pubs.acs.org/Langmuir © 2010 American Chemical Society
Adsorption of Binary Gas Mixtures in Heterogeneous Carbon Predicted by Density Functional Theory: On the Formation of Adsorption Azeotropes James A. Ritter* and Huanhua Pan Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208
Perla B. Balbuena Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843 Received May 10, 2010. Revised Manuscript Received August 3, 2010 Classical density functional theory (DFT) was used to predict the adsorption of nine different binary gas mixtures in a heterogeneous BPL activated carbon with a known pore size distribution (PSD) and in single, homogeneous, slit-shaped carbon pores of different sizes. By comparing the heterogeneous results with those obtained from the ideal adsorbed solution theory and with those obtained in the homogeneous carbon, it was determined that adsorption nonideality and adsorption azeotropes are caused by the coupled effects of differences in the molecular size of the components in a gas mixture and only slight differences in the pore sizes of a heterogeneous adsorbent. For many binary gas mixtures, selectivity was found to be a strong function of pore size. As the width of a homogeneous pore increases slightly, the selectivity for two different sized adsorbates may change from being greater than unity to less than unity. This change in selectivity can be accompanied by the formation of an adsorption azeotrope when this same binary mixture is adsorbed in a heterogeneous adsorbent with a PSD, like in BPL activated carbon. These results also showed that the selectivity exhibited by a heterogeneous adsorbent can be dominated by a small number of pores that are very selective toward one of the components in the gas mixture, leading to adsorption azeotrope formation in extreme cases.
Introduction Nonideal adsorbed solution behavior of gas mixtures, including adsorption azeotropes, is an interesting and important topic. It is an interesting topic because several studies have reported that adsorbed phase nonideality is related to differences in the sizes of the adsorbates,1,2 possible lateral interactions between them,3,4 and/or the heterogeneity of the adsorbent,5-9 with extreme cases of adsorbed phase nonideality resulting in the formation of adsorption azeotropes. It is an important topic because the design and development of gas phase adsorption processes rely on theoretical models to accurately predict the adsorbed phase behavior, including adsorption azeotropes. Many theoretical models have been developed for predicting multicomponent adsorption equilibria from gas mixtures. This *To whom correspondence should be addressed. E-mail:
[email protected]. (1) Sircar, S. Influence of Adsorbate Size and Adsorbent Heterogeneity on IAST. AIChE J. 1995, 41, 1135. (2) Siperstein, F. R. Determination of Azeotropic Behavior in Adsorbed Mixtures. Adsorption 2005, 11, 53. (3) Talu, O.; Zwiebel, I. Multicomponent Adsorption Equilibria of Nonideal Mixtures. AIChE J. 1986, 32, 1263. (4) Ritter, J. A.; Al-Muhtaseb, S. A. New Model that Describes Adsorption of Laterally Interacting Gas Mixtures on Random Heterogeneous Surfaces. 1. Parametric Study and Correlation with Binary Data. Langmuir 1998, 14, 6528. (5) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. Adsorption of Gas Mixtures: Effect of Energetic Heterogeneity. AIChE J. 1988, 34, 397. (6) Sircar, S. Role of Adsorbent Heterogeneity on Mixed Gas Adsorption. Ind. Eng. Chem. Res. 1991, 30, 1032. (7) Kaminsky, R. D.; Monson, P. A. An Analysis of the Statistical Model Adsorption Isotherm. AIChE J. 1992, 38, 1979. (8) Kaminsky, R. D.; Monson, P. A. Physical Adsorption in Heterogeneous Porous Materials: An Analytical Study of a One-Dimensional Model. Langmuir 1993, 9, 561. (9) Do, D. D.; Do, H. D. On the Azeotropic Behavior of Adsorption Systems. Adsorption 1999, 5, 319. (10) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; WileyInterscience: New York, 1984.
13968 DOI: 10.1021/la101865m
topic has been reviewed in several monographs.10-12 Of the more popular models, the ideal adsorbed solution theory (IAST) developed by Myers and Prausnitz13 is perhaps the most important model because it is a thermodynamically consistent formulation that assumes the adsorbed phase behaves like an ideal solution. So, any deviations from the IAST predictions exhibited by experimental data or other theoretical models can be immediately attributed to nonideal adsorbed solution behavior. Myers and Prausnitz13 suggested that the nonideality of the adsorbed phase can be incorporated into the activity coefficient of each component in the adsorbed phase. Talu and Zwiebel3 extended the IAST to account for adsorbed phase nonidealities, including adsorption azeotropes, using a spreading pressuredependent activity coefficient formulation. This nonideal AST alluded to lateral interactions between the adsorbates being the cause of adsorbed phase nonidealities. Another modification to the IAST accounted for the effect of adsorbent heterogeneity.5 This heterogeneous IAST also predicts adsorption azeotrope formation, thus suggesting that the cause of nonideal adsorbed solution behavior is due to adsorbent heterogeneity. Sircar1 studied the influence of adsorbate size and adsorbent heterogeneity on IAST. He showed that large differences in adsorbate sizes and the degree of adsorbent heterogeneity can cause the formation of an adsorption azeotrope. However, the models he used, namely the multisite Langmuir and heterogeneous Langmuir models, do not account for adsorbate lateral interactions. Ritter and Al-Muhtaseb4 developed a heterogeneous Langmuir model (11) Yang, R. T. Gas Separation by Adsorption Processes; Imperial College Press: London, 1997. (12) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (13) Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed Gas Adsorption. AIChE J. 1965, 11, 121.
Published on Web 08/16/2010
Langmuir 2010, 26(17), 13968–13975
Ritter et al.
Article
that included adsorbate lateral interactions and showed that these interactions can also cause adsorption azeotropes. In many cases, the effect of adsorbate lateral interactions can be stronger than that of adsorbent heterogeneity and may even overwhelm the usually opposing effects exhibited by adsorbent heterogeneity. Kaminsky and Monson7,8 studied binary adsorption equilibria in heterogeneous porous materials with a statistical mechanical model and found that pore size played a significant role in determining adsorption behavior in microporous systems, including the formation of adsorption azeotropes. Their one-dimensional model illustrated very nicely the effects of adsorbent heterogeneity on nonideal adsorbed solution behavior; however, since the molecular interactions were described by a simple square well potential, this limited their results to being only qualitative and thus not applicable to an actual adsorbent. Therefore, the objective of this work was to study nonideal adsorbed solution behavior in carbon materials, including adsorption azeotrope formation, using the more rigorous Kierlik and Rosinberg density functional theory (KRDFT) approach.14 In this case the molecular interactions are treated via LennardJones (L-J) or L-J type potentials for the fluid-fluid and fluidsolid interactions, along with Lorentz-Berthelot mixing rules, as done in many DFT approaches.14-16 Specifically, KRDFT was used to study binary gas mixtures adsorbed by BPL activated carbon with a known pore size distribution (PSD) and in single slit carbon pores of different widths. The effects of different L-J parameters (representing different sizes of molecules and different extents of molecular interactions) and of different pore sizes on adsorbed phase nonideality and the formation of adsorption azeotropes in heterogeneous carbon are disclosed. The degree of nonideal adsorbed solution behavior is revealed by comparing the KRDFT predictions to those obtained from the IAST.
Theory KRDFT has been used by several groups to study single gas adsorption in carbon materials.17-20 In many instances, the results agreed very well with those obtained by Monte Carlo simulation.16,18,20 KRDFT is perhaps the only DFT approach that is especially applicable to mixtures because its weighting functions are purely geometric and characteristic of each type of molecule.14 In this regard, KRDFT has also been used successfully by several groups to study gas mixture adsorption on carbon materials.17,18,20,22 DFT 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 mean-field approximation, and the (14) Kierlik, E.; Rosinberg, M. L. Free-Energy Density Functional for the Inhomogeneous Hard-Sphere Fluid: Application to Interfacial Adsorption. Phys. Rev. A 1990, 42, 3382. (15) Tarazona, P. Free-Energy Density Functional for Hard Spheres. Phys. Rev. A 1985, 31, 2672. (16) Davis, H. T. Statistical Mechanics of Phases, Interfaces, and Thin Films; VCH Publishers, Inc.: New York, 1996. (17) Kierlik, E.; Rosinberg, M. L. Density-Functional Theory for Inhomogeneous Fluids: Adsorption of Binary Mixtures. Phys. Rev. A 1991, 44, 5025. (18) Kierlik, E.; Rosinberg, M. L. Binary Vapor Mixtures Adsorbed on a Graphite Surface: A Comparison of Mean Field Density Functional Theory with Results from Monte Carlo Simulations. Mol. Phys. 1992, 75, 1435. (19) Balbuena, P. B.; Gubbins, K. E. Theoretical Interpretation of Adsorption Behavior of Simple Fluids in Slit Pores. Langmuir 1993, 9, 1801. (20) Pan, H.; Ritter, J. A.; Balbuena, P. Binary Isosteric Heats of Adsorption in Carbon Predicted from Density Functional Theory. Langmuir 1999, 15, 4570. (21) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. Fluid Behavior in Narrow Pores. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789. (22) Jiang, S.; Gubbins, K. E.; Balbuena, P. B. Theory of Adsorption of Trace Components. J. Phys. Chem. 1994, 98, 2403.
Langmuir 2010, 26(17), 13968–13975
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 is Z Z 1 0 dr dr0 Fi ðrÞFj ðr0 Þφattr Ω½fFi g ¼ FHS ½fFi g þ ij ðjr - r jÞ 2 Z þ dr Fi ðrÞ½vi ðrÞ - μi ð1Þ 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 Z ex FHS ½fFi g ¼ dr kTΦðfnR ðrÞgÞ ð2Þ 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 XZ ðRÞ dr0 Fi ðr0 Þωi ðr - r0 Þ ð3Þ nR ðrÞ ¼ i
The functional form of Φ is taken from scaled-particle theory (SPT)23 or, equivalently, from the Percus-Yevick compressibility equation of state24,25 as Φ ¼ - n0 lnð1 - n3 Þ þ
n1 n2 1 n2 3 þ 1 - n3 24π ð1 - n3 Þ2
ð4Þ
n0, n1, n2, and n3 are the reduced variables of the SPT, with X
nR ¼
ðRÞ
Fi Ri
ð5Þ
i
and ð0Þ
Ri
¼ 1,
ð1Þ
Ri
¼ Ri ,
ð2Þ
Ri
¼ 4πRi 2 ,
ð3Þ
Ri
¼
4 πRi 3 ð6Þ 3
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) as17 ð3Þ
ð7aÞ
ð2Þ
ð7bÞ
ωi ðrÞ ¼ θðRi - rÞ ωi ðrÞ ¼ δðRi - rÞ ð1Þ
ωi ðrÞ ¼
ð0Þ
ωi ðrÞ ¼ -
1 0 δ ðRi - rÞ 8π
1 00 1 0 δ ðRi - rÞ þ δ ðRi - rÞ 8π 2πr
ð7cÞ
ð7dÞ
(23) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Statistical Mechanics of Rigid Spheres. J. Chem. Phys. 1959, 31, 369. (24) Thiele, E. Equation of State for Hard Spheres. J. Chem. Phys. 1963, 39, 474. (25) Lebowitz, J. L. Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres. Phys. Rev. 1964, 133, 895.
DOI: 10.1021/la101865m
13969
Article
Ritter et al.
The fluid-fluid pair interaction uij is described by a cut and shifted Lennard-Jones (LJ) potential LJðrc Þ;
uij ðrÞ ¼ - εij - uij
LJðrc Þ; LJðrÞ - uij
¼ uij
¼ 0; with
r < 21=6 σij
21=6 σ ij
