Correspondence Comment on “Modeling the Mass-Action Expression for Bidentate Adsorption” SIR: A recent theoretical study of the equilibrium of ligand adsorption (1) employed both Monte Carlo simulation and quasi-chemical theory to show explicitly that the Langmuir adsorption isotherm does not express the equilibrium for bidentate ligands. As it was also shown in ref 1 that the Monte Carlo simulation and the quasi-chemical theory provide essentially identical results, we will not discuss the simulation. Instead, we propose to clarify the quasi-chemical theory in this context. The plan of this Comment consists of two parts. First, we restate the problem that is the concern of ref 1 and collect results from quasi-chemical theory that pertain thereto. Second, we address the conclusions of ref 1 in light of these results. The problem at hand concerns the reversible adsorption of multidentate ligands on a square lattice of sites. Here, each adsorbed ligand is an “m-mer” occupying exactly m sites, does not obstruct other empty sites, is immobile while on the surface, and does not interact with any other ligand except via “hard core” interactions in order to exclude multiple occupation of sites. The desorbed phase is either a gas or a liquid. The coverage is always submonolayer. Mixtures of different types of adsorbed ligands are also excluded here. The problem then is to determine the dependence of the equilibrium constant Km for the adsorption. This is not trivial because this process is heterogeneous (i.e., the fixed unoccupied sites are not an independent species), so the law of mass action employed for homogeneous reactions (2), which would give
Km ∝
Fm (1 - mFm)m
(1)
(with constant desorbed ligand fugacity) would not be expected to apply here. Langmuir derived an accurate expression for the fugacity z1 of adsorbed monomers on a lattice (3, 4) as
F1 z1 ) 1 - F1
(2)
where F1 is the number density of the adsorbed monomers. Holding constant the fugacity of the desorbed monomers, one may also write, as is commonly done (3),
K1∝
F1 1 - F1
(3)
Interestingly, the naive result in eq 1 agrees with eq 3 for monomers. The equilibrium for rigid adsorbed dimers, trimers, and so forth requires additional work. Recognizing that eq 1 is inappropriate for adsorption, some (5) have suggested that Km>1 might be expressed in some variation of eq 3, that is,
Km ∝
Fm 1 - mFm
(4)
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[email protected]. 10.1021/es0206137 CCC: $22.00 Published on Web 04/19/2002
2002 American Chemical Society
for any m g 1 (see also the references in refs 1 and 6-8). Nevertheless, as recognized by ref 1, the quasi-chemical approximation (QCA) obtained independently by Guggenheim and Bethe (3, 9, 10) provides the thermodynamics of adsorbed monomers, dimers, and so forth on a surface of ordered sites. Independent computer enumeration studies (11-13) (as well as the simulation of ref 1) have confirmed that the QCA is remarkably accurate for low and moderate densities (i.e., until the coverage approaches the packing limit). Neither we nor ref 1 are concerned with coverage in the packing limit, so for the purposes of this Comment, we take the QCA to be exact. The QCA provides the adsorbed ligand fugacity zm explicitly as
Fm(1 - Fm)m-1 , 1eme4 zm ) σm (1 - mFm)m
(5)
for the monomers (3), dimers (11), trimers (12), and tetramers (without loops) where the σm are constants that depend on shape. For m ) 1, this becomes Langmuir’s prediction in eq 2. One may leave these results in eq 5 as they are, but they are not in the form of a ratio of density of adsorbed ligand versus density of unoccupied sites, which would be convenient in order to obtain the expression for the equilibrium. To do this, the equilibrium first should be written as
Km ∝
Fm
(6)
(1 - mFm)νm
so that the question of the appropriate expression for the equilibrium comes down to the determination of νm. If the naive result were correct, then νm ) m, for all m. If the suggestion of refs 5-8 were correct, then νm ) 1, for all m. Nevertheless, after equating the variation of log zm in Fm with the variation of log Km, the νm given by the QCA is
ν1 ) 1
(7)
3 - 2F2 1 ν2 ) 1 e e2 2 2(1 - F2)
(8)
7 - 3F3 1 e3 ν3 ) 2 e 3 3(1 - F3)
(9)
1 13 - 4F4 ν4 ) 3 e e4 4 4(1 - F4)
(10)
This result places νm between the suggestions of eq 1 and eq 4. Turning now to ref 1, we first emphasize our agreement with two key observations therein: first, that eq 4 is not appropriate for bidentate (or multidentate) ligand adsorption, and second, that the QCA provides the appropriate theory for bidentate ligand adsorption. As ref 1 notes, no one choice works for all concentrations: “While the difference was not large, it was unambiguous, demonstrating that no single relationship can accurately predict [F2] based solely on the fractional surface coverage”. That statement would be true if ν2 were restricted to be a constant, independent of coverage. Nevertheless, the QCA shows that ν2 ranges from 3/2 to 2 with increasing coverage (eq 8); this now provides an accurate VOL. 36, NO. 10, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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and consistent prediction of the multidentate ligand adsorption equilibrium.
Acknowledgments We thank Mark Benjamin (U. Washington), Paul Meakin (INEEL), and Lawrence Pratt (LANL) for helpful discussions. Work at the INEEL was supported by the Office of Environmental Management, U.S. Department of Energy, under DOE-ID Operations Office Contract DE-AC07-99ID13727.
Literature Cited (1) Benjamin, M. M. Environ. Sci. Technol. 2002, 36, 307. (2) The rate of a homogeneous (uniform) chemical reaction at constant temperature is proportional to the concentration of the reacting substances. Hawley, G. G. The Condensed Chemical Dictionary, 10th ed.; Van Nostrand Reinhold: New York, 1987; p 251. (3) Hill, T. L. Introduction to Statistical Thermodynamics; AddisonWesley: Reading, MA, 1960; Chapters 7 and 14. (4) Denbigh, K. The Principles of Chemical Equilibrium; Cambridge University Press: New York, 1966; Chapter 14. (5) Morel, F. M. M.; Hering J. G. Principles and Applications of Aquatic Chemistry; John Wiley: New York, 1993; Chapter 8.
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(6) Schindler, P. W. Pure Appl. Chem. 1991, 63, 1697. (7) Filius, J. D.; Hiemstra, T.; VanRiemsdijk, W. H. J. Colloid Interface Sci. 1997, 195, 368. Kosmulski, M. ibid. 1997, 195, 395. LopezMacipe, A.; Gomez-Morales, J.; Rodriguez-Clemente, R. ibid. 1998, 200, 114. Evanko, C. R.; Dzombak, D. A. ibid. 1999, 214, 189. (8) Degenhardt, J.; McQuillan, A. J. Chem. Phys. Lett. 1999, 311, 179. (9) Feynman, R. P. Statistical Mechanics: A Set of Lectures; W. A. Benjamin: Reading, MA, 1972; Chapter 5. (10) Pratt, L. R.; LaViolette, R. A. Mol. Phys. 1998, 94, 909. Pratt, L. R.; LaViolette, R. A.; Gomez, M. A.; Gentile, M. E. J. Phys. Chem. B 2001, 105, 11662. (11) Gaunt, D. S. Phys. Rev. 1969, 179, 174. (12) Van Craen, J.; Bellemans, A. J. Chem. Phys. 1972, 56, 2041. (13) Evans, J. W. Rev. Mod. Phys. 1993, 65, 1281.
Randall A. LaViolette* and George D. Redden Idaho National Engineering and Environmental Laboratory, P.O. Box 1625 Idaho Falls, Idaho 83415-2208 ES0206137
