Mass Action Expressions for Bidentate Adsorption ... - ACS Publications

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Mass Action Expressions for Bidentate Adsorption in Surface Complexation Modeling: Theory and Practice Zimeng Wang and Daniel E. Giammar* Department of Energy, Environmental and Chemical Engineering, Washington University, St. Louis, Missouri, United States ABSTRACT: The inclusion of multidentate adsorption reactions has improved the ability of surface complexation models (SCM) to predict adsorption to mineral surfaces, but variation in the mass action expression for these reactions has caused persistent ambiguity and occasional mishandling. The principal differences are the exponent (α) for the activity of available surface sites and the inclusion of surface site activity on a molar concentration versus fraction basis. Exemplified by bidentate surface complexation, setting α at two within the molar-based framework will cause critical errors in developing a self-consistent model. Despite the publication of several theoretical discussions regarding appropriate approaches, mishandling and confusion has persisted in the model applications involving multidentate surface complexes. This review synthesizes the theory of modeling multidentate surface complexes in a style designed to enable improvements in SCM practice. The implications of selecting an approach for multidentate SCM are illustrated with a previously published data set on U(VI) adsorption to goethite. To improve the translation of theory into improved practice, the review concludes with suggestions for handling multidentate reactions and publishing results that can avoid ambiguity or confusion. Although most discussion is exemplified by the generic bidentate case, the general issues discussed are relevant to higher denticity adsorption.

1. INTRODUCTION Adsorption affects the fate, transport and bioavailability of contaminants and nutrients in both natural and engineered aquatic systems. Over the past forty years surface complexation modeling (SCM) has emerged as a powerful tool for describing adsorption processes at solid-water interfaces.1−5 In contrast to the constant-Kd, Langmuir and Freundlich models, SCMs have enabled predictions of the impact of solution chemistry (e.g., pH and ionic strength) on the binding of inorganic aqueous solutes to solid surfaces with a single set of model parameters.6,7 SCMs calculate adsorption equilibrium through mole balance and mass action equations. Coulombic correction factors that involve calculations of the surface charge and potential are included to account for the contribution of electrostatic interactions to the overall energetics of adsorption of sorbate molecules to surfaces. SCMs have been incorporated in widely used chemical equilibrium software programs8−12 and reactive transport models.13−16 The information of surface speciation simulated by SCMs has also been used in developing kinetic models for various chemical processes at solid-water interfaces.17−19 The determination of the stoichiometry and mass action equations of surface complexation reactions is an essential step in model formulation.20 With the developments of density functional theory (DFT) calculation and spectroscopic techniques, particularly extended X-ray absorption fine structure (EXAFS), molecular-scale evidence for multidentate surface complexes has been observed for a wide range of adsorbates (e.g., heavy metals and metalloids) and adsorbents (e.g., metal oxides and clays).21−26 © XXXX American Chemical Society

The spectroscopically determined structures of surface complexes are useful constraints in developing predictive SCMs for adsorption processes.20,27,28 Consequently, multidentate surface complexation reactions are increasingly important in the stillgrowing SCM literature (Figure 1). Multidentate surface complexes form through binding of an adsorbate to two or more adjacent functional groups. While the molecular-scale structure of surface complexes can be complicated depending on the mineral surfaces involved,29−31 pragmatic practices of surface complexation modeling treat the surface as a collection of equivalent functional groups or divide them into a small number of groups (i.e., strong and weak sites or based on structures of specific crystal faces3,5). As bidentate binding is the most common scenario, this review focuses on bidentate surface complexation, although the issues discussed are also relevant to tridentate and higher denticity adsorption. A generic representation of bidentate surface complexation is S2 + B ⇌ S2 B

KB =

{S2 B} {S2 }{B}

(1)

(2)

Received: December 18, 2012 Revised: April 2, 2013 Accepted: April 3, 2013

A

dx.doi.org/10.1021/es305180e | Environ. Sci. Technol. XXXX, XXX, XXX−XXX

Environmental Science & Technology

Critical Review

Figure 1. Upper panel: The growth of the SCM literature and the increasing interest in multidentate surface complexation over the last three decades. Lower panel: Timeline of the key references that involved or addressed the issue of the mass action law for multidentate surface complexation reactions. The literature survey was performed using Google Scholar as of March 2013. Searching criteria were set with the keyword “surface complexation model” for the entire SCM literature. Additional keywords of bidentate, tridentate, or multidentate were then used to identify the subset of references on this subject. In this literature survey, not all publications returned by the search engine actually implemented multidentate surface complexation models; some may have just discussed the issue while implementing monodentate surface complexation. Regardless of the extent of implementation, the survey illustrates increasing attention and interest in the coordination denticity of surface complexes in adsorption modeling.

Equation 2 is the mass action expression, which relates the equilibrium constant (KB) to the activities (denoted by curly brackets) of free sorbate molecules (B), unoccupied bidentate surface sites (S2) and bidentate surface complexes (S2B). Each bidentately adsorbed molecule occupies two surface sites (S), so the mole balance on surface sites in terms of molar (mol/L) concentration (denoted by square brackets) is given as eq 3. The mole balance on sorbate B is given as eq 4. [S] + 2[S2 B] = [S]tot

(3)

[B] + [S2 B] = [B]tot

(4)

K=

(6)

As discussed in detail later, the value of α is determined by the specific assumption for relating the amounts of S2 and S. Solving eqs 2−4 to determine equilibrium speciation requires (1) a definition of standard states for relating the activities of surface species to their concentrations and (2) a numerical relation between the amounts of available monodentate sites and bidentate sites. Different treatments of these two issues have led to different and sometimes confusing formulations of multidentate surface complexation models (Table 1). Although there have been several insightful theoretical discussions on this issue from the perspectives of geochemists and interface scientists (Figure 1), mishandling and ambiguity in multidentate adsorption modeling still persists. Several significant publications on this subject that appeared in the geochemistry literature have not been widely cited in the implementation of models by the more applied community of environmental scientists and engineers. The objective of this critical review is to help bridge the gap between research on the theory of adsorption reactions and research that applies surface complexation modeling to interpret adsorption data. Review of previous theoretical investigations delineates the evolution of our understanding of multidentate adsorption modeling. A compilation of recent SCM publications is presented with an evaluation of the appropriateness of their SCM presentations. The implications of selecting an approach are then illustrated with a previously published data set and model. Practical suggestions for SCM

Unoccupied sites are usually treated as having pH-dependent protonation states (e.g., SOH, SOH2+, and SO−), but for the purposes of a focused examination of the mass action expression, surface site protonation states are not considered here. Bidentate complexes could also involve one sorbate molecule reacting with two sites of different types, which are often invoked in more advanced surface complexation models.32 These particular scenarios are not specifically discussed here, where we consider the simplest case of only one type of surface site, but the present analysis has general applicability for bidentate surface complexation. Even if the concentration of monodentate sites S is known, which is a challenge in itself, the concentration of S2 cannot be readily quantified because it involves combinations of two specific S sites. Therefore the adsorption reaction and mass action expression are generally written as33 2S + B ⇌ S2 B

{S2 B} {S}α {B}

(5) B



Critical Review

Table 1. Summary of Mass Action Expressions Revieweda approach

numerical scale of activity

mass action equationb (for n-dentate)

adsorption isothermc (for bidentate n = 2)

model 1

molarity or molality

K1 =

[SnB] [S]n [B]

q1 =

model 2

molarity or molality

K2 =

[SnB] [S][B]

q2 =

model 3

mole fraction

K3 =

[SnB]/[S]tot ([S]/[S]tot )n [B]

q3 =

model 3

coverage fraction

K3′ =

n[SnB]/[S]tot ([S]/[S]tot )n [B]

q3′ =

model 3d

mole fraction and referenced to a fixed hypothetical N* and A*

Kθ =

[SnB] (NA)n n − 1 Cs [S]n [B] N *A*

equivalent to q3 and expressed with equilibrium constant Kθ that is independent of N and A.

QCA modele

mole fraction

K QCA =

[SnB]/[S]tot ([S]/[S]tot )α [B]

4[B]K1[S]tot −

8[B]K1[S]tot + 1 + 1

8[B]K1[S]tot [B]K 2 2[B]K 2 + 1 4[B]K3 −

8[B]K3 + 1 + 1 8[B]K3

2[B]K3′ −

qQCA =

4[B]K3′ + 1 + 1 4[B]K3′

4[B]K QCA −

4[B]K QCA + 1 + 1

2(4[B]K QCA + 1)

a

For simplicity of presentation, conditions of unity activity coefficient and non-electrostatic interaction are assumed. b[] indicates concentrations in mol/L of solution. S indicates the monodentate site. B is the adsorbate molecule. SnB is a generic form of n-dentate surface complex of adsorbed B. [S]tot is the total concentration of monodentate sites determined by the sorbent concentration, site density, and specific surface area. cThe adsorption isotherm is expressed in a form of sorption density q = [S2B]/[S]tot as a function of the free (i.e., dissolved) concentration of B at equilibrium. Isotherms for n > 2 have much longer formulas and are not presented here. Figure 2(c and d) generated from the example model illustrate the different trends of the isotherms. dThe equation shown here is not strictly the mass action equation, but the final form of analytical relationship between the 1.0 mol/L and the new standard states (see derivations in Sverjensky 2003). N and A are site density and specific surface area of the sorbent, respectively. As originally suggested by Sverjensky, N* = 10 × 1018 sites/m2 and A* = 10 m2/g. eThe exponent α is a function of q in a form of α = n − (n − 1) log (1 − q)/log(1 − nq). q is constrained to be from 0 to 1/n and thus the value of α increases with increasing surface coverage from (n2 − n+1)/n to n. For n = 2, 1.5< α 0.1). The isotherms simulated at a lower sorbent loading again illustrated the artifact of model 1 (Figure 2d). Models 2 and 3 both produced isotherms independent of sorbent concentrations, but model 1 greatly underestimated the extent of adsorption. Consistent with eq 10, the initial slope of the isotherm simulated by model 1 in this example is proportional to the sorbent concentration. 4.4. Practical Suggestions for SCM Practitioners. The present discussion highlights the practical significance of the earlier theoretical studies on multidentate adsorption modeling and points to actions that can reshape future SCM literature. Technical Clarity of Publication. Publications should clearly explain the mass action expression used for individual models in addition to merely tabulating the equilibrium constants given by software programs. Complete clarity of the mass action law in SCM must include explicit information on the standard state (i.e., the numerical scale used for activity of surface species), the value of exponents of the activity of available monodentate sites (1 or n), and the relevant settings in the software program used (on a basis of molarity, mole fraction, or coverage fraction). Precaution for Model 1. Direct application of model 1 for multidentate surface complexation modeling (i.e., molar-based standard states with α = denticity) should be avoided. While the use of model 1 may provide a satisfactory fit to a data set at a single sorbent concentration, it cannot be used for other sorbent concentrations. Modelers need to be aware of the possible repercussions and take precautions when referring to previous SCM publications. Acceptable Use of Model 1: Conversion to Model 3. The use of model 1 is only acceptable if the explanatory text for the model specifies the conditions of sorbent loading at which the reported molar-based equilibrium constants are obtained and provides information about how to apply the model at different solid concentrations (Example refs 74, 104, 106, and 110). For example, the information of eq 26 should be presented for bidentate surface complexation reaction

Table 4. U(VI)−Goethite Surface Complexation Model As an Illustrative Example log Ka reactions in the example model

model 1 model 2

model 3

surface complexation reactionsb SOH + H+ ⇌ SOH2+ 7.0c 7.0c c − + SOH ⇌ SO + H −10.0 −10.0c 2SOH + UO22+ ⇌ (SO)2UO2 + 2H+ 0.4c, −3.61d f aqueous reaction of uranyl UO22+ + H2O ⇌ UO2OH+ + H+ −5.2 −10.3 UO22+ + 2H2O ⇌ UO2(OH)2(aq) + 2H+ −19.2 UO22+ + 3H2O ⇌ UO2(OH)3− + 3H+ −33.1 UO22+ + 4H2O ⇌ UO2(OH)42− + 4H+ −2.7 2UO22+ + H2O ⇌ (UO2)2(OH)3+ + H+ −5.6 2UO22+ + 2H2O ⇌ (UO2)2(OH)22+ + 2H+ −11.9 3UO22+ + 4H2O ⇌ (UO2)3(OH)42+ + 4H+ −15.6 3UO22+ + 5H2O ⇌ (UO2)3(OH)5+ + 5H+ −31.0 3UO22+ + 7H2O ⇌ (UO2)3(OH)7− + 7H+ −21.9 4UO22+ + 7H2O ⇌ (UO2)4(OH)7+ + 7H+

7.0c −10.0c 0.70−log Cse

a

The tabulated equilibrium constants were all based on the standard state of 1.0 mol/L as input in FITEQL. bDiffuse double layer model: specific surface area = 35 m2/g, pHzpc = 8.6, site density = 2.2 μmol/m2 = 1.32 sites/nm2. [Note there is a corrigendum145 associated with the original article,111 where the unit of density was corrected.] cFrom the original model in Missana et al.111 The mass action expression for this bidentate surface complexation reaction is as model 1, for a solid concentration of 2 g/L. dObtained by fitting the adsorption edge data set at 2 g/L goethite (Figure 2a) using FITEQL with the mass action expression of model 2. eCs is the concentration of goethite (g/L). For Cs = 0.16 g/L, log K = 1.50. Model 1 was converted to model 3 by modifying the FITEQL-input equilibrium constants in inverse proportion to sorbent concentration. fThe constants used in the model published by Missana et al.111 Some constants are different from the most updated values from the database published by the Nuclear Energy Agency.146

Both models 2 and 3 can be used to provide a good fit to the experimental data. Model 1 can only provide a good fit at a single sorbent concentration. With the approach of Model 1, we used FITEQL 49 to reproduce the Missana et al. model111 to fit the experimental edge data set at 2 g/L sorbent loading (Figure 2a). We then implemented model 2 by modifying the mass action coefficient of [SOH] from two to one and redetermined K by optimizing the fit of the experimental data at 2 g/L goethite. Model 1 was converted to model 3 by modifying the FITEQL-input equilibrium constants in inverse proportion to sorbent concentration. At a fixed sorbent concentration, models 1 and 3 are equivalent. With a different log K, the approach of model 2 could also simulate the experimental data (Figure 2a). The artifact of model 1 discussed earlier using eqs 9 and 10 arises when the model is implemented at a different goethite concentration (Figure 2b). At a lower sorbent concentration (0.16 g/L), the adsorption predicted using model 1 decreases significantly more than was observed in experiments. In the original publication, exactly the same behavior of underestimating adsorption at decreased solid-to-solution ratio was observed (See Figure 9 in Missana et al.111); however, this was interpreted as an indication of “the importance of validating surface complexation models on the widest range of experimental condition for reliable application” and not by recognizing that it was an artifact of using model 1. In our present analysis, models 2 and 3 both yield good agreements

K3 = K1[S]tot = K1(NACs) J

(26)



Critical Review

Figure 2. SCM simulations of adsorption edges (panels a and b) and isotherms (panels c and d) in this example. Symbols are experimental data extracted from the original Missana et al. publication.111 As done in the original publication, adsorption edge simulations were performed at a total U(VI) concentration of 0.44 μmol/L at two different goethite concentrations (2 and 0.16 g/L). Simulations of isotherms were performed at pH 7.0. The ionic strength for all simulations was set at 0.1 mol/L.

Comparison of Models 2 and 3. Both models 2 and 3 can avoid the artifact associated with the sorbent concentration effect. The underlying assumptions and mathematical nature of models 2 and 3 lead to similar behaviors of the model at low surface coverage but to predictions of greater adsorption extents in model 2 as saturation is approached. For most natural systems, where the surfaces of adsorbing minerals are often far from saturation, models 2 or 3 can be used with equal success. However, for engineered adsorption processes where surface sites are highly utilized, the difference between the two approaches may become significant. Model 3 agrees better with the QCA theory and Monte-Carlo simulation, and it complies with the new more rigorous standard state framework for surface species.

where K3 is the instrinc constant and K1 is the constant for model 1 that is dependent on the sorbent loading (N = site density in mol/m2, A = specific surface area in m2/g, Cs = sorbent concentration in g/L). Therefore the value of K1, which is the input equilibrium “constant” in software programs with molar-based activity system, should vary in inverse proportion to Cs. Implementation of Model 3. Application of model 3 within a molar-based standard state system requires equilibrium constants to be dynamically updated if the amounts of adsorbing materials vary along a flow path or in a treatment process. Alternatively, as in updated versions of PHREEQC, Visual MINTEQ, ECOSAT, and TOUGHREACT, its implementation has already been enabled by modifying the quantification scale of the activity of surface species. If the mass action formulation of the SCM used in the program is fraction-based, clear and consistent representation of equilibrium constant expressions (as with model 3) should be included (example refs 102, 103, and 124), instead of implicitly assuming that the entire readership is knowledgeable about the programs used in every study. Accounting for the Variations of Site Density and Specific Surface Area. If the studies (for example, ref 110) involve comparison of sorbents with the same identity but different specific surface areas or site densities and the models are developed based on 1.0 mol/L standard state, then the conversion equations proposed by Sverjensky63 (e.g., eqs 17−19) can be applied to obtain instrinsic equilibrium constants regardless of the amounts and properties of the sorbent. The conversion equations are also useful for applying constants from a model in one study to a related model in a different study (for example, ref 125). Application of these conversion equations for multidentate adsorption also avoids the artifact of model 1 by inherently converting model 1 to model 3.

5. CONCLUSIONS The appropriate formulation of mass action expression for multidentate surface reaction is admittedly not a new topic in SCM research. In fact, as this Critical Review summarized, it has been discussed by adsorption phenomena theoreticians along with progressive developments and extension of surface complexation theory for at least the last two decades. The primary theoretical issues are on (1) the inherent limitation of the traditional 1.0 mol/L standard state framework for surface species and (2) the relationship between the quantity of available multidentate sites to that of available monodentate sites. The insights provided by these theoretical studies have shaped the literature of SCM, resulting in a variety of mass action formulations for multidentate surface complexation reactions. However, these new insights have not been fully reflected in recent SCM publications so that mishandling and ambiguity of multidentate adsorption modeling are still continuously seen. There are practical steps and guidelines K



Critical Review

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that can be followed that can lead to greater improvements in SCM practice based on advances in SCM theory.

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AUTHOR INFORMATION

Corresponding Author

*Address: Campus Box 1180, One Brookings Drive, St. Louis, MO 63130. Phone: 314-935-6849. Fax: 314-935-7211. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The critical comments and constructive suggestions of as many as seven anonymous reviewers (three on this review article and four on an earlier version submitted as a research article) substantially improved the depth, comprehensiveness, and clarity of the manuscript. We appreciate communications with Michelle Scherer and David Dzombak that provided us motivation and suggestions in writing this Critical Review based on the earlier research article version. We also thank Ruben Kretzschmar for his efforts in handling the present manuscript. Discussions with Philippe Van Cappellen (University of Waterloo) and Wenming Dong (Lawrence Berkeley National Laboratory) were instructive. This research was supported by the U.S. Department of Energy, Office of Science, Subsurface Biogeochemical Research Program (DESC0005324). Partial funding was provided by the National Science Foundation (BES 0608749).

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REFERENCES

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