Environ. Sci. Technol. 2002, 36, 307-313
Modeling the Mass-Action Expression for Bidentate Adsorption MARK M. BENJAMIN* Department of Civil and Environmental Engineering, Box 352700, University of Washington, Seattle, Washington 98195-2700
The number of bidentate binding sites on a pristine surface (i.e., sites comprising two adjacent monodentate sites plus the space between them) can be substantially larger than the maximum number of bidentate molecules that can adsorb to the surface. When bidentate sorption occurs, the number of available bidentate sites decreases in response to direct occupation of some sites, but an even more significant loss results from the fact that several unoccupied sites immediately surrounding each adsorbed molecule can also become unavailable. Recognition of this phenomenon allows development of a model for the adsorption process that matches simulated data from Monte Carlo (MC) modeling extremely well. The model also explains the observation that, on a given surface with a given fractional occupation, the number of available bidentate sites depends on whether the occupied sites are populated by monodentate or bidentate adsorbed species. A model developed more than 60 years ago but not widely recognized by modern adsorption modelers also fits the MC simulations very well. The simulated data are also reasonably approximated by assuming that the number of available bidentate sites on a surface is proportional to the square of the number of available monodentate sites, although the fit is not as good as with the models mentioned above. By contrast, approximating the number of available sites as proportional to the number of monodentate sites to the first power yields predictions that do not match the simulations. The results have implications for modeling of both multidentate adsorption reactions and monovalent-divalent ion exchange.
Introduction The possibility of multidentate adsorption, i.e., binding of a single adsorbate molecule to multiple sites on the surface of an adsorbent, has been widely accepted for years (e.g., refs 1-4), and in recent years the existence of multidentate surface complexes has been confirmed via surface spectroscopy (5-7). However, the best approach for estimating the concentration of multidentate sites available on partially occupied surfaces is not agreed upon, and as a result, several different formulations have been proposed for the mass action expression describing multidentate adsorption. Reactions in which an adsorbate binds to only one surface site, but adjacent sites become unavailable due to steric factors, have the same stoichiometry as multidentate sorption reactions and hence can be modeled by the same formalism (and are subject to the same uncertainties). * Phone: (206)543-7645; fax: (206)685-9185; e-mail: markbenj@ u.washington.edu. 10.1021/es010936n CCC: $22.00 Published on Web 01/04/2002
2002 American Chemical Society
FIGURE 1. Schematic of two arrays of surface sites. Each circle represents a potential monodentate binding site. (a) Bidentate sites (two horizontally aligned monodentate sites separated by a short distance) do not overlap; each monodentate site has only one partner with which to form a bidentate site. (b) Overlapping bidentate sites; each monodentate site can combine with any of its nearest neighbors to form a bidentate site. Lines represent adsorbed molecules. Multidentate binding sites on a surface are often represented as groups of monodentate sites having a specified pattern (e.g., straight chain, bent, branched, etc.) and spacing. The case that is most relevant for adsorption of small inorganics (e.g., metal ions or oxyanions) is bidentate surface complexation, and that case is the focus of this paper. The terminology used to describe multidentate adsorption is, of course, based on an analogy with multidentate complexation in solution. While that analogy is often apt, a multidentate surface ligand differs from a soluble multidentate ligand in subtle but important ways. For instance, in solution, bidentate ligands comprise two functional groups that are covalently linked to one another so that each functional group has a uniquely identifiable partner with which it forms the bidentate ligand. This situation contrasts with the case in which two independent dissolved ligand molecules bind to a single metal ion to form a di-ligand complex (MeL2). Like the functional groups of a soluble bidentate ligand, the two functional groups that form a bidentate surface ligand (i.e., the two adjacent monodentate sites) are covalently bonded to one another (through the solid). In some cases, the arrangement of these two sites is such that each can form a bidentate site only with the other, as shown schematically in Figure 1a. However, in other cases, a monodentate site might be capable of joining with any of several neighboring sites to form a bidentate site (Figure 1b), with the exact number of such sites depending on the surface structure and the chemistry of the adsorbate. This multiplicity of potential partners with which a given monodentate site might combine to form a single bidentate site makes the formation of a bidentate surface complex something of a hybrid between the two types of solution-phase reactions described above and has important implications for the term representing available bidentate sites in the mass action expression. VOL. 36, NO. 3, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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This paper explores four approaches for estimating the value of that term: two approaches that are common in current environmental literature; one that was developed more than 60 years ago but has been overlooked by most modern researchers; and one based on a new derivation presented below. The predictions of all four models are then compared with the results of Monte Carlo simulations of the surface loading process.
Conceptual and Mathematical Representations of Multidentate Adsorption According to the surface complexation model of adsorption, the bidentate adsorption of solute Bd to an adsorbent with individual (monodentate) surface sites designated by ≡S can be represented by the following reaction and equilibrium constant expression:
≡S2 + Bd T ≡S2Bd KBd )
a≡S2Bd a≡S2aBd
)
c≡S2Bd/c≡S2Bd,std
(1) γ≡S2Bd
(c≡S2/c≡S2,std) (cBd/cBd,std) γ≡S2γBd
(2)
where ≡S2 is an unoccupied bidentate surface site; ≡S2Bd is a bidentate surface complex; KBd is the equilibrium constant for the reaction; ai, ci, and γi are the chemical activity, concentration, and activity coefficient of species i, respectively; and the subscript std refers to species in their standard states. The analysis to be presented here applies regardless of the charge on the surface or the adsorbate and regardless of whether previously bound species are released when a bidentate adsorption reaction occurs, so those factors are left out for simplicity. When evaluating KBd according to eq 2, standard state concentrations are chosen, assumptions are made so that the activity coefficients can be quantified, cBd is analyzed directly, and c≡S2Bd is evaluated based on a mass balance on total Bd in the system (c≡S2Bd ) cBdtot - cBd). However, c≡S2 cannot be evaluated experimentally, so it must be estimated based on some modeling assumptions. Ad Hoc Assumptions for Estimating c≡S2. In the adsorption literature of the past 25 years, two approaches have been used to estimate c≡S2, both based on an assumed relationship to c≡S. The first approach, favored by Davis, Morel, and their co-workers (4, 8-10), assumes that c≡S2 is directly proportional to c≡S:
c≡S2 ) k1c≡S
(3)
This approach treats ≡S2 sites identically to dissolved bidentate ligands, with each ≡S site corresponding to a functional group in the ligand. An implicit corollary is that the concentration of available bidentate sites on a surface is one-half of the concentration of available monodentate sites, i.e., that k1 ) 0.5. The second approach posits that c≡S2 is proportional to c≡S2, based on an argument that bidentate sites are generated by a combination of two independent ≡S sites via a process that can be represented as a pseudo-reaction:
2≡S T ≡S2
(4)
c≡S2 ) k2c≡S2
(5)
One conceptual justification for this approach is that, even though surface functional groups are fixed in space on the adsorbent surface, the vacancies on the surface can be considered mobile if the adsorbed species are mobile. That 308
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is, if adsorbed ions can diffuse along the surface, the locations of vacant ≡S sites will change from one instant to the next; if these vacancies are treated as independent monodentate ligands, the equilibrium concentration of adjacent pairs of such ligands (i.e., ≡S2 sites) might reasonably be modeled as being proportional to the square of the concentration of monodentate sites. Equation 5 was used by a number of early workers to model bidentate adsorption onto oxides (1-3) and has been used widely to model binding of divalent ions on ion-exchange resins (11, 12). Morel and Hering (10) have criticized the use of eq 5 because, if site concentrations are expressed in moles of sites per liter of solution, it predicts that doubling the concentration of particles in a suspension without altering the surface speciation would quadruple c≡S2, whereas logic demands that c≡S2 double in such a scenario. This problem arises because, according to a strict interpretation of reaction 4, a bidentate site could be formed by a combination of monodentate sites on two different particles, which is inconsistent with the conceptual model. The problem can be overcome by applying the assumption that leads to eq 5 to the mass action expression written in terms of mole fractions rather than molar concentrations, i.e.:
c≡S2 c≡Stot
) k2′
( ) c≡S c≡Stot
2
c≡S2 c≡S2 ) k2′ c≡Stot
(6)
(7)
where c≡Stot is the total concentration of monodentate sites in the system. Equation 7 yields values of c≡S2 that are logically consistent when the total concentration of particles in the system changes. Hiemstra and Van Riemsdijk (13) have recently argued in support of this approach for modeling bidentate adsorption onto oxides. Theoretical Analysis Based on Surface Structure: Nonoverlapping Bidentate Sites. As an alternative to making ad hoc assumptions about the form of the mathematical relationship between c≡S2 and c≡S, the relationship can be derived based on idealized geometrical models for the surface site distribution. For instance, if the sites are assumed to be independent and nonoverlapping (the scenario shown in Figure 1a), the concentration of available bidentate sites is simply
c≡S2 ) c≡S2,max - c≡S2Bd
(8)
where c≡S2,max is the maximum possible concentration of available bidentate sites, i.e., the value of c≡S2 when the surface is completely unoccupied. Furthermore, in such a system, the only unoccupied monodentate (≡S) sites are those that are the end-members of open ≡S2 sites, so
c≡S2 ) 0.5c≡S
(9)
This result is identical to eq 3 with k ) 0.5 and therefore provides a justification for using that equation to model systems with nonoverlapping bidentate sites. Overlapping Sites: The Chang Model. If the surface comprises overlapping bidentate sites (such as the scenario shown in Figure 1b), the analysis is more complex. In 1939, using arguments grounded in statistical mechanics, Chang (14) developed an approximate equation to characterize bidentate adsorption onto a rectangular array of identical monodentate sites, each of which could combine with any of its nearest neighbors to form a bidentate site. His result was subsequently extended by other researchers (15-17) to
higher order multidentate adsorption of a species X, yielding
(
c≡Sr ) c≡Srmax 1 -
2(r - 1)f rz
)
1-r
(1 - f )r
(10)
where r is the number of monodentate sites occupied by a single adsorbed X molecule; z is the number of nearest neighbors surrounding each monodentate site, and f is the overall fractional surface coverage. Applied to bidentate adsorption, eq 10 becomes
c≡S2 ) c≡S2,max
z (1 - f )2 z-f
(11)
Equation 11 can be rewritten to show c≡S2 explicitly as a function of c≡S and parameters that depend on the surface structure but not the extent of coverage by substituting 1 c≡S/c≡Stot for f:
c≡S2 )
c≡S2,max
z c 2 c≡Stot (z - 1)c≡S + c≡S ≡S tot
(12)
Overlapping Sites: A Mean Occupation Model. Equations 10-12 characterize the concentration of available r-dentate sites as a function of the surface occupation of such sites by r-dentate adsorbate molecules, for a case where those sites overlap. It is of interest to ascertain whether the same equations apply if the sites are occupied by other adsorbates (especially monodentate adsorbates, Md). To address that issue, a different approach to the problem was developed that explicitly considers whether the sites that are occupied are populated by Md or Bd molecules. The key to the alternative approach, which will be referred to as the mean surface occupation (MSO) model, is the evaluation of the number of bidentate sites lost when a single Md or Bd molecule adsorbs. Consider first the number of bidentate binding sites on a pristine surface, N≡S2,max. For this calculation, it is useful to consider a surface comprised of a rectangular m × n array of monodentate sites that can combine in various ways to form overlapping bidentate sites. The number of ≡S sites on such a surface is mn. Since each bidentate adsorbate occupies two such sites, at most mn/2 Bd molecules can adsorb. However, an adsorbate molecule attached to a given ≡S site can form a bidentate surface complex by bridging to any of that site’s nearest neighbors, and each of these orientations represents an independent binding possibility. Therefore, the number of bidentate sites on the pristine surface is considerably larger than mn/2; simple geometric considerations indicate that, on an m × n array of sites, N≡S2,max is given by:
N≡S2,max ) 2mn - m - n
FIGURE 2. Loss of available bidentate sites accompanying adsorption of one molecule under various conditions. (a) Adsorption of a monodentate or bidentate adsorbate molecule onto the pristine surface; x’s indicate the sites that become excluded. (b) Binding of a bidentate adsorbate molecule (black circles connected by line) when several nearby sites have already been occupied by bidentate molecules (gray circles connected by solid lines). Three of the sites that would be excluded by the newly arriving molecule were already excluded, so only three more sites (marked by x’s) are excluded when the reaction occurs. excluded by adsorption of an Md or Bd molecule on the interior of a pristine surface is four or six, respectively. Because fewer sites are excluded if the molecule binds on the perimeter of the site array, the average number of sites excluded per adsorbing molecule when all possible adsorption sites are considered is slightly less than four or six, respectively. The exact value, referred to below as nex,Md or nex,Bd, respectively, depends on the size and shape of the site array. However, it is useful to note (again, based on geometric considerations) that for a rectangular m × n array:
nex,Md ) 4 -
2 2 m n
(14a)
nex,Bd ) 6 -
3 3 m n
(14b)
(13)
The corresponding expressions for other site geometries (e.g., hexagonally arranged sites) are different, but in all cases the number of bidentate sites on the unoccupied surface is substantially larger than the maximum number of bidentate molecules that can ultimately bind. Now consider the effect on the number of available bidentate sites when a single molecule adsorbs somewhere in the middle of the surface described above. If the adsorbate is monodentate, the reaction decreases the number of available bidentate sites by four, as indicated by x’s in Figure 2a. If the adsorbate is bidentate, seven bidentate sites are loststhe occupied site plus six surrounding sites; this scenario is also shown in Figure 2a. In the following discussion, bidentate sites that are unoccupied but are nevertheless unavailable because one or both of their end-members are occupied are referred to as being excluded. Thus, the number of bidentate sites
Combining eq 13 and 14 with the fact that N≡Stot ) mn yields the following relationships:
nex,Md nex,Bd N≡S2,max ) N≡Stot ) N≡Stot 2 3
(15)
In a given system, all the N values in eq 15 would be divided by the same value to convert them to molar concentrations, so the same relationships can be expressed as
c≡S2,max )
nex,Md nex,Bd ) c c 2 ≡Stot 3 ≡Stot
(16)
Now consider the situation when a molecule adsorbs to the surface after it is partially occupied. The bidentate sites that share an end-member with the newly adsorbing molecule are excluded from the pool of available ≡S2 sites, just as they VOL. 36, NO. 3, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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are on the pristine surface. However, some of those sites might already have been excluded as a result of previous adsorption reactions, so the incremental loss of available sites is less than on the pristine surface. An example of such a situation, in which adsorption of a Bd molecule reduces the number of available bidentate sites by only four (the occupied site plus three surrounding sites) is shown in Figure 2b. In the limit of a highly covered surface, molecules would adsorb primarily to sites whose nearest neighbors were all already occupied. In that case, adsorption of a Bd molecule would reduce the number of available bidentate sites by only one, and adsorption of an Md molecule would not reduce that number at all. Exact calculations for the rate of consumption of bidentate sites as a function of surface coverage become very complicated once even a few molecules are adsorbed. However, that rate can be approximated analytically if certain assumptions are made, as follows. Define f ′ as the likelihood of occupation of an ≡S site that is a nearest neighbor of a newly adsorbing molecule. For a system in which only monodentate molecules have adsorbed, the sites near a newly arriving Md molecule are just as likely to be occupied as are sites elsewhere on the surface, so f ′ equals the overall fractional surface coverage f. As noted above, when a Md molecule adsorbs, the bidentate sites that utilize the occupied site as one of their end-members (nex,Md sites, on average) are all excluded. If each of those sites had a probability f of already being excluded, the number of newly excluded sites would be nex,Md(1 - f). Thus, the change in the concentration of available ≡S2 sites accompanying adsorption of a differential concentration of Md molecules can be expressed as
dc≡S2 ) -nex,Md(1 - f ) dc≡SMd
(17)
Noting that c≡SMd ) c≡Stot f, so dc≡SMd ) c≡Stot df, eq 17 can be integrated to yield
nex,Mdc≡Stot c≡S2 ) (1 - f )2 + c1 2
(18)
where c1 is a constant of integration. Applying the boundary condition that c≡S2 ) c≡S2,max when f ) 0 and utilizing the fact that c≡S2,max ) nex,Md/2c≡Stot leads to the following simple results:
c≡S2 ) c≡S2,max(1 - f )2 )
c≡S2,max c≡S2 c≡Stot c≡Stot
(19)
(20)
Note that, if c≡S2,max/c≡Stot is identified as k2′, eq 19 is identical to eq 7. Thus, this analysis indicates that the ad hoc assumption leading to eq 7 (i.e., that the concentration of available bidentate sites is proportional to the square of the concentration of available monodentate sites, when both concentrations are expressed as mole fractions) can be justified based on geometric considerations if the surface is occupied by a monodentate adsorbate. If an analogous analysis is carried out for a system in which only Bd molecules adsorb, the expression corresponding to eq 17 is
dc≡S2 ) -{1 + nex,Bd(1 - f ′)} dc≡S2Bd
(21)
The first term in brackets in eq 21 accounts for the fact that bidentate sites that are actually occupied are guaranteed to be lost when a Bd molecule adsorbs, and the second term accounts for the exclusion of bidentate sites surrounding the adsorbing molecules. 310
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Unlike the case for monodentate adsorption, the value of f ′ in eq 21 cannot be equated with the fractional surface coverage f, because the ≡S sites surrounding an open ≡S2 site are slightly less likely to be occupied than ≡S sites elsewhere on the surface. The basis for this statement is that an ≡S site that is not adjacent to an open ≡S2 site might be occupied by a Bd molecule bridging between it and any of the sites around it; for a rectangular site array, there are four such sites. By contrast, an ≡S site that is adjacent to an open ≡S2 site can be occupied in one less way (i.e., three ways for a rectangular array); the space bridging the ≡S site to the open ≡S2 site cannot be occupied, or the ≡S2 site would no longer be open (it would be excluded). Thus, on a rectangular array, the ≡S sites adjacent to an open ≡S2 site are only approximately three-fourths as likely to be occupied as other ≡S sites on the surface. Under the circumstances, for a rectangular array, f ′ can be approximately related to f by
f′)
0.75c≡S2 + c≡S2Bd + c≡S2,ex f c≡S2 + c≡S2Bd + c≡S2,ex
(22)
where c≡S2,ex is the concentration of excluded sites. For other site configurations, the value of 0.75 might be different, but the general form of the equation would be identical. When eq 22 is inserted into eq 21, the differential equation cannot be integrated analytically. However, it can be integrated numerically (in conjunction with the boundary condition that c≡S2 ) c≡S2,max when c≡S2Bd ) f ) 0), to yield an expression for c≡S2 as a function of surface coverage by bidentate adsorption, i.e.
c≡S2 ) c≡S2,max -
∫
c≡S2Bd,fin
0
{1 + nex,Bd(1 - f ′)} dc≡S2Bd (23)
Summary of Equations for Estimating c≡S2. Summarizing the preceding discussion, eqs 3, 7, 12, 19, and 23 all purport to characterize the concentration of bidentate sites as a function of surface coverage on surfaces with overlapping bidentate sites. Equations 3 and 7 have been postulated based on conceptual arguments and are in common use, but they have never been derived formally. If those conceptual arguments are correct, then the equations should apply regardless of whether the occupied sites are filled by monodentate or bidentate adsorbates. Equation 12 was derived by Chang (14) for a surface occupied by bidentate adsorbate molecules, and its applicability to a system with monodentate adsorption has not been addressed. Equations 19 and 23 have been derived explicitly for surfaces occupied by monodentate and bidentate adsorbates, and the different results suggest that the concentration of open bidentate sites on a surface with a given fractional occupation differs depending on whether the occupied sites are populated by mono- or bidentate adsorbates. Interestingly, even though eq 3 has been used to characterize surfaces with overlapping sites, it can be derived from geometric arguments only if the sites are assumed to be nonoverlapping. Also, although eq 7 has been applied in systems with bidentate adsorbates, it is derivable from geometric arguments only if the adsorbates are monodentate. The quantitative predictions of the various equations for c≡S2 have never been compared with one another, and none of the equations has been validated by comparison with an independent data set. Therefore, neither the accuracy of the equations nor the magnitude of the errors associated with their use when they are not completely correct has ever been evaluated. In the following section, such comparisons are made.
FIGURE 3. Frequency distribution for the number of ≡S2 sites available on a 100 × 100 array of ≡S sites, after 1000 Bd molecules have adsorbed (2000 ≡S sites occupied, 8000 remaining). The abscissa shows the upper value of each range. The average value of c≡S2 from 100 simulations of this level of surface coverage was 13324.
Monte Carlo Simulation of the Availability of Bidentate Sites on a Surface Although the concentration of available ≡S2 sites in a system cannot be determined experimentally, it can be monitored in simulations of the surface loading process, allowing the relationship between c≡S2 and c≡S to be explored directly. To carry out such an analysis, Monte Carlo (MC) simulations were run in which adsorbate molecules were randomly allocated to binding locations on a 100 × 100 Cartesian array of ≡S sites. Adsorption of only one type of molecule (either Md or Bd) was considered in each simulation. When adsorption of bidentate molecules was simulated, Bd molecules were assumed to be capable of binding to a pair of vertically or horizontally aligned sites but not to sites along a diagonal. For the given assumptions, the unoccupied surface comprised 10 000 monodentate and 19 800 () mn - m - n, with m ) n ) 100) bidentate sites. In the simulations, all of these sites were numbered, from M1 to M10 000 and from B1 to B19 800, respectively. To simulate binding of the first bidentate molecule to the surface, a random number between 1 and 19 800 was selected, and an adsorbate molecule was assigned to the corresponding bidentate site. The two monodentate end-members of that site were then identified, along with all of the other bidentate sites that involved those end-members. These bidentate sites were precluded from future occupation (i.e., they became excluded sites). Next, another random number between 1 and 19 800 was selected. If the corresponding bidentate site was not already occupied or excluded, an adsorbate molecule was assigned to it, and the newly excluded sites were added to the list of such sites. The whole process was repeated until the desired number of molecules had been assigned to sites. If a random number was generated that would have assigned a molecule to an occupied or excluded site, no assignment was made, and another random number was generated. To simulate adsorption of Md molecules, the procedure was the same as described above, except that the random numbers selected ranged from 1 to 10 000. After each increment of 200 sites had been occupied (i.e., after adsorption of every 100 Bd or every 200 Md molecules), the number of available bidentate sites remaining on the surface was counted. The surface loading process was then continued until 90% surface coverage was achieved. At that point, a new simulation was initiated. One hundred simulations were run for each type of adsorbate. Although the simulations did not account for either desorption or surface diffusion directly, the results can be considered to apply to systems where those processes occur,
because each simulation could be interpreted as the result of such processes instead of adsorption alone. That is, for each level of surface coverage, the 100 simulations considered 100 different arrangements of a given number of molecules on the surface. In the simulations, those arrangements were generated by loading an empty surface with adsorbate molecules 100 times. However, the same arrangements could be obtained if the molecules were loaded onto the surface only once and then were allowed to diffuse randomly around the surface, or if they were loaded once and then were allowed to desorb and resorb as a result of dynamic equilibrium with the solution. Thus, the results can be interpreted equally legitimately as representing the average surface state of 100 particles onto which irreversible adsorption of nondiffusing molecules has occurred, or as the average surface state of any single particle over time in systems where adsorption is reversible or where the adsorbate molecules are able to diffuse along the surface. Probability Distribution of Bidentate Site Availability for Partially Occupied Surfaces. Logically, the number of available bidentate sites on a partially occupied surface is expected to depend not only on the fractional surface coverage but also on the pattern of surface occupation. For instance, if the adsorbed molecules happen to be tightly clustered on one part of the surface, the number of bidentate sites open on the rest of the surface will be larger than if the adsorbed molecules are more uniformly distributed. While the former distribution is statistically unlikely, it is possible, and the MC simulations allow that possibility. As a result, the number of available bidentate sites varied from one simulation to the next, even for the same surface coverage. The results for 20% surface coverage are shown as a frequency distribution in Figure 3. The distribution was extremely narrow, with all 100 simulations yielding values for the number of available bidentate sites in the range 13 330 ( 50, corresponding to 67.3 ( 0.3 % of the 19 800 sites on the pristine surface. Converting the values from numbers of sites to molar concentrations does not change their ratios, so the conclusion is that 20% surface coverage reduces c≡S2 by almost 33%. The frequency distributions for other levels of surface coverage were comparably narrow. Interestingly, when the surface was 90% occupied, the average surface state had only 33 open bidentate sites, accounting for 66 open ≡S sites. The remaining 934 open ≡S sites were isolated, meaning that they were completely surrounded by ≡S sites that were occupied by one end of an adsorbed bidentate complex. This result does not imply that the surface could never become 95% or even 99% occupied by bidentate complexes; however, it does mean that, if such VOL. 36, NO. 3, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 4. Monte Carlo simulations for the number of ≡S2 sites remaining available on a surface with a 100 × 100 array of ≡S sites, as a function of the number of monodentate or bidentate molecules sorbed. (a) Surface coverage
