11718
Langmuir 2008, 24, 11718-11721
Eigen Kinetics in Surface Complexation of Aqueous Metal Ions Herman P. van Leeuwen Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ReceiVed May 8, 2008. ReVised Manuscript ReceiVed July 24, 2008 The mechanism of chemisorption of aqueous metal ions at surfaces has long been a topical issue in such fields as soil chemistry and bioenvironmental science. Here it is quantitatively demonstrated for the first time that release of water from the inner hydration shell is the rate-limiting step in inner-sphere surface complexation. The reactive intermediate is an outer-sphere complex between metal ion and surface site, with an electrostatically controlled stability defined by Boltzmann statistics. Using tabulated dehydration rate constants for metal ions, the resulting scheme allows for prediction of rates of sorption of aqueous metal ions at any type of complexing surface.
1. Introduction The majority of metal complex formation reactions in aqueous solution follow the Eigen mechanism, in which release of water from the inner coordination sphere of the metal is the rate-limiting step. For complexation of aqueous metal ions with ligand sites on surfaces, it has so far only been noted that there seems to be a “correlation” between the rate of surface complex formation and the inner-sphere dehydration rate of the metal. To date, a comprehensive mechanistic basis for surface complexation kinetics in aqueous media has not been available.1-3 Here we explore a 2D analogue of the Eigen complexation kinetics in which the formation of a precursor surface outer-sphere complex and dehydration of the metal ion are the crucial steps. The ensuing surface complexation rates will be set against experimentally determined rates for metal ion adsorption at well-defined oxide surfaces. The main purpose is to finally resolve the basic question concerning the applicability of Eigen complexation kinetics to metal sorption at surfaces. The result will facilitate understanding and predictability of kinetic features of aqueous metal ion binding by surfaces, which is of fundamental relevance for such disparate fields as corrosion science, (bio)reactor technology, ecosystem dynamics, and many others.
2. Theory 2.1. Eigen Scheme for Surface Complexation. The majority of metal complex formation reactions in aqueous solution follow the Eigen mechanism.4 It comprises the formation of a precursor ion pair between the hydrated metal ion and the complexing agent. The more common situation is that the ion pair formation is fast compared to the subsequent substitution of a water molecule from the inner hydration sphere for the coordinated ligand.5 An analogue of the Eigen scheme for the complexation of a hydrated metal ion M(H2O)62+ with a surface ligand site >–S- can be formulated as4 (1) Stumm, W. Chemistry of the Solid-Water Interface; Wiley: New York, 1992. (2) Vakros, J.; Kordulis, C.; Lycourghiotis, A. Langmuir 2002, 18, 417–422. (3) Pinheiro, J. P.; Minor, M.; van Leeuwen, H. P. Langmuir 2005, 21, 8635– 8642. (4) Eigen, M. Pure Appl. Chem. 1963, 6, 97–115. (5) Buffle, J.; Zhang, Z.; Startchev, K. EnViron. Sci. Technol. 2007, 41, 7609– 7620.
S KOS
>–S- + M(H2O)62+ {\}
>–S·M(H2O)6+ outer sphere surface ion pair
(fast) (1)
kw
>–S·M(H2O)6+ 98 >–SM(H2O)5+ + H2O (rate-limiting) inner-sphere surface complex
(2) Figure 1 sketches the setting of the outer-sphere surface complex, which functions as the precursor in the eventual innersphere surface complex formation process. Except perhaps some complexation reactions of the metal ions with extremely fast dehydration rates, the ion pair formation is the faster step in the overall process (see Buffle et al.5 for a recent discussion). The first step, given by eq 1, therefore, usually obeys the equilibrium condition
Ksos )
{>–S·M(H2O)6+} mol-1 · m3] 2+ - [ M H O >–S [ ( 2 )6 ]{ }
(3)
s where Kos is the thermodynamic stability constant of the surface outer-sphere complex, {>–S · M(H2O)6+} its surface concentration (mol m-2), [M(H2O)2+ 6 ] the volume concentration of M in adjacent solution (mol m-3), and {>–S- } the surface concentration of free ligand sites (mol m-2). Similar to the known situation for 3D complexation reactions, the rate constant kw for release of H2O from the inner coordination sphere of M is essentially independent of the nature of the ligand site >–S- . This implies that the rate of surface complex formation, Rsa, is given by kw and {>–S · M(H2O)6+}. Using eq 3, Rsa follows as
Rsa ) kwKsos[M(H2O)62+]{>–S-} [mol · m-2 · s-1]
(4)
s kwKos
in which the product can be identified with the surface complex formation rate constant, ksa, or, in the jargon of interfacial chemistry, the metal ion adsorption rate constant, kad:
kwKsos ≡ ksa ≡ kad [mol-1·m3·s-1]
(5)
In the case of dispersions of particles with ligand sites at their surfaces, the rate of adsorption, Rsa, is often given per unit volume with the surface concentration {>–S- } represented by its 3D equivalent, i.e. [>–S- ]
[>–S-] ) {>–S-}A/V
(6)
where A/V is the total surface area of the particles per unit volume of dispersion.
10.1021/la8014332 CCC: $40.75 2008 American Chemical Society Published on Web 09/23/2008
Surface Complexation of Aqueous Metal Ions
Langmuir, Vol. 24, No. 20, 2008 11719
Usos )
Figure 1. Outer-sphere surface ion pair of a metal ion M2+ with an intact inner-sphere hydration shell and a negatively charged surface ligand site >–S-. The center-to-center distance between M and S is denoted ros.
2.2. Stability of the Surface Outer-Sphere Complex. Following the classical approach by Fuoss,6 the stability constant s of the surface outer-sphere complex, Kos , can be defined by applying Boltzmann statistics to the distribution of M2+ over its free form, M(H2O)62+, in bulk solution and its surface ion pair form, >–S · M(H2O)6+. Writing the number ratio of the latter over the former as nos/nsln, we have
nos gos exp(-U sos/kT) ) nsln gsln
(7)
s where Uos is the potential free energy for an individual surface ion pair with respect to the standard state of M2+ in aqueous solution, and gos and gsln represent the degeneracies for the surface outer-sphere complex form and the solution form of M2+, respectively. These degeneracies are proportional to the pertaining volumes:
gos ) nSV sos/Vsln gsln
(8)
s where Vos is the volume occupied by an individual metal ion, including its inner hydration sphere, in its outer-sphere surface complex, nS is the total number of free sites >–S- , and Vsln is the volume of the aqueous solution. Realizing that [M(H2O)62+] equals NAvnsln/Vsln (NAv is Avogadro’s number) and combining eqs 7 and 8 with 3 leads to
K sos )
NAvnSV sos exp(-U sos/kT) (nS - nos)
(9)
which, for the case of a sufficiently large excess of surface ligand sites over metal ions M, with (nS - nos) ≈ nS, reduces to
K sos ≈ NAvV sos exp(-U sos/kT) s Uos ,
(10)
In order to define let us consider a planar surface with a regular distribution of immobile surface sites >–S- . The primary Coulombic term in the formation of the pair>–S · M(H2O)6+ is defined by the respective charges, zM and zS (here +2 and -1), and the center-to-center distance, ros, between M and S with one hydration layer included (see Figure 1). There are secondary electrostatic terms for the attraction/repulsion of M(H2O)62+ by adjacent charged sites on the surface and for screening by the electrolyte solution and the substrate phase. Thus, the total electrostatic energy involved in the formation of an individual surface ion pair>–S · M(H2O)6(zM+zS) from M(H2O)6zM and >–S zS constitutes the following elements: (6) Fuoss, R. J. Am. Chem. Soc. 1958, 80, 5059–5061.
e2zMzS + surface charge term + screening term 4πε0εros (11)
Depending on the distribution of negative and positive charges on the surface, the surface charge term may be either attractive or repulsive. In the smeared-out approximation, it may be formulated as a double layer correction term that accounts for merely bringing the M(H2O)6zM+ to a distance ros from the surface. Stumm,7,8 and others after him,9-13 confused this double layer correction on the local activity of M(H2O)6zM+ with the complete s Kos for an indiVidual ion pair with a particular >–S- . The final term in eq 11 stands for the electrostatic screening of the >–S · M(H2O)6(zM+zS) surface ion pair by the electrolyte in solution and, if applicable, ionic conduction in the substrate phase. For relatively small surface potentials, the ionic composition in the double layer at the solution side of the interface is not too different from that in the bulk solution. Under these conditions, this screening term may be approximated on the basis of semi-infinite Debye-Hu¨ckel electrostatics, although the exact formulation for an ion pair at an interface between two phases with different permittivities and conductivities is more involved. For details, see the pertaining Debye-Hu¨ckel type analysis by Netz.14,15 Note that the surface charge term and the screening term are of a different origin. With surface charge densities approaching zero, the double layer correction term vanishes, whereas the screening term remains. For practical significance and elaboration of the two terms, see section 3 below. 3. Application. Let us now analyze some representative experimental findings for which sufficiently detailed and quantitative data on the nature of the surface complexation reaction are available. The study by Grossl et al.9 on the kinetics of adsorption of Cu2+ on crystalline goethite particles (R-FeOOH) comprises the data necessary for the present comprehensive reconstruction purposes. It measures the temporal change of the surface concentration of Cu2+ bound to oxidic sites, {>– OCu+}, upon perturbation of adsorption/desorption equilibrium by a pressure jump technique. 3.1. Cu2+ Binding at Goethite Surface Sites. At the studied pH, the relevant oxidic Cu2+-binding sites on the goethite surface are present at a surface density of 1.7 × 10-6 mol m-2, i.e. about 1 ligand site per nm2.16 On the whole, the surface has an appreciable positiVe charge density, giving rise to a local potential of approximately +110 mV at the position of the Cu2+ ion in its outer-sphere ion pair with the surface site.17 The goethite dispersions studied had an area-to-volume ratio, A/V, of 5 × 105 m-1, and the characteristic adsorption time τad was found to be in the range of 20 to 40 ms, with solid evidence that diffusion of Cu2+ in solution is fast on this time scale.18 Since the (7) Stumm, W. Colloid Surf., A. 1997, 120, 143–166. (8) Stumm, W. Colloid Surf., A. 1993, 73, 1–18. (9) Grossl, P. R.; Sparks, D. L.; Ainsworth, C. C. EnViron. Sci. Technol. 1994, 28, 1422–1429. (10) Hachiya, K.; Sasaki, M.; Saruta, Y.; Mikami, N.; Yasunaga, T. J. Phys. Chem. 1984, 88, 23–27. (11) Hachiya, K.; Sasaki, M.; Ikeda, T.; Mikami, N.; Yasunaga, T. J. Phys. Chem. 1984, 88, 27–31. (12) Jeon, B.-H.; Dempsey, B. A.; Burgos, W. D.; Royer, R. A.; Roden, E. E. Water Res. 2004, 38, 2499–2504. (13) Ataliglou, T.; Bourikas, K.; Vakros, J.; Kordulis, C.; Lycourghiotis, A. J. Phys. Chem. B 2005, 109, 4599–4607. (14) Netz, R. R. Phys. ReV. E. 1999, 60, 3174–3182. (15) Netz, R. R. Eur. Phys. J. E 2000, 3, 131–141. (16) Hiemstra, T.; de Wit, J. C. M.; van Riemsdijk, W. H. J. Colloid Interface Sci. 1989, 133, 105–117. (17) Hiemstra, T. Personal communication, 2008. (18) Pohlmeier, A.; Lustfeld, H. J. Colloid Interface Sci. 2004, 269, 131–142.
Van Leeuwen
11720 Langmuir, Vol. 24, No. 20, 2008
experimental data were obtained for a large excess of sites over Cu(II), the surface complexation reaction rate Rsa (cf. eq 4) is pseudo-first-order and comes to
Rsa )
d{>–OCu+} ) kad′ [Cu2+](V/A) dt
(12)
where
kad′ ) kwK sos{>–O-}(A/V) [s-1]
(13)
Calculation of kad′ on the basis of the 2D Eigen approach proposed above will be carried out with kw ≈ 109 s-119,20 for a (1:1) background electrolyte concentration of 10-2 M. The surface of crystalline goethite (R-FeOOH) generally comprises three types of crystal faces (100, 010, 001) which occur in different proportions. Within the framework of the CDMUSIC model,16 the relevant surface sites for the type of goethite used by Grossl et al.9 are FesOH1/2-, Fe2dOH, and Fe3tO1/2-. For details on the meaning of the formal charges, the site densities, etc., see Hiemstra et al.16 Around pH 6, the FesOH1/2- is fully protonated to FesOH21/2+, and about 30% of the Fe3tO1/2- is protonated to Fe3tOH1/2+, whereas the protonation/deprotonation of Fe2dOH is hardly significant. The resulting surface charge density is approximately +100 mC m-2, composed of +300 mC m-2 due to FesOH21/2+ and -200 mC m-2 due to Fe3tO1/2-. For the outer-sphere ion pair formation with Cu2+, the Fe2dOwould be the electrostatically most attractive type of site. However, due to protonation, its density is very low, which still makes it kinetically unimportant at pH around 6.21 As a consequence, the Fe3tO1/2- is the most important site for ion pair formation with Cu(H2O)62+. The Fe3tO type of site has an average total surface density of 1.5 nm-2; hence, at pH around 6 the {Fe3tO1/2-} is approximately 1 nm-2, that is 1.7 × 10-6 mol m-2. The eventual inner-sphere complex may very well be of a bidentate nature.17,22 For the kinetics of the complex formation, this aspect may be immaterial if the formation of the 1:2 complex is faster than that of the preceding 1:1 complex, which is the usual case for 3D complexation.23 This level of detail in the rate of Cu2+ binding by goethite still awaits experimental confirmation. 3.2. Predicted 2D-Eigen Rate of Cu2+/Goethite Surface s Complex Formation. Vos for Cu2+/Goethite. The volume, Voss, available for an individual Cu(H2O)62+ in the approximate hemisphere around an Fe3tO1/2- type of site at the goethite surface is
2 V sos ≈ π(ros3 - rO3) 3
(14)
where ros and rO are the Cu2+ · · · O center-to-center distance in Cu2+(H2O)-O and the radius of the oxygen site, respectively. The value for ros is estimated as 0.57 nm while rO (tO1/2-) is s about 0.16 nm.24 Thus, for Vos we obtain 0.38 nm3. We note that, for the relevant site density of about 1 nm-2, the hemispherical outer-sphere volumes of adjacent Fe3tO1/2- groups show a slight overlap. Within the framework of the present exercise, this detail can be ignored. s s Uos for Cu2+/Goethite. The primary Coulombic term in Uos 1 (cf. eq 11) is computed on the basis of a 2+, /2- interaction (19) Morel, F. M. M.; Hering, J. G. Principles and Applications of Aquatic Chemistry, 2nd ed.; Wiley: New York, 1993. (20) Margerum, D. W.; Cayley, G. R.; Weatherburn, D. C.; Pagenkopf, G. K. In Coordination Chemistry; Martell, A. E., Ed.; ACS Monograph 174; American Chemical Society: Washington, DC, 1978; Vol. 2, pp 1-220. (21) van Leeuwen, H. P.; Town, R. M.; Buffle, J. J. Phys. Chem. A 2007, 111, 2115–2121. (22) Vakros, J.; Bourikas, K.; Perlepes, S.; Kordulis, C.; Lycourghiotis, A. Langmuir 2004, 20, 10542–10550.
for the Fe3tO1/2- site. The surface charge term is appropriately estimated on the basis of a smeared-out double layer correction on the activity of Cu2+ at the reaction site of outer-sphere complex formation. The potential at this site may be expected to be somewhere in between the electrokinetic zeta potential, ζ, and the surface potential, ψ0, of the goethite/solution interface. For pH around 6 and 0.01 M NaNO3 background electrolyte, the former is measured at some +50 mV,25 whereas the proton binding data yield a ψ0 of approximately +200 mV.16 On the basis of an extended Stern layer model for the double layer, the potential in the plane of nearest approach for single-shell hydrated cations is computed to be +110 mV versus bulk solution.17 As a consequence, the activity of Cu2+ at the position ros is reduced by a factor of 6 × 103 as compared to bulk solution. For a first estimation of the magnitude of the electrostatic screening term in eq 11, we use a semi-infinite Debye-Hu¨ckel term e2zMzS/4πε0εros × κros/2(1+κros), where κ is the reciprocal Debye length of the electrolyte solution. The estimation merely serves to obtain an idea of the extent of screening of the surface ion pair, since effects of the substrate and the local electrolyte concentration near the highly charged surface are not included. For a κ-1 of 3 nm and the given ros for the Cu2+/goethite system, the Debye-Hu¨ckel screening term is calculated to modify log s s Kos by -0.3 units. The real reduction in Kos and the ensuing surface complex formation rate constant will be larger. According to eq 10, the overall results of the computation of s s s Vos and Uos lead to a value for log Kos (m3 mol-1) of -6.8. Using eq 13, this yields a value for log kad′ (s-1) of 2.1 for the Cu2+/ goethite dispersion specified above. In view of the significant uncertainties in the various parameters used, this result may be considered to be in convincing agreement with the experimentally found regime of τad which corresponds to log kad′ values around 1.5. It is well-established that, for a given type of surface, the adsorption rate constants for different metal ions with the same charge number correlate with their inner-shell dehydration rate s constants, kw.10,11,26,27 The computations of Kos on the basis of eqs 10 and 11 confirm that the predicted surface complexation rate constants are rather insensitive to the precise charge distribution in the intermediate outer-sphere ion pair.19 In light of the present result, this implies that the applicability of the Eigen mechanism to surface complexation of aqueous metal ions is of a rather general nature and underscores the utility of the comprehensive 2D Eigen approach developed herein. Acknowledgment. This research was performed within the ECODIS project (518043) funded by the European Commission’s sixth framework program, subpriority 6.3 “Global Change and Ecosystems”. J. Buffle, T. Hiemstra, R.R. Netz, J.-P. Pinheiro, and R. M. Town are gratefully acknowledged for helpful discussions.
Appendix kad
Symbols and AbbreViations metal ion adsorption rate constant (mol-1 m3 s-1)
(23) Puy, J.; Cecı´lia, J.; Galceran, J.; Town, R. M.; van Leeuwen, H. P. J. Electroanal. Chem. 2004, 571, 121–132. (24) Bengley, B. J. Phys.: Condens. Matter 1989, 1, 2395–2408. (25) Juang, R.-S.; Wu, W.-L. J. Colloid Interface Sci. 2002, 249, 22–29. (26) Zhang, P. C.; Sparks, D. L. Soil Sci. Soc. Am. J. 1989, 53, 1028–1034.
Surface Complexation of Aqueous Metal Ions s Kos
kw nos nsln nS ros
stability constant for the surface outer-sphere complex (dm3 mol-1) rate constant for water substitution in the inner sphere (s-1) number of metal ions in surface ion pair form number of free metal ions in solution number of free surface sites center-to-center distance between M and S in the outersphere complex (m)
Langmuir, Vol. 24, No. 20, 2008 11721
Rsa s Uos s Vos
rate of surface complex formation (mol m-2 s-1) potential free energy for a surface ion pair (J) volume available for outer-sphere complex formation at an individual surface site (m3)
LA8014332 (27) Stumm, W.; Sigg, L.; Sulzberger, B. In Chemical and Biological Regulation of Aquatic Systems; Buffle, J., de Vitre, R. R., Eds.; Lewis Publishers: Boca Raton, 1994; pp 45-89.
