 Thus, although the plots in Figures 10b-12b could not be compared directly to those results, some similarities were observed. For the CO2-C2H4 system, the shapes of the isotherms predicted by KRDFT (Figure 10b) were similar to those obtained experimentally for the same mixture adsorbed on Nuxit-AL carbon at 293 K.33 Both results showed a near symmetric behavior of the isotherms with crossing at about x2 ) 0.5, suggesting ideality of the system. For the CH4 - C2H6 system, the shapes of the isotherms were also similar to those obtained experimentally for the same mixture on Nuixt-AL activated carbon at 293 K.33 Both results showed asymmetric behavior of the isotherms, suggesting nonideality of the system. For the CH4-C3H8 system, however, a comparison was not possible because no experimental isotherm data were available at a fixed pressure with different compositions. The CO2 (1)-C2H4 (2) system exhibited ideal mixture behavior in this heterogeneous carbon, similar to that observed in the homogeneous carbon. The qi’s of the components in the mixture increased (or decreased) monotonically and almost linearly from the value for the single component to the value at infinite dilution. However, two notable differences were observed between the qi’s in

the homogeneous and heterogeneous carbons. First, in contrast to both qi’s increasing with increasing x2 in the homogeneous carbon, both qi’s decreased with increasing x2 in the activated carbon, which was typical of a heterogeneous adsorbent since the total loading increased with increasing x2 (Figure 10b). This was the case even though the partial loading and qi for CO2 both increased with decreasing x2. Second, in contrast to the case for the homogeneous carbon, the qi’s in the heterogeneous carbon were lower than the qi°’s at the same partial loading. This behavior manifested itself as an opposing component loading effect, with competition for the various pore sizes within the very heterogeneous activated carbon giving rise to the observed behavior. For example, as x2 f 0, CO2 occupied some of the higher energy (smaller) pores that would have otherwise been occupied by C2H4 and vice versa as x2 f 1.0; so both of the qi’s were reduced relative to qi°’s. These two differences were also observed at certain x2 ranges for the CH4-C2H6 and CH4-C3H8 systems, but in general, the behavior of the CH4-C2H6 and CH4-C3H8 systems were uniquely different from the CO2-C2H4 system and even from themselves. As mentioned above, the CH4-C2H6 and CH4-C3H8 systems were considered to be nonideal, with the CH4-C3H8 system considered to be the most nonideal system. The qi for C3H8 in the CH4-C3H8 system exhibited trends similar to both components in the CO2-C2H4 system, which implied that as x2 decreased, CH4 occupied more of the pores that would have otherwise been occupied by C3H8. The opposite behavior was exhibited, however, by the qi for C2H6 in the CH4-C2H6 system and also by CH4 in the CH4-C3H8 system as x2 f 1.0. As discussed for the

(33) Szepesy, L.; Illes, V. Acta Chim. Hung. 1963, 35, 245. (34) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Son: New York, 1954. (35) Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth-Heinemann: Newton, 1985.

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case of Figure 6, this behavior suggested that the presence of CH4 instead of C2H6 (or the presence of C3H8 instead of CH4) around the same amount of C2H6 (or CH4) gave rise to stronger interactions between the molecules in the mixture than those that existed in the pure state, and this effect resulted in qi∞ > qi°. In these more nonideal systems, the differences in the molecular nature, reflected in the effective energy and size parameters, gave rise to these subtle changes in the qi’s relative to the qi°’s; these interactions also must have overwhelmed the simple (ideal) opposing component loading effect. As explained earlier, the presence of CH4 (or C3H8) in a pore containing the same amount C2H6 (or CH4) relative the pure state made the effective pore size smaller, which led to larger qi’s than qi°’s in single pores at the same partial loading. Moreover, the CH4 in both the CH4-C2H6 and CH4C3H8 systems shown in Figures 11 and 12 exhibited similar trends. As x2 increased in both cases, qi for CH4 exhibited a minimum where qi e qi°; for a specific 0 f x2min range, this corresponded to opposing component loading effects, similar to those exhibited by the ideal system. For CH4C2H6, beyond x2min, the presence of the heavy component caused qi for CH4 to increase toward the qi∞ value. For the CH4-C3H8 system, the effect was so great that qi∞ exceeded qi° as x2 f 1.0. The explanation for this behavior was given above; i.e., the small CH4 molecule at increasingly more dilute concentrations in the mixture was effectively being adsorbed in pores of a completely different character and size, due to the presence of the other larger component. Conclusions The nonlocal density functional theory of Kierlik and Rosinberg was used to predict the binary isosteric heats

Pan et al.

of adsorption (qi) of three binary gas mixtures, CO2-C2H4, CH4-C2H6, and CH4-C3H8, in a homogeneous carbon comprised of the same sized pores and in a heterogeneous activated carbon comprised of differently sized pores. The temperature and pressure were fixed at 350 K and 1bar. The qi’s of the components in the mixture were different from their pure state qi°’s because of two different effects, which differed depending on the nonideality of the adsorbate system and the heterogeneity of the adsorbent. The first effect was an opposing component loading effect where the presence of one component caused a decrease in the qi of the other component relative to its qi° at the same partial loading by competing for the available higher energy sites (smaller pores). The second effect was caused by differences in the intermolecular forces between the components in the mixture. In the homogeneous carbon, only the second effect was important since the adsorption sites were homogeneous; in this case, the presence of one component usually caused an increase in the qi of the other component relative to its qi° at the same partial loading. In the heterogeneous carbon, very complex behavior was exhibited, even where the presence of one component caused both increases and decreases in the qi of the other component relative to its qi° at the same partial loading, depending on the composition of the mixture. Acknowledgment. This material is based upon work supported by the U.S. Department of Energy under Cooperative Agreement No. DE-FC02-91ER75666. LA981511Q