Journal of Colloid and Interface Science 266 (2003) 1–18 www.elsevier.com/locate/jcis

Hydrous ferric oxide: evaluation of Cd–HFO surface complexation models combining CdK EXAFS data, potentiometric titration results, and surface site structures identified from mineralogical knowledge Lorenzo Spadini,a,∗ Paul W. Schindler,b Laurent Charlet,a Alain Manceau,a and K. Vala Ragnarsdottir c a Environmental Geochemistry Group, LGIT-IRIGM, University Joseph Fourier and CNRS, B.P. 53, 38041 Grenoble, France b Institute for Inorganic Chemistry, Freiestrasse 3, 3012 Bern, Switzerland c Department of Earth Sciences, University of Bristol, Bristol BS8 RJ, UK

Received 30 July 2002; accepted 23 April 2003

Abstract The surface properties of ferrihydrite were studied by combining wet chemical data, CdK EXAFS data, and a surface structure and protonation model of the ferrihydrite surface. Acid–base titration experiments and Cd(II)–ferrihydrite sorption experiments were performed within 3 < − log[H+ ] < 10.5 and 0.5 < [Cdt ] < 12 mM in 0.3 M NaClO4 at 25 ◦ C, where [Cdt ] refers to total Cd concentration. Measurements at −5.5
log[Cdt ]
−1.4 at fixed pH completed the wet chemical data set. The acid–base titration data could be adequately modeled by +1/2

≡Fe–OH2

− H+ ↔ ≡Fe–OH−1/2 ,

log k(int) = −8.29,

assuming the existence of a unique intrinsic microscopic constant, log k(int) , and consequently the existence of a single significant type of acid–base reactive functional groups. The surface structure model indicates that these groups are terminal water groups. The Cd(II) data were modeled assuming the existence of a single reactive site. The model fits the data set at low Cd(II) concentration and up to 50% surface coverage. At high coverage more Cd(II) ions than predicted are adsorbed, which is indicative of the existence of a second type of site of lower affinity. This agrees with the surface structure and protonation model developed, which indicates comparable concentrations of highand low-affinity sites. The model further shows that for each class of low- and high-affinity sites there exists a variety of corresponding Cd surface complex structure, depending on the model crystal faces on which the complexes develop. Generally, high-affinity surface structures have surface coordinations of 3 and 4, as compared to 1 and 2 for low-affinity surface structures.  2003 Elsevier Inc. All rights reserved. Keywords: Cadmium; Cd; HFO; Ferrihydrite; Adsorption; Surface; Structure; Titration; Equilibrium analysis; Goethite; EXAFS

1. Introduction Sorption and desorption reactions of trace elements on reactive mineral surfaces are of major importance in natural systems, as they regulate the bioavailability, transport, and toxicity of metal ions in soils, surface waters, and groundwaters. The description of the sorption process can be based on mass balance approaches [1] involving exclusively “macroscopic” observation parameters such as pH, Eh, and sorbent and sorbate concentrations. The macroscopic character * Corresponding author.

E-mail address: [email protected] (L. Spadini). 0021-9797/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00504-6

of this method does not allow the observation of individual molecular-scale surface complex structures. The surface stoichiometries obtained from data fitting are consequently ambiguous and may represent a moiety of a series of different particular species. The predictive character of the models obtained is thus limited to the investigated concentration range [2]. Improving the prediction of sorption processes thus requires the integration of the macroscopic and microscopic scales. In the past decade, particular attention has been given to discrepancies between the macroscopic and microscopic approaches to mineral surface chemistry. For example, the measurement of cation adsorption on the hydrous ferric oxide (HFO) ferrihydrite in a low sorbate concentration range

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led researchers to postulate the occurrence of two cation adsorption sites of different affinities [3,4]. In more recent years data analysis has integrated proton and ion affinities as determined from bond valence [5,6] and molecular static [7,8] calculations. These calculations are based on the mineral surface and bulk structure of the sorbent material, whereas the former investigations were purely macroscopic, as they were based on a generalized surface complexation model. This progress developed in parallel with the improvement of spectroscopic methods, which now allow an investigation of the structures of surface complexes at the molecular level. These methods include extended X-ray absorption fine structure (EXAFS) spectroscopy [9–12], Fourier transform infrared (FTIR) spectroscopy [13–15], and X-ray photoelectron spectroscopy (XPS) [16,17]. These spectroscopic approaches have allowed new insights into surfacecatalyzed redox processes [10], structure-dependent cation site affinities [9,16], and surface-structure-promoted nucleation processes of cation adsorption [12,16]. In the present paper the surface properties of ferrihydrite were studied by Cd(II) adsorption experiments. The equilibrium analysis of the wet chemical data set was combined with a surface structure and protonation model of ferrihydrite surface and with CdK –ferrihydrite EXAFS findings [9]. The structural model is based on recent findings relative to the ferrihydrite bulk structure [18,19]. The combination of these different data leads to original insights into the relation between coordination and stoichiometry, site affinity, and the variety of possibly existing surface complex structures. First, the potentiometric titration data are presented and modeled; these data are combined with the surface structure model presented in the second part of the paper.

2. Material and methods 2.1. Stock solutions All solutions were prepared from bidistilled water. HClO4 stock solution: Perchloric acid (Merck p.a. 70–72%) was diluted and standardized iodometrically [20]. NaClO4 stock solution: NaClO4 sicc. (Fluka puriss.) was dried at 120 ◦ C for several hours and then dissolved without further purification to obtain a 0.3 M stock solution. Cd(ClO4 )2 solution: An excess of anhydrous CdO (Merck, p.a.) was dissolved in a 1 dm3 volumetric flask containing approximately 0.2 mol of standardized HClO4 . The solution was then filtered through a 0.02-µm Millipore filter in order to remove any remaining CdO particles and diluted with 0.3 M NaClO4 to obtain the Cd stock solutions used in the experiments. 2.2. Preparation of ferrihydrite A 0.2 M solution of Fe(ClO4 )3 ·9H2 O (Fluka pract.) was filtered and precisely analyzed for Fe(III) by iodometry [20].

The precise amount of NaOH (Merck, Titrisol) necessary to precipitate Fe(OH)3(s) was added ([OH]tot /[Fe(III)tot] = 3.00). During the preparation argon or nitrogen gas was bubbled through the solution to minimize the introduction of CO2 gas into the system. After dilution to an Fe(III) concentration of 0.1 M, the suspension was aged for 3 weeks. Thus the PZNPC (point of zero net proton charge) of the solid corresponds to the quiescent ph of the ferrihydrite suspension (h = [H+ ], i.e., proton concentration, as compared to H = (H+ ), an activity). It was found to be ph 8.2. The Fe(III) and NaOH concentrations have to be precisely known, as a ±1% error in base addition corresponds to an approximate PZNPC pH shift of ±1. The specific surface area S, determined by the BET method using N2 was found to be S = 244 ± 12 m2 g−1 . This value is only indicative, as the reactive surface area is strongly different for this amorphous microporous substrate [3]. To minimize changes in surface area during aging of the suspension, all titrations were performed within the fourth week. After 3-month aging of the stock solution at pH 8.2 a color change from the brown-red ferrihydrite color to a yellow-orange goethite-like color occurred. 2.3. [Hs ] sorption curves and pH calibration Measurements and ph calibrations were carried out as a series of titrations in a constant ionic medium, 0.3 M Na(ClO4 ). The parenthesis (ClO− 4 ) indicates that the anion concentration was fixed to 0.3 M. Working in a constant ionic background leads to an almost invariable ionic strength, which in turn fixes the activity coefficients. pH calibration toward [H+ ] concentrations instead of (H+ ) activities could thus be performed by titrating 40 ml of a 1 mM HClO4 solution with 40 ml of 2 mM NaOH solution in 0.3 M Na(ClO4 ). The HClO4 stock solution was prepared from perchloric acid (Merck p.a. 70–72%) and standardized iodometrically for [H+ ] [20]. Thus in the following ph = − log[H+ ]. Titration vessels were immersed in a water bath held at 25 ± 0.05 ◦ C. Stirring was achieved with a hanging Teflon propeller. Argon gas, presaturated with H2 O by passing through 0.3 M NaClO4 , was used to maintain an inert atmosphere. The titrations were performed with an automatic system for e.m.f. measurement. The total concentration of hydrogen [Ht ] was varied by adding sodium hydroxide in 0.3 M NaClO4 from a Metrohm 665 Dosimat. The concentration of free hydrogen ions, h, was determined with a combined glass electrode (Ag/AgCl reference electrode). [H+ ] was calculated based on the modified Nernst equation E = E ◦ − 59.157ph,

(1)

where E is the electrode potential (mV) and E ◦ (mV) is a constant which was determined before each titration by Gran’s method [21]. The logarithm of the ionic product of water, log Kw , was fixed at −13.70 due to ionic strength considerations. Equilibrium was assumed to be achieved when

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

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the drift of the electrode potential was less than 0.1 mV in 4 min. Sorption measurements were based on a 0.0444 mol/dm3 (corresponding to 4.75 g/dm3 of Fe(OH)3 ) Fe(III) solution, obtained by mixing 20 ml of the ferrihydrite stock suspension at the PZNPC ph and 25 ml of acidified cadmium stock solution. Starting ph was about 3. In these experiments the total Cd concentration [Cdt ] was varied within 0
[Cdt ]
11.2 mM and the surface proton deficit [Hs ] determined with the equation [Hs ] = h − Ht − Kw h−1 ,

Hs (PZNPC) = 0.

(2)

2.4. Cd isotherm experiment A Cd isotherm was established in order to validate the speciation model also for low total Cd concentrations. Individual batch experiment runs at ph 6.7 ± 0.2, 25 ± 0.05 ◦ C were performed with the total Cd concentrations varying within 24 µM
[Cdt ]
43 mM. The samples were equilibrated during 48 h; the dissolved fraction of Cd was determined by AAS (atomic absorption spectroscopy) after filtration. In this experiment pH was maintained by dropwise addition of a base. 2.5. Cd titration curve In a specific Cd–ferrihydrite titration experiment the concentration of the dissolved fraction of Cd was determined. To 20 ml of ferrihydrite stock solution 25 ml of acidified Cd(II) stock solution was added, yielding a starting pH of 3, a [Cdt ] of 5.72 mM, and a ferrihydrite total concentration of 4.75 g/l in a 0.3 M Na(ClO4 ) background solution. The solution was titrated in steps of 2 ml of added base (0.02 M OH− in 0.3 M Na(ClO4 )). After equilibration for each step, 1 ml of the slurry was retrieved, filtered, and analyzed for Cd by AAS after dilution and acidification.

3. Results and discussion 3.1. Raw wet chemical data The acid–base titration data of the ferrihydrite surface without Cd(II) are given in Fig. 1a. The titrations at initial Cd total concentrations [Cdt ] = 0, 0.558, 1.12, 2.79, 5.58, and 11.2 mM are given in Fig. 1b. In both figures [Hs ] represents the surface proton deficit in mol/l relative to the analytically determined PZNPC, the ph at which the net proton charge is equal to 0 (see Eq. (2)). Ht [mol/l] is the total acid concentration added to the solution. Figure 2 corresponds to a Cd isotherm curve recorded at pH 6.7 ± 0.2 and at 24 µM
[Cdt ]
43 mM, obtained from batch experiments. The curve will make it possible to assess at low [Cdt ] the speciation models presented below. Figure 3 corresponds to one particular titration curve in which [Hs ] and the soluted Cd concentrations were determined.

Fig. 1. (a) HFO acid–base data (✸) and the 2 pK fits 3, 8, 9, 10, and 11 of Table 1 (solid lines). Fit 3 corresponds to the mathematical solution in which all acid–base reactive functional groups have the same intrinsic reactivity. Solutions 1 to 7 fit very closely to curve 3 and are thus not individually shown. Fit 8 is the solution with the highest error squared sum V (Y ) considered acceptable; this solution (and solutions 1, 2, and 4 to 7) relates to a concept of almost equivalent acid–base functional groups. In contrast, the alternative curves 9 to 11 are not compatible with this concept and differ significantly from the experimental values. Curve 11 corresponds to a nonelectrostatic model fit. (b) Potentiometric titration data of the Cd–HFO–H+ system at varying Cd total concentrations, 2 pK fits (solid lines), and 4 pK fits (dotted lines). [Hs ] is the surface proton concentration released to solution relative to the PZNPC.

3.2. 2 pK model, acid–base titration Reactions presented in this paper were based on the generalized equation p ≡n SHn/2 + qCd2+ + rH+ (2q+r)+

↔ (≡n SHn/2 )p Cdq Hr

,

n

βpqr .

(3)

The superscript n in n βpqr and ≡n SH indicates the n pK site. The n pK site corresponds to a unit on the ferrihydrite surface which exchanges n acid–base reactive protons and to which, correspondingly, n macroscopic acid–base reaction constants are associated. pqr refers to the stoichiometry coefficients. Surface potential effects were corrected with the constant capacitance (CC) model [22]. More details of the fit proce-

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L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

Table 1 2 pK fits of the ferrihydrite surface Fit

κ [nF m−2 g−1 ]

log 2 β101(int)

log 2 β10-1(int)

W

[≡2 St ] [mM]

PZNPC

V (Y )

1 2 3 4 5 6 7 8 9 10 11

1.5 1.6 1.66 1.7 1.8 1.9 2.0 2.1 2.5 3.5 100

20.96 8.14 7.99 7.87 7.72 7.62 7.54 7.48 7.17 6.84 6.27

4.37 −8.45 −8.59 −8.72 −8.87 −8.97 −9.04 −9.10 −9.25 −9.55 −10.27

25.33 −0.31 −0.60 −0.85 −1.15 −1.35 −1.50 −1.62 −2.08 −2.71 −4.00

4.99 5.12 5.12 5.13 5.02 4.90 4.79 4.70 4.64 4.43 4.06

8.30 8.30 8.29 8.30 8.30 8.30 8.29 8.29 8.29 8.28 8.27

5.12 5.09 5.09 5.08 4.99 4.89 4.90 5.10 10.2 41 250

Notes. κ: surface capacitance. W = log(2 β101(int) 2 β10-1(int) ). [≡2 St ]: total concentration of the 2 pK surface sites. Specific surface area S = 244 m2 /g. Concentration of solid Fe(OH)3 = 4.75 g/l.

dure are given in Appendix A. Modeling of the chemical equilibrium was first based on a conventional 2 pK model concept which means that a surface site is defined as an unit exchanging two acid–base reactive protons. For this case the generalized equation becomes p ≡2 SH + qCd2+ + rH+ (2q+r)+

↔ ≡2 SHp Cdq Hr

2

,

βpqr ,

(4)

and the surface acid–base exchange is correspondingly given by the two reactions ≡2 SH + H+ ↔ ≡2 SH+ 2, +

2 −

≡ SH − H ↔ ≡ S , 2

Fig. 2. Cd–HFO sorption isotherm with experimental data (+), 2 pK (solid line, model 3 of Table 1), and 4 pK (dotted line, model c of Table 3) fits. The two models fit the data at low Cd total concentrations and up to − log[Cdads ] ∼ −2.7, corresponding to about 50% surface coverage.

Fig. 3. Cd–HFO adsorption edge data (), 2 pK fit (solid line, model 3 of Table 1), and 4 pK fit (dotted line, model c of Table 3).

2 2

β101(int) ,

β10-1(int) .

(5) (6)

Fit results based on this model are given in Table 1 and in Fig. 1a (solid lines). Eleven fits are compared in which κ, the surface capacitance term, was systematically varied and 2 β101(int) , 2 β10-1(int) , and the total concentration of sites [≡2 St ] were adjusted. Fits 1 to 8 do not differ significantly, as is indicated by both Fig. 1a and the close error square sum V (Y ) in Table 1. The agreement is less and considered not acceptable for Fits 9, 10, and 11. In these fits the κ values are high. This parameter is inversely proportional to the surface potential within the scope of the CC model. In this respect Fit 11 equals a nonelectrostatic model fit. Attention has thus to be focused on the acceptable fits 1 to 8. In these fits [≡2St ] varies within 4.70
[≡2 St ]
5.13 mM. The [≡2 St ] range is thus relatively close and apparently acceptable. But below pH 3 the extrapolated model curve tends to become horizontal (Fig. 1a). This tendency relates to the saturation of the model site and thus to [≡2 St ]. In this range, at pH < 3, no experimental data exist: the solid dissolves below pH 3.3 and above pH 11, thus constraining the experimental range. It is thus not clear whether the fitted [≡2 St ] relates to an effectively existing buffering minimum of acid– base reactive functional groups or not. In contrast to [≡2 St ], the fitted values of the two constants 2 β101(int) and 2 β10-1(int) vary strongly. For the acceptable solutions (Fits 1 to 8) the variation covers 28.4 log units for 2 β10-1(int) and, respectively, 27.0 log units for 2 β10-1(int) . At the same time, the

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

PZNPC value calculated by  1
log 2 β10-1(int) − log 2 β101(int) = −PZNPC = −8.29 (7) 2 remains constant. This thus indicates a strong correlation of the parameters 2 β101(int) , 2 β10-1(int) , and κ. The question to address is whether as the fitted constants 2 β101(int) and 2 β10-1(int) reflect different or unique binding energies of individual protons. As is outlined in King [23], the relation between the macroscopic equilibrium constants 2 β101(int) , 2 β10-1(int) , and microscopic constants of individually reacting protons are of a statistical order. In a system where all protons are equivalent, the microscopic constants of individual protons k(int) , defined as ≡s−1/2 + H+ ↔ ≡s–H+1/2 ,

k(int) ,

(8)

relates to the n macroscopic constants of the n pK system as n

Ki(int) = kint (n − i + 1)/i,

(9)

n/2−i+1 n/2−i ≡ S–Hn−i+1 ↔ H+ + ≡n S–Hn−i , n Ki(int) (i = 1, . . . , n), n

(10)

where ≡s represents one acid–base reactive functional group, which exchanges for one individual proton, and ≡n S represents a larger unit exchanging n protons. For the given 2 pK system this relation simplifies to + 2 0 ≡ S–H+1 2 ↔ H + ≡ S–H , 2 K1(int) , 2 K1(int) = 2kint (i 2 0 + 2 −1 2

≡ S–H ↔ H + ≡ S , 1 2 K2(int) , 2 K2(int) = kint 2

= 1),

(11)

(i = 2).

(12)

It results that 2 K1(int) / 2 K2(int) = 4. Thus in a system in which all protons are equivalent the macroscopic constants of a 2 pK system as defined in (11) and (12) are not identical but differ by a factor of 4. This statistical dependence can be comprehensively outlined as follows (23): k(int)

≡2 S– y Hz+

≡2 Sz−1 ,

≡2 S– y H x Hz+1 k(int) 2

2

(13)

k(int)

≡2 S– x Hz+

k(int)

[H+ ][≡2 S– y Hz+ ] [H+ ][≡2 S– x Hz+ ] + [≡2 S– y H x Hz+1 ] [≡2 S– y H x Hz+1 ] = 2kint,

K1(int) =

K2(int) =

(14)

[H+ ][≡2 Sz−1 ]

[≡2 S– y Hz+ ] + [≡2 S– x Hz+ ] [H+ ][≡2 Sz−1 ] 1 = kint. = 2[≡2 S– y Hz+ ] 2

(15)

x H+ and y H+ represent the two acid–base reactive protons of the diprotic surface site ≡S– y H x H.

5

For the alternative definition of the 2 pK system given in (5) and (6), the same relation of microscopic to macroscopic constants becomes 2

β101(int) = 1/(2kint),

(16)

2

β10-1(int) = (k/2int).

(17)

The ratio 2 β101(int) 2 β10-1(int) = 1/4 expresses consequently the same situation of equivalent protons for this alternative set of 2 pK reactions. The term W = log(2 β101(int) 2 β10-1(int) ) consequently relates to proton interdependency, and the specific value Weq = log(1/4) = −0.6 designs a system in which all protons are equivalent. The values of W for the 11 discussed fits are given in Table 1. They vary in the very broad range −4
W
25.33. Weq is within this range. The 11 fit solutions thus match three different situations W > Weq , W = Weq , and W < Weq , which differ with respect to the proton interdependence: (i) Solutions with W > Weq imply that the protons leaving the reaction sites first promote a stronger binding of the remaining proton. This situation is senseless with respect to surface charge considerations, the corresponding solutions 1 and 2 can thus be disregarded (Table 1). (ii) Solutions W = Weq imply that all protons are equivalent; this corresponds to solution 3. In this case protonation reactions are assessed by the unique microscopic reaction constant given by ≡s−1/2 + H+ ↔ ≡s–H+1/2 ,

log k(int) = −8.29.

(18)

The values of the macroscopic constants are correspondingly 7.99 and 2 β 101(int) = 1/(2kint) = 10 10-1(int) = (kint /2) = −8.59 10 . (iii) Finally, solutions with W < Weq infer that the protons leaving the surface first induce a stronger binding of the remaining protons, or that the two protons of the site have different microscopic constants. Both cases can occur and have thus to be considered. The solutions 4 to 8 correspond to this case and furthermore fit the data set conveniently. Thus the fit does not discriminate between the cases of all functional groups being equivalent (3) or not (4 to 8). But the pK values of all these acceptable solutions are close, within the ranges 7.48
log 2 β101(int)
7.99 and −9.1
log 2 β10-1(int)
−8.59, and thus within 0.51 log units. This indicates that the surface protons, if not equivalent, nevertheless have close reaction constants. The difference W8 − Weq = −1 can be interpreted as the maximum admitted difference of the microscopic pK constants of two populations of protons which have the same total concentrations. This pK gap is small and thus supports a near-equivalency of the involved protons. This result can also alternatively be interpreted as indicative of the presence of two populations of different total concentrations which have more different pK values. But in this case one of the two populations becomes minor in concentration and rapidly insignificant. Thus the result shows that the individual reactivities of proton-exchanging sites on the HFO surface are close. From this it can be concluded that (i) acid–base protons almost all have the same structural environment, and 2β

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L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

furthermore (ii) that these groups react mutually independently; i.e., the sorption of one proton does not influence the affinity of the neighboring protons. The fit results are thus not compatible with the classical concept of a surface oxygen losing both of its protons one after one, ↔ ≡OHz+ + H+ ↔ ≡Oz−1 + 2H+ , ≡OHz+1 2

(19)

as in this case the mutual interactions of the protons are expected to be strong. It was shown by Hiemstra et al. [24] that for this case the two protons are not likely to be released both into the solution within the investigated pH range. This concept is thus not applicable to our experiments; the alternative reaction scheme,    OH ↔ O + H+ ↔ O + 2H+ (20) + OH O− OH is considered the better stoichiometric concept of the 2 pK site. The two protons are here associated each to one independent surface oxygen, which thus strongly diminishes their mutual interactions. The reaction equations (5) and (6) apply to both site definitions (19) and (20). But the stoichiometric interpretation of the adsorption process differ depending on the site definition: a Cd(II) ion adsorbed on the site as defined in (19) is considered monodentate, whereas adsorbed on (20) it is interpreted as a bidentate surface complex. This difference in interpretation depends only on the conceptual definition of the reactive site, and not on the reaction equations and the pK values, which are the same in both cases. This simply illustrates that acid–base sorption concepts may lead to wrong structural interpretations if not carefully and critically investigated. 3.3. Adsorption of Cd2+ —2 pK model Equilibrium modeling is based on the Hs data set given in Fig. 1b. The modeling involves a single type of Cd reactive site and equivalent acid–base reactive protons. Thus the fit is based on the fixed parameters [≡2 St ], κ, 2 β101(int) , and 2 β10-1(int) of Fit 3 of Table 1. Initial [Cdt ] varies within 0.558
[Cdt ]
11.2 mM or, respectively, 11% to 219% surface loadings. For acceptable solutions, the first step in the Cd(II) adsorption reaction starts by coordination of a Cd(OH2 )2+ 6 ion on a neutral site without release of any proton into the solution. Such adsorption has been used by Gunneriusson [25] (Pb2+ on αFeOOH), Stadler and Schindler [26] (Cd2+ on montmorillonite), and Ludwig [27] (Cu2+ on TiO2 ). This surface complexation mechanism appears at first sight somewhat unusual. However, it appears to be a common feature of metal ion adsorption on oxide mineral surfaces. The model Cd2+ + H2 O ↔ CdOH+ + H+ , log 2 β01-1 = −10.3,

(21)

Cd2+ + ≡2 SH ↔ ≡2 SH–Cd2+ , log 2 β110(int) = 5.96,

(22)

Cd2+ + ≡2 SH + H2 O ↔ ≡2 S–CdOH + 2H+ , log 2 β11-2(int) = −11.78,

(23)

fits conveniently the data set up to [Cdt ] = 5.58 mM or 109% surface loading (Fig. 1b, solid line) with the minimum number of the two given Cd surface species. This model was then applied to the data of the Cd isotherm curve (Fig. 2, solid line). Here also the model explains adequately the low-concentration Cd data and fits within a range (−4.7
log[Cdads ]
−2.7 or, respectively, 0.062% to ∼40% surface coverage). Thus both Hs and Cd measurements appears to indicate that the model is valid at low Cd loadings and up to 40% of surface coverage. The measurement of a Cd adsorption experiments at variable pH and at high [Cdt ] of 5.74 mM or, respectively, 112% surface coverage definitely states that the model does not satisfy high Cd loadings: the adsorption of the soluted Cd is complete which contradicts the model assumption of [≡St ] lower than [Cdt ]. Thus other sorption mechanisms, or possibly sorption sites of lower affinity and/or lower stoichiometry, interfere at these high loadings. The topic of this paper concerns exclusively the link between macroscopic and microscopic data. Thus the following discussion centers on the equilibrium model valid up to 40% surface coverage and to the link of this model to molecular level information. 3.4. Goethite bulk and mineral surface structure The ferrihydrite surface structure and its acid–base properties are intimately related to the bulk structure which thus needs to be discussed first. Two-line ferrihydrite, commonly referred as “hydrous ferric oxide,” HFO [3], is an X-ray amorphous substrate. The last relevant study was performed on 6-line ferrihydrite [18,19] and shows that the bulk Fe3+ ions are octahedrally coordinated by three O2− and three OH− groups as in goethite α-FeOOH. As explained in [9], the ferrihydrite structure includes goethitelike microdomains which involve four or more Fe octahedra, thus forming Cd–goethite-like surface complex structures (gray ellipses in [9, Fig. 13]). The ferrihydrite bulk can thus be viewed as a mosaic of single and double octahedral chains linked by corners which compare to shortened goethite chains. This similarity supposedly prevails not only in the bulk but also on the surface: as for goethite, edges and corners available for cation adsorption should concomitantly exist. The surface structure of these two solids thus should compare. This expected similarity in the surface structure lets us take the goethite bulk as a model for the ferrihydrite bulk. This prevents the discussion of the complex ferrihydrite bulk, which contains probability terms of vacancy occupations that do not allow the development of a conceptual acid–base model. The goethite bulk consists of double chains of edgelinked Fe octahedra that are cross-linked by corner linkages. OH groups are located within the double chains whereas O groups border the double chains (Fig. 4a). Cutting this bulk

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

Fig. 4. (a) Bulk structure of α-FeOOH and associated (O, OH) groups. (b) Lamellar habit of α-FeOOH crystals with the dominant {110} and the {021} chain cut face. (c) α-FeOOH surface structure and surface complex structures (SCS) of adsorbed cations. The SCS are given twice, as coordinated octahedra (gray shaded) and as coordinated surface complex. In these latter, (c), (•), and (Fe) symbolizes the adsorbed central cation, the 1st shell (O, OH) anions shared between the adsorbed and surface cations, and the 2nd shell Fe(III) cations. For example, the 2 E{021} 3 SCS has three 1st shell (O, OH) groups shared with two edge linked 2nd shell Fe neighbors, the 1 E 2 C{021} SCS has three (O, OH) groups shared with three Fe neighbors 3 (1 edge and 2 corner linked). The two {110} SCS 1 C{110} 1 and 2 C{110} 2 have lower 1st shell or surface coordination numbers (SCN) of 1 and 2, respectively, and do not share edges but only corners as compared to the {021} SCS.

structure along specific (hkl) planes generates the goethite faces (Fig. 4b). The dominant face is the {110} face, which runs parallel to the c axis. The chains are terminated essentially by the {021} face [28,29]. This “chain cut face” is of minor importance with respect to the surface area. This leads to the needle shape habit of the goethite crystals (Fig. 4b). Adsorption of octahedrally coordinated cations (‘c’ in Fig. 4c) on crystal growth sites leads to the formation of face-specific Cd–Fe surface complex structures (SCS): on {021} 1 E–2 C{021} 3 and 2 E{021} 3 SCS form, whereas on {110} 1 C{110} and 2 C(110) SCS are generated (Fig. 4c). These 1 2 symbols express the first and second shell structure and coordination of an adsorbed Cd octahedron: The brackets relate to the face on which the SCS form. Letters and exponents relate to the second (surface Fe) coordination shell: Capital letter(s) denote the octahedral linkages shared between the

7

adsorbed Cd and vicinate Fe surface octahedra: vertex (V:  Cd–O–Fe ∼ 180 ◦ ), corner (C:  Cd–O–Fe ∼ 130 ◦ ), edge (E:  Cd–O–Fe ∼ 90◦ ), and face (F:  Cd–O–Fe ∼ 70◦ ). The exponent gives the number of such linkages involved. The sum of these exponents is the second shell coordination number, i.e., the number of surface Fe octahedra linked to an adsorbed Cd octahedron. Thus in 1 E–2 C{021} 3 the Cd octahedron is linked to three Fe octahedra; one Fe octahedron shares an edge and the two other each share a corner with the Cd octahedron. In 2 E{021} 3 the Cd ion has only two Fe neighbors, which both share one edge with the Cd octahedron. The subscript at the right indicates the first shell surface coordination number, SCN, i.e., the number of (O, OH) groups shared between the Cd and surface Fe octahedra. Note that first and second shell coordination number may differ. Thus the two given 1 E–2 C{021} 3 and 2 E{021} 3 {021} SCS both share three O, OH groups (SCN = 3) but have two and three Fe neighbors, respectively. Despite their individuality, these two {021} SCS share some common characteristics which differentiate them from the {110} SCS. Thus the two {021} SCS involve edges (and corners), whereas the two {110} SCS involve only corners. Also, the two {021} SCS have both a relatively high SCN of 3 as compared to the lower {110} SCN of 1 and 2. On goethite, the {021} face is of minor importance, which indicates that this face was more reactive during crystal growth as compared to the dominant {110} face. Consequently the two 1 E–2 C{021} 3 and 2 E{021} 3 Fe–Fe SCS which developed in the crystal growth process were more reactive than the two Fe–Fe 1 C{110} 1 and 2 C(110)2 SCS. The two {021} SCS are characterized by relatively high SCN. Thus on this solid the SCN could potentially be the determinant parameter controlling the stability of adsorbed complexes, such as in solution the chelating effect associate high stability to a high number of ligand “dents.” In the following the SCN will consequently be taken as determinant parameter with respect to site reactivity for adsorption of both Fe3+ (adsorption during crystal growth) and Cd2+ (adsorption) ions. This concept does not distinguish the stability of the two {021} SCS, as they both have the same SCN of 3. On {110}, instead, the 2 C(110) SCS should form preferentially as compared to the 2 1 C{110} SCS. This qualitative concept thus considers the 1 SCN, and not the linkage type or respectively interatomic distances, as the parameter of primary importance with respect to stability. Face and vertex linkages, and thus extreme Me–O–Me distance variations that affect the stability, are not considered within this work; the SCN-stability concept may thus be valid within these limits. More generally, for Cd adsorption, this concept predicts that the {021} Cd–Fe SCS form preferentially and thus dominate the speciation at low Cd loadings, whereas close to saturation of the surface sites the later-forming but more abundant {110} SCS predominate. This concept compares to the CdK EXAFS results of Cd(II)–goethite adsorption samples presented in [9]. EXAFS spectroscopic investigations are particularly sensitive to in-

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teratomic distances and consequently to the linkage types involved in the formed SCS. At low Cd coverage (10%) Cd–Fe edges and corners are observed, which compares to the {121} SCS; at higher Cd loadings only corners existing on the {110} SCS are observed. This thus indicates that Cd effectively link to Fe(III) crystal growth sites and that both Fe and Cd SCS compare in relative reactivity. Consequently the SCN-affinity concept is validated for both, Fe and Cd adsorption. This discussion is based on the previous work [9] in which goethite delivered from Bayer was used for Cd(II) adsorption. More recent studies [30] did not confirm the occurrence of edge linkages on goethite at low Cd(II) surface coverage. Instead, Cd–Fe corners only were observed in the whole range of investigated surface coverage as existing on the dominant {110} face. We suggest that this relates to the use of different goethite substrates: Chain cut faces are supposedly relatively more developed in the Bayer product as compared to substrates synthesized in the laboratory, and consequently the {021} SCS are too few to be detected by EXAFS on synthetic substrates. Figure 5 shows the two goethite faces completely covered by Cd(II) ions. The maximum site density, calculated as number of Cd ions adsorbed per nm2 surface spanned by the unit cell vectors, is 3.0 on {110} (Fig. 5a left and top right) and 3.7 on {021} (Fig. 5b). The {021} face in Fig. 5b is covered by 1 E– 2 C{021} SCS. The face could alternatively be completely 3 covered by 2 E{021} 3 SCS at the same density. Instead, occurrence of both SCS on this face leads to a diminution of the calculated maximum site density. As indicated, inserting a single 2 E{021} 3 SCS (Fig. 5b right) in a 1 E–2 C{021} 3 saturated face (Fig. 5b left) leads to formation of Cd–O– Cd links to three neighbored Cd octahedra (Fig. 5b circles). Such links are not indicated from EXAFS results and also do not compare to the surface complexation model presented above. Thus three 1 E–2 C{021} 3 octahedra need to be removed to place a single 2 E{021} 3 octahedron. The overall maximal Cd density is consequently diminished. Regular alternation of both SCS reduces the site density to a “minimum maximal value” of 1.85 sites nm−2 (half of the highest possible maximal density). On the {110} face the maximum site density can also be reduced to half (1.5 sites nm−2 ) of the previous value by placing the Cd2+ octahedra differently on the available corners (Fig. 5a bottom left). The effective geometry of site occupation is not known. On goethite the {110} face is the predominant one and thus controls the overall site density, which we consider to be within 2 to 3 sites nm−2 . 3.5. HFO surface structure model Considering the likely similarity of the HFO and goethite surface structures, the goethite bulk will be taken as model for the HFO bulk. This simplification is necessary in order to develop a coherent HFO surface structure, charge, and protonation model. The conceptual identity of the HFO and goethite bulk does not implicate an identity in terms of ex-

Fig. 5. Maximum surface coverage model of the {110} (a) and {021} (b) faces obtained from coordination of the maximum possible number of Cd octahedra to available surface sites. The formation of Cd–(O, OH)–Cd links (connected Cd octahedra) is not indicated from EXAFS results, thus representing the limiting criteria in loading the available sites with Cd(II) ions. The thereby obtained maximum coverage depends from the distribution of Cd on available sites. On {110} the density of the top right distribution is twice that of the bottom right distribution. The two distributions have consequently different Cd densities, but both have the maximum allowed load with respect to the limiting criteria. On {021} covering the face with only one (of two possible) SCS yields the highest possible coverage. Inserting, for example, a single 2 E{021} 3 octahedron in the 1 E–2 C{021} 3 saturated surface yields the formation of Cd–O–Cd links (circles in Fig. 5b) to three neighboring 1 E–2 C{021} 3 SCS. The necessary elimination of these three neighbors results in an overall decreased site density. For this face the Cd density of a surface covered with a unique SCS is twice that of a surface where regular alternation of both SCS is assumed. In (b) the surface spanned by the unit cell vectors (a, b, (1/2)c) is given by the dotted line.

isting faces and their relative importance, and the associated SCS. First, the goethite needle shape habit of the goethite crystals resulting from the relatively slow crystal growth process is not indicated for an X-ray amorphous compound. On HFO the relative extend of chain cut faces should consequently be more important. Second, beyond the already presented {021} face alternative chain cut faces ({hkl}, l = 0), abbreviated further on to {hkl}, most probably exist. Their exact individual extent cannot be determined, but they will be shown to share common specific characteristics which differ from those of the {hk0} faces running parallel to the c axes, among which is {110}. Figure 6 presents a hypothetical “needle dip” formed from the four chain-cut faces {021}, {001}, {101}, and {121} with the face-specific Cd–Fe SCS detailed in Table 2. All in all 10 Cd–Fe SCS are formed on the four faces, of which 8 are nonequivalent with respect to SCN and involved linkage types. Edge linkages are involved in all these surface complex structures. Chain-cut faces thus can generally be considered as edge-forming faces. The SCN of all these SCS are 3 and 4, except for the 1 E{001} 2 SCS (SCN = 2). But this particular SCS is in competition with

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

Fig. 6. The {hkl} model faces {021}, {121}, {101}, and {001} and the Cd–HFO SCS expected to form on these faces. A total of 10 SCS may be discerned from which 8 are nonequivalent with respect to the surface coordination number SCN and the involved linkages (Table 2).

the 2 E–2 C{001} 4 SCS of higher SCN; the low SCN SCS thus should not form in significant amounts. The SCN of the SCS expected to form on HFO chain-cut {hkl} faces are thus 3 or 4 depending on the face (Table 2). The major {hkl} face on goethite is the {021} face which forms SCS with maximum SCN of 3. Faces forming SCN = 4 SCS potentially exist on goethite but are of neglectable extent as compared to the {021} face. This fact once more indicates the validity of the SCN-based affinity concept: faces of smallest extent are of highest affinity and concomitantly of highest SCN. The situation is less complicated for the alternative {hk0} faces. The investigation can be limited to the {110} and {010} face, as no other nonequivalent SCS will be found on other faces. Also, the {100} SCS equals the {110} face in linkage types, SCN, relative occurrence (same for both SCS), and, as will be shown below, in charge. The SCS forming on the two faces are thus equivalent and can be given as 2 C{110,100} and 1 C{110,100} SCS. Only these two SCS are 2 1 expected to form in significant number on {hk0} faces. The Table 2 {hkl, l = 0} Cd–HFO SCS Face

Symbol

001

2 E–2 C{001}

001

1 E{001}

021

2 1 E–2 C{021}

021

2 E{021}

Involved terminal groups 4

2 2

3

2 2

101

3 2 E–1 C{101}

3

1

101

1 E–1 C{101}

3

3

121

2 E–1 C{121}

4

3

121

1 E–3 C{121}

4

3

121

1 E–1 C{121}

3

3

121

2 E{121}

3

2

9

only alternative SCS relates to a triple corner 3 C(010)3 SCS which may form exclusively on the {010} face. This face is not expected to be important, as it involves a particular arrangement of two stacked double chains. The SCN of the two dominant 2 C(hk0)2 and 1 C{hk0} 1 are 2 and 1 and thus comparatively low. Consequently the conclusions drawn for the two goethite faces {021} and {110} also apply to the extended set of HFO {hkl} and {hk0} faces: first {hkl} SCS share at least one edge with surface Fe octahedra and possibly concomitantly also corners, whereas {hk0} SCS share only corners. Second {hkl} faces have higher SCN numbers (3 or 4), than {hk0} faces (SCN of 1 or 2). Thus the SCN-based affinity concept designs {hkl} as faces of relatively high affinity as compared to {hk0} faces of generally lower affinity. As high-affinity chain-cut faces are abundant on HFO, high-affinity SCS are abundant too. Low-affinity SCS of {hk0} faces are thus expected not to be expressed in significant amounts except at high Cd loadings. This concept of site heterogeneity is supported from Cd– HFO CdK EXAFS results [9]. In the CdK EXAFS spectra of Cd–HFO samples, edges and corners form in all samples independent of the Cd loadings, which varied between 1% and 90%. This result thus indicates that edge forming {hkl} chain-cut faces are effectively abundant on that solid, conforming to the crystal chemical considerations. A marked decrease of the CdK EXAFS amplitude at high Cd loadings attests to an increased structural order, which could be indicative of the Cd binding on sites of low affinity, such as the 1 E{001} 2 or the {hk0} SCS. These SCS are also characterized by a generally lowered second shell Fe coordination number, which further decreases the CdK EXAFS signal of the Fe neighbors. It becomes thus clear that the data recorded from EXAFS experiments relate not to one particular SCS but to series of SCS, most of them characterized by edges. In summary, these conclusions conform to the finding (i) that numerous different individual Cd SCS exists on the HFO surface, (ii) that this heterogeneity can be ordered into two groups of affinity which conform to the SCN and consequently the {hkl} and {hk0} face family, and (iii) that high affinity {hkl} sites are relatively abundant. The Cd– HFO equilibrium analysis model presented above has to be compared to these findings. The model is based on the assumption that a unique type of Cd receiving site and affinity exist. With this we succeed in fitting a large Cd concentration range up to about 60% of the maximum surface coverage calculated from experimental values. Suggestively the model can thus be considered as representing sorption on the abundant high-affinity {hkl} SCS. This concept thus differs from the HFO model presented by Dzombak and Morel [3], which implies that two types of sites exist, with the fraction of high affinity sites representing only 2.5% of the total site concentration. We do not agree with this concept and believe this ratio to be much higher. At this point the site densities obtained from the wet chemical data set and from the model structures can be compared. For this, {hk0} faces are assumed to be of minor extent and thus not considered. Maximum theoretical

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Cd site densities of the model faces {001}, {021}, {101}, and {121} are respectively 4.4, 3.7, 3.6, and 3.1 Cd/nm2; the mean value is 3.7 Cd/nm2. Experimental values should thus be within 3.7 to 1.85 Cd/nm2, the lower value being half of the maximum value for reasons related to site geometry presented above. The site density from the wet chemical data complexation model equals 1.3 sites/nm2 , calculated from the fitted [≡2 St ] of 5.12 mM, the 44.4-mM slurry of precipitated Fe, and the surface area estimate of 600 m2 /g of an 89 g/mol Fe Fe2 O3 ·H2 O (or FeOOH) solid [3]. This calculated site density is slightly below the expected theoretical minimum value and approximately three times lower than the theoretical maximum value. Suggestively this gap relates to the underestimated [≡2 St ] value found to represent only about 60% of the total adsorbed Cd in the isotherm at maximal load. Also, the surface area of 600 m2 /g is an estimate, its determination does further not relate to the HFO preparation protocol used in this work which includes a 3-week aging time. 3.6. Surface protonation model of α-FeOOH and HFO The next step consists in determining the proton load of the surface functional groups. The concept consists in linking Fe octahedra to a goethite bulk structure taken as model for the HFO bulk. The “link unit” is a electrostatically neutral Fe(OOH)(H2 O)4 octahedra (Fig. 7a). The linkage process consists of clogging a series of such octahedra, oriented as given in Fig. 7b, together. This process leads to the formation of Fe–(O, OH)–Fe bridges with the necessary release of one H2 O group per linking process into the solution (Fig. 7b, inset). The key is to remove only neutral water molecules to preserve the neutrality of the generated mineral. After termination of the linkage process a mineral bulk with the required goethite (O, OH) distribution and structure (Fig. 4a) results. The crucial point is that the linkage process leads also to electrostatically neutralized surface planes (Fig. 7c). The overall surface proton load thus corresponds to the PZNPC proton load. The load is face-specific and is given in Fig. 7c for the three example faces {001}, {100}, and {010}. Surface protons are outlined as white dots within black circles, each dot corresponding to one proton. The black circles represent the surface functional groups: large black circles represent terminal functional groups (surface oxygens coordinated to one Fe(III) ion) and small black circles relate to bridged groups (surface oxygens coordinated to two Fe(III) ions). Thus a small black circle with no white dot is a bridged surface oxygen, a large black circle with two white dots is a terminal H2 O group, etc. Triply coordinated groups are not particularly designed. They have all the same coordination and proton load as bulk anions; they are consequently considered as not being acid–base reactive and thus are not discussed further. The specific proton distribution as given in Fig. 7c results from a formal linkage process, thus only the overall PZNPC proton load is relevant but not the obtained distribution of

protons over individual groups. Thus Fig. 7d represents an alternative proton distribution with the same overall proton load. For example, on {001} protons were simply displaced from terminal groups (Fig. 7c) to unoccupied bridged groups (Fig. 7d). Rules for the distribution of protons need consequently to be developed. Surface octahedra presumably tend to minimize their octahedral charges, such as the bulk Fe(III) octahedra: Bulk Fe octahedra can be defined as neutral by addition of the positive charge of the central Fe(III) ion and the negative partial charges of the first shell anions. In the bulk the first Fe shell consists of 3 oxygens and 3 hydroxo groups. The anions are coordinated themselves to three Fe(III) ions in their first shell. They thus contribute −1/3 of their charge to the neutralization of each of the three cations. The formal contribution to neutralization, or bond valence, is thus υOH = −1/3 for the bulk OH− and υO = −2/3 for the bulk O2− groups. As one central Fe(III) is surrounded by 3 OH− and 3 O2− groups in the first coordination shell the positive charge cFe = +3 is neutralized by the six partial charges of its first shell anions: cFe + 3υO + 3υOH = 0. The same calculation procedures are applied to surface octahedra, and the protons are distributed so that the octahedral charge is closest to zero. For this the bond valences of the doubly and singly coordinated surface functional groups need to be calculated. Thus the bond valence of a bridged OH− is consequently υ OH–B = −1/2, those of a bridged υ O–B = −2/2 and those of terminal hydroxo and water groups are, respectively, υ OH–T = −1/1 and υ H2 O–T = −0/1. For the example {001} face the octahedral charge of individual octahedra are nonzero, namely −1/2 and +1/2 for the distribution given in Fig. 7c, but equals zero for all octahedra in the distribution Fig. 7d. Thus this second distribution is considered as representative of the PZNPC proton distribution on {001}. Figure 7e finally gives the proton distributions at minimum charge on the alternative {hkl} faces. On all these faces minimum charges are nonzero with the bridged groups remaining hydroxylated. More importantly, the {001}, {021}, and {101} distributions are characterized by alternate protonation of the terminal groups: half are hydroxylated and the other half hydrated. This finding is determinant with respect to the association of the SCS with reaction stoichiometry, as will be shown more clearly below. The presented concept is based on a coherent charge model: the total surface charge is obtained by summing up the individual octahedral charges of surface and bulk octahedra. This concept differs from the bond valence approach developed for predicting interatomic forces and distances [5,31,32]. Applying such models to surface functional groups without equilibration may lead to total surface charge deficits or excesses which will not fit the global PZNPC charge balance on which this model is based. The distribution of the octahedral charges on individual sites is rather formal; clearly the octahedral charge is not the unique parameter affecting proton distribution, as implicitly assumed.

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

11

Fig. 7. PZNPC surface proton load model. Singly coordinated (terminal) groups and doubly coordinated (bridged) (O, OH, H2 O) groups are designed by black circles; the white dots symbolize the associated protons. Triply coordinated groups have the same coordination and proton load as bulk anions; thus they are considered nonreactive and not specifically designated. Neutral Fe(H2 O)4 (OH)(O) octahedra (a) are combined (b) to the goethite-like HFO model bulk structure (c) by displacing only neutral water molecules (b, inset). This formal linkage process leads to (i) an electrostatically neutralized goethite bulk structure, and (ii) a formal proton load specific to the given {001}, {100}, and {010} faces representative of electrostatically neutral conditions, i.e., representative of the PZNPC proton load (c). (d) represents the same overall load but redistributed within each face following the octahedral charge distribution rule. (e) finally shows the same rules applied to the alternative hkl faces {021}, {101}, and {121}. OC refers to octahedral charge.

The charge distribution needs consequently to be verified. In some cases ambiguous distributions result: thus on {100} the charge is minimal for the distribution given in Fig. 7d, where both bridged and terminal groups are hydroxylated. But bridged groups are known to have a higher acidity. Consequently the model given in Fig. 7c could relate to the effective PZNPC proton distribution. The minimum octahedral charge model may thus not predict the PZNPC proton distribution in this case. On the {hkl} faces instead, only relevant with respect to this work, the distributions are unambiguous, the bridged groups all hydroxylated, and the proton distribution model therein considered valid. Fine charge delocalization calculations are beyond the topic of this work and thus not addressed. Also, these effects seem not to be relevant when the acid–base titration curve is considered: the statistical treatment of these data indicates that the individual microscopic reaction constants are all very close and thus independent of charge effects.

3.7. Surface coordination and surface stoichiometry The next step consists of linking proton load and SCN to reaction stoichiometry. The Cd adsorption reactions (22) and (23) are based on the bidentate Cd binding site given in Eq. (20). This means that one Cd(II) ion coordinates two acid–base reactive (O, OH) groups. The mean surface coordination number SCN obtained from crystal chemical considerations on {hkl} faces is higher, namely between 3 and 4, thus indicating that some of the coordinating O, OH groups are not acid–base reactive. Acid–base reactive and nonreactive coordinating groups need thus to be identified. A Cd octahedra coordinates to bridged and terminal groups; in principle both may be acid–base reactive. But the equilibrium analysis indicates that the individual acid–base constants are almost the same, thus pointing to a close structural environment of most reactive groups. Thus supposedly only one of the two types of functional groups, either bridged or terminal groups, are acid–base reactive. We consider the

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Fig. 8. The pK acid–base reaction concept: The macroscopic reaction (a), the reaction stoichiometry (b), and the example SCS 2 E{021} (c) and 1 E–2 C{021} (d) site. In (c) the octahedral charges (OC) of the two involved surface octahedra sums up to 0 at the PZNPC thus comparing to the site charge model and the overall charge balance. In (d) the apparent sum of the three octahedral charges is not zero but sums up to −1/3 at the PZNPC. This relates to the fact that two of the three surface Fe octahedra (the upper and lower octahedra in (d) with octahedral charges of −1/3 each (Fig. 7e) are shared with two neighboring 1 E–2 C{021} Cd sites. Half of their charge has thus to be associated to these neighboring sites; the sum of the thus corrected octahedral charges (OC ) becomes c zero at the PZNPC as required. This formal problem relates to the conceptual association of charges to octahedra. The division of octahedral charges between sites questions the existence of charge delocalization and its effects.

terminal groups as the acid–base reactive ones: their undersaturation with respect to the bulk structure is strongest and their reactivity is thus expected to be more important. Also, these groups are exactly half hydroxylated and half hydrated at the PZNPC on the relevant {100}, {021}, and {110} faces (Figs. 7d, 7e) which compares to the equilibrium model (OH, H2 O) site given in (20). Finally, considering alternatively the bridged groups as the acid–base reactive ones leads to improbable proton distributions on the HFO surface not compatible with (i) the octahedral charge minimization rule and with (ii) the postulated (OH, H2 O) stoichiometry. Thus at high pH terminal unreactive water groups had to exist, and at low pH concomitantly bridged water groups and terminal hydroxo groups would form. Thus bridged groups cannot be independently acid–base reactive. For these reasons terminal groups are proposed as being the acid–base reactive ones. For this likely case reaction equation, stoichiometry and surface structures of the acid–base exchange (Fig. 8) and Cd(II)–HFO complexation (Fig. 9) are compared. Figures 8a and 9a give the respective reaction equations and Figs. 8b and 9b the stoichiometries. Figures 8c, 8d and 9c, 9d finally show the associated SCS for the example {021} face (1 E–2 C{021} 3 and 2 E{021} 3 SCS). As can be seen, a Cd complexing site involves two terminal acid–base reactive groups (large black circles) which exchanges two protons as required from the stoichiometry model. The octahedral charges of the two involved octahedra sums up to −1, 0, +1 in the deprotonated, neutral, and protonated state (Figs. 8c, 8d). The charge of the “macroscopic” reaction equation thus fits the octahedral charge concept and consequently also the

overall structural and face-specific charge balance. Cd coordinates in both cases (Figs. 9c and 9d) to two acid–base reactive groups and coordinates on this face additionally one bridged acid–base unreactive group (small black circles). This relates to the SCN of three of these two {021} SCS. The site charge and the number of two coordinating acid–base groups coincide also for the two {001} SCS (not shown) which both fit the given reaction stoichiometry. The correlation is less straightforward for the SCS formed on the alternative {101} and {121} faces. Thus on {101} two SCS form with, respectively, one and three terminal groups involved (Table 2). The stoichiometry–structure relation is thus not senso stricto preserved on this face. Nevertheless the mean number of coordinating acid–base reactive groups is still 2; the “averaged” stoichiometry thus remains valid. Macroscopic data fits have the potential to reveal mean but not SCS-specific stoichiometries. The SCN of both SCS is 3, which means that they both should compare in reactivity, suggestively preventing preferential formation of one of these two structures. The situation is definitely different on the {121} face. Three terminal groups are involved in all SCS (Table 2); thus this face does not fit the stoichiometry– structure relation outlined above. This face incorporates a structural edge which can be visualized in Fig. 7e. The pattern suggests that this face is of minor importance as compared to the other ones. More generally, these detailed investigation shows that the stoichiometric information obtained from wet chemical data modeling analysis do not relate to particular SCS but rather correspond to a mean representation of a series of individual structures and associated stoichiometries. The stoichiometric relation given in Fig. 9b thus

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

13

Fig. 9. The 2 pK Cd–HFO sorption concept: The macroscopic reaction (a), the reaction stoichiometry (b), and the two {021} example SCS (c, d). As can be observed, two terminal and thus acid–base reactive groups are involved in the coordination process.

has to be considered as a mean concept, specifically valid only for the {100} and {021} SCS.

≡SH2 + H+ ↔ ≡SH+ 3, 4

+

(25)

−

≡SH2 − H ↔ ≡SH ,

3.8. Introducing the 4 pK model

4

In Fig. 9 a proton is released into solution during complexation of a Cd2+ species on a neutral surface site. This release is obligate, else Cd–H2 O–Fe coordinating groups form, which are not indicated for mineralogical and electrostatic reasons. This does not compare favorably to the reaction (22) obtained from equilibrium modeling which requires complexation of a Cd2+ ion on a neutral site without release of protons into the solution. Equilibrium and crystal chemical considerations thus do not compare in this respect, the problem is linked to the presence of terminal hydrated groups at the PZNPC. This can be overcome by considering the proton during Cd complexation not as displaced into solution but as displaced to a Cd-free hydroxylated terminal group. In terms of complexation modeling this can be achieved by enlarging the site to a double-sized 4 pK site exchanging four protons and two Cd(II) ions. At the PZNCP such a site ≡4 SH2 involves two hydroxylated and two hydrated terminal groups as given in Fig. 10a. The first Cd2+ ion may thus coordinate to two hydroxyl groups without release of protons into solution as required from reaction (22). The second Cd2+ ion then coordinates to two water groups obligatorily releasing two (and not one) protons into the solution following reaction (23), as depicted in Fig. 10b. Figures 10c and 10d gives the corresponding structures for the two {021} SCS. The [Hs ] data (Fig. 1a) was consequently fitted with a 4 pK model. The acid–base characteristics of this 4 pK model equals those introduced in the 2 pK model: all individual acid–base reactive groups are supposed to have the same reactivity given by Eq. (18). The four acid–base reaction constants ≡SH2 + 2H+ ↔ ≡SH2+ 4 , 
2  4 β102(int) = 1 6k(int) = 1015.8 ,

β101(int) = 2/(3k(int) ) = 108.11,

(24)

β10-1(int) = (2/3)k(int) = 10−8.47, +

≡SH2 − 2H ↔ ≡S 4

2−

(26)

,

2 β10-2(int) = (1/6)k(int)

= 10−17.4,

(27)

were consequently calculated based on Eqs. (10) and (18). The fit with these constants and the fixed parameters κ = 1.65, [≡2 St ] = 2.56 mM, solid concentration = 4.75 g/l Fe(OH)3 yielded a fit merit of V (Y ) = 4.98. The Hs (ph) curve of this fit coincides with that of #3, Table 1. The total concentration of sites [≡4St ] is half that of [≡2 St ] as the first exchanges a twofold number of protons and Cd(II) ions. Then the Cd data was fitted by evaluating different Cd sorption models. Three different 4 pK Cd–HFO models were found to be in adequate agreement with our experimental data. They are presented in Table 3 (models a to c). Statistics (in terms of V (Y )) do not discriminate between the Table 3 Modeling of the 4 pK Cd2+ adsorption ≡SH2 + Cd2+ ≡SH2 + 2Cd2+ − 2H+ ≡SH2 + 2Cd2+ − 3H+ ≡SH2 + 2Cd2+ − 4H+

Model a: V (Y ) = 18.79 ↔ ≡SCdH2+ log 4 β110(int) = 6.43 2 ↔ ≡SCd2+ log 4 β12-2(int) = −5.16 2 + ↔ ≡SCd2 H−1 log 4 β12-3(int) = −14.77 ↔ ≡SCd2 H−2 log 4 β12-4(int) = −33.56

≡SH2 + Cd2+ ≡SH2 + 2Cd2+ − 2H+ ≡SH2 + 2Cd2+ − 3H+ ≡SH2 + 2Cd2+ − 6H+

Model b: V (Y ) = 19.06 ↔ ≡SCdH2+ log 4 β110(int) = 6.43 2 ↔ ≡SCd2+ log 4 β12-2(int) = −5.19 2 + ↔ ≡SCd2 H−1 log 4 β12-3(int) = −14.62 4 ↔ ≡SCd2H2− −4 log β12-6(int) = −43.13

Model c: V (Y ) = 18.50 ↔ ≡SCdH2+ log 4 β110(int) = 6.42 ± 0.02 ≡SH2 + Cd2+ 2 log 4 β12-2(int) = −5.13 ± 0.03 ≡SH2 + 2Cd2+ − 2H+ ↔ ≡SCd2+ 2 ≡SH2 + 2Cd2+ − 4H+ ↔ ≡SCd2 H−2 log 4 β12-4(int) = −23.88 ± 0.03

14

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

Fig. 10. The 4 pK acid–base and Cd sorption concept. The site exchanges four protons and two Cd(II) ions. (a) gives the reaction equations, (b) relates the sorption of the first (without proton release) and second (with two protons released) Cd ions, and (c) depicts this concept for pairs of adsorbing 2 E{021} and 1 E–2 C(021) SCS.

three models. In all of them, the first step is the adsorption of a Cd(H2 O)2+ 6 ion onto a neutral site without proton release into the solution; in the second step the second Cd(II) ion is adsorbed and concomitantly two protons are released into the solution (Fig. 10b). As pH increases further, more deprotonation occurs (third and fourth reaction steps). These protons may belong to water molecules that were coordinated to the adsorbed Cd ions. We prefer model c for its simplicity. Based on the assumption that the uptake of each Cd2+ species is controlled by the same microscopic stability constant, 4 KCd1(int) is calculated from the experimentally determined 4 KCd2(int): ≡4 SH2 Cd2+ ↔ ≡4 SH2 + Cd2+ , 4

KCd2(int) = 1/ 4 β110(int) = 10−6.42,

4 2+ ≡ SH2 Cd4+ + Cd2+ , 2 ↔ ≡ SH2 Cd 4 KCd1(int) = 4 4 KCd2(int) = 10−5.82 .

(28)

4

(29)

Combination of the equations ≡4 SH2 + Cd2+ ↔ ≡4 SH2 Cd2+ , log 4 β110(int) = 6.42, ≡ SH2 Cd 4

2+

+ Cd

2+

↔≡

(30) 4

SCd2 H4+ 2 ,

− log KCd1(int) = 5.82, 4

2+ + 4 ≡ SCd2 H4+ 2 − 2H ↔ ≡ SCd2 , log 4 βb(int) = −17.36,

(31)

4

(32)

results in 2+ + 4 ≡4 SH2 + 2Cd+ 2 − 2H ↔ ≡ SCd2 ,  log 4 β12 -2(int) = −5.12,

(33)

which is in full agreement with the fitted log 4 β12-2(int) value (Table 3, model c). The fact that the combination of 4β 2 4 4 110(int) , KCd1(int) , and βb(int) gives β12-2(int) suggests 4 that β10-2(int) correctly describes the deprotonation of terminal water groups in the presence of adsorbed Cd(II). The two Cd(II) ions thus bind independently. The model is in good agreement with the experimental data (Fig. 1b, dotted line). This model obtained exclusively from [Hs ] data (Fig. 1) was then compared to the Cd(II) isotherm recorded at pH 6.7 ± 0.2 (Fig. 2, dotted line) and to the Cd(II) adsorption edge curve (Fig. 3, dotted line). As in the previously presented model (21), (22) general good agreement of experimental data and model can be observed at low (Cd isotherm) and median (Cd isotherm and Cd adsorption edge curve) Cd coverage. Instead at high Cd loadings both Cd isotherm (Fig. 2) and Cd adsorption edge curve (Fig. 3) do not agree with the model obtained from the [Hs ] data (Fig. 1b): the measured Cd(II) adsorption is higher than predicted. The excess adsorbed Cd relates suggestively to adsorption of Cd onto the low SCN {hk0} sites, which are of low reactivity and not considered in both equilibrium and structure models. The validity of the model at the low Cd concentrations

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

of the Cd isotherm nevertheless indicates the model being valid for a large range of soluted Cd concentrations. 3.9. Alternative models The given 4 pK model fits adequately the experimental data and compares to the {hkl} proton load model. The question has to be addressed inasfar as the 4 pK model can be taken as representative for the complexity of the HFO surface structure as a whole. For example, the model implicitly assumes that most of the Cd receiving sites are high-affinity sites which would relate to the 60% Cd surface loading considered by the model. Conceptually these sites would correspond to the {hkl} SCS with SCN of 3 and 4. The {hk0} face SCS, with SCN of 1 or 2, then would form the low-affinity SCS not considered in this model. The SCN-based affinity concept rather promotes the definition of four sites associated to the four SCN instead of this single site model. Also, adsorption of Cd(II) ions without release of protons into solution may be questioned in terms of electrostatics [33,34]. To conform or reject these reservations three further 2 pK models will be briefly presented and discussed. The first considers that the surface is divided into two different Cd sorption sites ≡2 X and ≡2 Y of equal total concentrations [≡2 Xt ] = [≡2Yt ] = [≡2 St ]/2 = 2.56 mM. Surface acid– base properties and Cd hydrolysis were fixed as given above. The model ≡2 XH + Cd2+ ↔ ≡2 XH–Cd2+ , log 2 βa = 6.37, ≡ XH + Cd 2

2+

(34) +

+ H2 O ↔ ≡ X–CdOH + 2H , 2

log βb = −11.82, 2

≡ YH + Cd 2

2+

(35)

↔ ≡ YH–Cd 2

2+

,

log βc = 3.92, 2

≡ YH + Cd 2

2+

(36) +

+ H2 O ↔ ≡ Y–CdOH + 2H , 2

log βd = −11.76, 2

(37)

leads to a fit of comparable quality as compared to the 2 pK and 4 pK models for all wet chemical data available (Figs. 1–3, fits not shown). This indicates that the data may also be fitted when considering the existence of sites of different affinities. The alternative set of reaction constants ≡2 XH + Cd2+ ↔ ≡2 XH–Cd2+ , log 2 βe = 6.39, ≡ XH + Cd 2

2+

(38) +

+ H2 O ↔ ≡ X–CdOH + 2H , 2

log 2 βf = −12.79, ≡ YH + Cd 2

2+

(39) +

+

↔ ≡ Y–Cd + H , 2

log βg = −3.17, 2

≡ YH + Cd 2

2+

+ 2H2 O ↔ ≡

log βh = −20.69, 2

(40) 2

Y–Cd(OH)−1 2

+

+ 3H , (41)

15

also yield a comparable fit quality. In this model all four species have different Cd/H stoichiometries, the Cd ions exchange 0, 2 protons, and, respectively, 1, 3 protons when adsorbing on the ≡X and ≡Y site. At low pH the adsorption of Cd on ≡X sites is naturally preferred. Preferential adsorption of Cd(II) here thus relates not only to SCN but also to the possibility of displacing or releasing protons. The third tested model concerns a model where the displacement of protons at the surface is not supposed to occur; i.e., the sorption of a Cd2+ ion leads obligatorily to the exchange of at least one proton with the solution. The corresponding model ≡2 YH + Cd2+ + H2 O ↔ ≡2 Y–CdOH + 2H+ , log 2 βi = −12.25, ≡ XH + Cd 2

2+

(42) +

+

↔ ≡ X–Cd + H , 2

log 2 βj = −2.21, ≡ XH + Cd 2

2+

+ 2H2 O ↔ ≡

log βk = −21.16, 2

(43) 2

X–Cd(OH)−1 2

+

+ 3H , (44)

leads to an almost acceptable fit for the Hs and the Cd curve (fit not shown) but fails in explaining the Cd isotherm data set (Fig. 2, fit not shown). At log[Cd2+ ] = −5.4 the predicted log [Cdads ] equals −5.2 instead of ∼ −4.7; the predicted [Cdads ] is thus clearly too low. The interest of this model concerns the demonstration that the Cd/H exchange ratio may influence the [Cd2+ ]/[Cdads] ratio at different Cd total concentrations. Thus possibly the discussed Cd/H stoichiometry of 2 pK and 4 pK species relates rather to the aim of limiting the number of species than to effectively existing species. Thus all in all little confidence in classical equilibrium model remains when considering the complexity of the X-ray amorphous HFO substrate and the formed SCS at the microscopic scale. It is clearly not possible to involve the total number of species in the data modeling process: eight different SCS may form only on the more closely inspected {hkl} faces, and to each of these SCS a series of protonation reactions relate. Supposedly more SCS are added when the effective HFO bulk structure is considered, more complicated than the goethite model used. It follows that this and other HFO equilibrium analysis models are susceptible to represent a moiety of different individual existing species, and that conclusions related to surface site heterogeneities and Cd/H exchange ratio rather relate to the resolution limits of the investigation methods than to effectively existing particular species or sites. We consider the existence of sites of different Cd affinity on HFO. But the crystal chemical considerations do not confirm the existence of two particular distinctive sites of different affinity in a concentration ratio of 40 as postulated by Dzombak and Morel [3]. The present crystal chemical model rather indicates that on this and generally on other X-ray amorphous substrates sites of high SCN and thus supposedly high affinities are abundant. We thus do not consider the model given in [3] as based

16

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

on physical reality even if the model succeeds in predicting the solid/solution fractionation of Cd and other cations. More generally the study shows that the combination of different investigation methods for a surface structure model is necessary to decipher the individual structures and reactions involved in sorption processes. The classical equilibrium modeling method clearly fails in determining an unambiguous set of equilibrium reactions. CdK EXAFS spectroscopy helps in deciphering the linkage types involved in the SCS due to its potential to determine precisely interatomic distances. Instead, determination of coordination numbers with this method are not straightforward at all, the potential of the method in determining individual SCS is thus limited. The combination of this information in a detailed investigation of the bulk and surface structure is thus decisive for a critical evaluation of the apparatus response in terms of SCS and, respectively, equilibrium reactions and the parameters controlling site affinities.

4. Summary and conclusions The present study compares in a consistent manner macroscopic and microscopic data related to the sorption of Cd on HFO. The reactivity and distribution of acid–base reactive groups and the Cd surface complex structures SCS are investigated by comparing titration data to a HFO surface structure model. The goethite bulk structure is taken as a model for the HFO bulk; the chain-cut {hkl} faces considered abundant on HFO are more precisely analyzed. The statistical analysis applied to the acid–base titration data shows that the microscopic reactivity constants of the individual acid–base protons are very close. The environment of most acid–base reactive functional has thus to be unique, and consequently not both bridged and terminal groups may be acid–base reactive. The terminal water groups are identified as the reactive ones by considering the surface structure model and the Cd/H exchange ratios. Two such terminal groups are involved in almost all {hkl} Cd–Fe SCS. The Cd stoichiometry determined in this and other works is thus 2. The individual Cd–Fe surface structures are nevertheless numerous and differ in stability as (i) varying numbers (0, 1, 2) of acid–base unreactive bridged groups link to the sorbed Cd ions, and as (ii) the involved Cd–Fe linkage types (edges and corners) vary. 8 different Cd–Fe SCS, nonequivalent with respect to SCN and involved linkage types, were thus determined on the more precisely investigated {hkl} faces. To these 3 further {hk0} Cd–Fe SCS add. To a unique microscopic acid–base protonation constant thus relates a high number of different Cd receiving sites. The relationship between SCS and Cd affinity is a second addressed topic. The potentially forming SCS have surface coordination numbers (SCN) of 1 to 4. On goethite, during the Fe(III) crystal growth process, SCS with high SCN form on high-affinity {hkl} faces and SCS of low SCN form on low-affinity {hk0} faces. CdK EXAFS shows that adsorbing

Cd(II) ions compare in this respect to the Fe(III) ions coordinated during crystal growth. Thus the SCN is potentially the controlling parameter of affinity for Cd–goethite SCS, and, in analogy, also for Cd–HFO surface complexes. Up to 60% Cd loading the Cd–HFO equilibrium data set could be modeled assuming the existence of a single Cd site and thus a unique Cd affinity. This suggestively relates to the high-affinity sites of the {hkl} faces (SCN of 3 and 4), which should be abundant on that solid. We thus consider, in opposition to the model of Dzombak and Morel [3], that high-affinity sites are abundant and compare in concentration to the low-affinity sites. We relate the low-affinity sites to the {hk0} SCS (SCN of 1 and 2). More generally, the work presented here shows that wet chemical methods may lead to ambiguous results with respect to site affinity by simply comparing alternative good fitting models. Another point relates to the charge balance. The charge of the Cd-receiving sites is directly obtained from a formal linkage process of neutral bulk FeOOH octahedra. The sum of the charges of the individual surface sites thus fits the overall charge balance. We consider this point as important and problematic that it is not taken into account in actual state-of-the art models. The distribution of the octahedral charge on individual sites is not quantitatively assessed in this paper; we consider nevertheless the effect on stability as minor considering the almost unique microscopic acid–base protonation constant. From these results we conclude that neither spectroscopic nor wet chemical investigation methods can identify the variety of existing SCS on HFO. Classical wet chemical Cd– HFO sorption models, such as that given in [3] limit the number of Cd binding sites to two, a number we consider related to the resolution limit of the investigation method rather than to existing individual SCS. Similarly EXAFS spectroscopic investigations determine linkage patterns such as edges and corners which are common to series of different SCS. Wet chemical and spectroscopical findings thus need to be combined to crystallographic knowledge to identify potentially existing individual structures and their variability. Thus rather than presenting an overall applicable surface complexation model the present work points to the difficulties related to the comparison of microscopic and macroscopic information. Our results show that the combination of crystal chemical, spectroscopic, and wet chemical information is necessary to understand sorption processes on the microscopic scale. We guess that some of the topics discussed, such as the coherency of charge calculations and the relations of stoichiometry to SCS and SCN need generally to be studied in more detail in studies and sorption models that relate the macroscopic and microscopic scale.

Acknowledgments We are indebted to Professor R. Giovanoli for the X-ray diffraction analysis of the ferrihydrite powder. We further

L. Spadini et al. / Journal of Colloid and Interface Science 266 (2003) 1–18

thank Professor Staffan Sjöberg for critical comments and valuable discussions and also Bernhard Trusch for analytical support. Financial support provided by the European Environmental Research Organization (EERO) is gratefully acknowledged. K.V.R. acknowledges financial support from the Centre National des Recherches Scientifiques (CNRS) while working in Grenoble during the summer of 1999.

Appendix A. Equilibrium analysis details The surface site definition is based on the assumption that the ferrihydrite surface contains bridged hydroxylated groups (≡Fe2 –OH in the protonated state) and terminal water groups (≡Fe–OH2 for the protonated state). Only the terminal water groups are acid–base reactive within the titration pH range. Several such neighboring reactive groups are combined at the site ≡n SHn/2 , which exchanges n terminal protons within the titration range. In the data treatment, n will be 2 or 4, depending the site definition. The general equilibrium between the three components ≡n SHn/2 , Cd2+ , and H+ is given by p ≡n SHn/2 + qCd2+ + rH+ (2q+r)+

↔ (≡n SHn/2 )p Cdq Hr

,

n

βpqr .

(A.1)

The n acid–base reactive anion groups of a site are in a fixed position relative to each other. It follows that p needs to be
1 [23]. The acid–base equilibrium is then defined by ≡SHn/2 + rH+ ↔ ≡SHr+ (n/2+r) ,

n

βp0r ,

p

[Cdt ] =

q

p

[Ht ] =

r

q

p

(A.3) ] [H ] , (A.4)

2+ q

+ r

r

 p

q βpqr [≡SHn/2 ] [Cd n

q

r n βpqr [≡SHn/2 ]p [Cd2+ ]q [H+ ]r . (A.5)

r

The conditional constants in Eqs. (A.5)–(A.9) need to be corrected for the coulombic energy of the charged surface to obtain the corresponding intrinsic constants n

βpqr(int) = n βpqr e((2q+r)F ψ/RT ) for all species with p = 0,

(A.6)

where ψ is the acting surface potential. The surface charge is given by  Tσ = (2q + r) n βpqr p=0 q

or by σ=

Tσ F (C/m2 ), Sa

(A.8)

where S is the specific surface area (m2 g−1 ) and a is the concentration of solid (g dm−3 ). The data was modeled with the constant capacitance model [22], adequate for the equilibrium analysis in heterogeneous systems in which attention is focused on the formation of inner-sphere surface complexes. This model relates the surface charge and the surface potential by σ = κψ.

(A.9)

Combining Eqs. (A.8) and (A.9) gives ψ=

Tσ F . Saκ

(A.10)

The fit of the potentiometric data was performed with the program FITEQL version 2.0 [35]. The evaluation was based on the proton balance (A.5). The program minimizes the weighted sum of error squares  (YH /σYH )2 , (A.11) where YH = Ht(calc) − Ht(exp) .

(A.12)

References

(A.2)

with −n/2
r
+n/2 (n = 2, 4, 6, . . .). The total concentrations [≡St ], [Cdt ], [Ht ], of the components are correspondingly given by  p n βpqr [≡SHn/2 ]p [Cd2+ ]q [H+ ]r , [≡St ] = 

17

r

× [≡SHn/2 ]p [Cd2+ ]q [H+ ]r (molc /dm3 )

(A.7)

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