For experiments conducted in the absence of CO₂, all reagents were prepared with freshly distilled and deionized water collected under a nitrogen atmosphere using a portable microwave still. The water was then transferred to a nitrogen atmosphere glove box and used to make FeCl₃, SrCl₂, KOH, and NaOH stock solutions from solids that were purged for 20 to 30 minutes under a nitrogen stream and weighed in the nitrogen atmosphere glove box.

For experiments conducted in the presence of CO₂, stock solutions were made by dissolving reagent grade NaCl, SrCl₂, FeCl₃ and Fe(NO₃)₃ solids in distilled and deionized water. Commercial high purity NaOH, HNO₃, and HCl stock solutions were also used to adjust the pH of the sorption experiments. Solutions were stored in sealed containers and were not continually exposed to the atmosphere. We define dissolved carbonate as the sum of dissolved aqueous carbon species in this paper. Sources for dissolved carbonate in the sorption experiments include diffusion of atmospheric CO₂ when the suspensions were prepared, dissolved carbonate present in the NaOH stock solution used to adjust solution pH, and possibly carbonate sorbed to goethite when synthesized at atmospheric pCO₂.

The amorphous silica used in the sorption experiments was synthetic silica gel (Mallinckrodt Silica), 100–200 mesh, (lot # 6512), with an average pore diameter of 150 Å. The gel was repeatedly cleaned ultrasonically with distilled and deionized water until the suspension yielded a clear supernatant after 10 minutes of settling. Cleaned gel was dried at 40°C for 24 hours and stored in a plastic container at room temperature. Surface area was 277 m² g^-1 determined by BET nitrogen gas adsorption. No effort was made to exclude atmospheric CO₂ in the cleaning and drying procedure for amorphous silica.

Four lots of goethite were synthesized following protocols outlined in Schwertmann and Cornell 29 . For CO₂-free sorption experiments, goethite (Lot 1) was synthesized using KOH and FeCl₃•6H₂0 in a nitrogen atmosphere from reagents dissolved in CO₂-free distilled and deionized water. After initial formation of iron hydroxide, the suspension was purged with nitrogen for 60 hours at 70°C to transform the hydroxide to goethite. It was then rinsed repeatedly to remove chloride using dialysis tubing and CO₂-free distilled and deionized water. Goethite was dried under a nitrogen stream at 40°C and stored in a nitrogen atmosphere glove box. Mineralogy was confirmed by XRD. For sorption experiments prepared in atmospheric CO₂, goethite (Lots 2 and 3) was synthesized following the same protocol except that no effort was made to exclude CO₂. For Lot 4, goethite was prepared from Fe(NO₃)₃ instead of FeCl₃•6H₂O and no effort was made to exclude CO₂. Surface areas determined by BET nitrogen gas adsorption were: Lot 1 = 37.9 m² g^-1; Lot 2 = 38.2 m² g^-1; Lot 3 = 37.9 m² g^-1; and Lot 4 = 27.7 m² g^-1. An average surface area of 37.8 m² g^-1 was used to model the strontium sorption to goethite in NaCl solutions.

Strontium sorption was measured in amorphous silica and goethite suspensions prepared in the presence and absence of atmospheric CO₂ at 25°C from pH 6 to 10. Tables 1, 2, 3, 4, 5, 6 list total surface area, sorption atmosphere, ionic strength, and initial and final solution composition for each experiment. Amorphous silica or goethite was mixed with a freshly prepared SrCl₂/NaCl or Sr(NO₃)₂/NaNO₃ solution of the desired concentration in polycarbonate test tubes. After the pH was adjusted, the tubes were sealed, shaken vigorously by hand, and then reacted for 2 or 14 days in a constant temperature orbital-shaker water bath at 200 rpm. The 14-day experiments were conducted in goethite suspensions to see if additional reaction was needed to precipitate strontium carbonate at higher pH. At the end of the experiment, the final pH of each solution was measured, a sample (2.5 ml) was taken, filtered (4.1 nm pore size), acidified with high purity HCl or HNO₃ to prevent SrCO₃ precipitation, and analyzed for strontium by inductively coupled plasma atomic emission spectrometry (ICP-AES) (detection limit = 10^-7 molal; precision ± 2%). For many of the experiments prepared in the presence of atmospheric CO₂, total dissolved carbonate was measured from a filtered sample using a carbon analyzer with an IR detector (detection limit = 5 × 10^-5 molal). With this technique, dissolved carbon is purged with 11 N phosphoric acid and nitrogen gas. The resulting concentrations are under saturated with respect to atmospheric CO₂ at higher pH. Additional control experiments with no solid present were done to check for strontium sorption to vessel walls and for precipitation of SrCO₃(s). Strontium sorption uncertainty is calculated from the analytical uncertainty of the initial and final solution concentrations and the uncertainty associated with a small amount of strontium inherent in the substrate measured from substrate control experiments. For the experiments prepared in the absence of CO₂, preparation, sampling, and reagent storage were done in a nitrogen atmosphere glove box.

Strontium sorption to amorphous silica experiments from pH 8 to 10 with dissolved CO₂ and initial strontium concentrations of 10^-3 M were analyzed with EXAFS. After reaction and centrifugation, supernatant liquids were removed and sorption samples were loaded as wet pastes into teflon sample holders with Kapton windows just prior to XAS analysis. For sorption samples collected at cryogenic temperatures, wet samples were quenched by immersion in liquid nitrogen and then placed in a helium cryostat in the beamline hutch.

For strontium sorption samples, EXAFS spectra were collected on wiggler beamline IV-3 at the Stanford Synchrotron Radiation Laboratory (SSRL). The incident beam was detuned to 50–70% of the maximum incoming intensity to reject higher-order harmonic reflections. The mid-point of the absorption edge of SrCO₃(s) reference compound (set to 16105 eV) was used for energy calibration. Spectra were collected in fluorescence mode using a 13-element germanium array detector. For each sorption sample, 20–40 scans were collected to achieve an adequate signal.

Spectra in the EXAFS region were analyzed with the program EXAFSPAK 30 . Reference phase shift and amplitude functions used in non-linear least-squares fitting of experimental spectra were calculated using the ab initio program FEFF6 31 32 33 . In non-linear least-squares fits of the sorption sample spectra, bond distance (R), backscatterer number (N), and the disorder or Debye-Waller term (σ ²) were treated as adjustable parameters. The difference between theoretical and experimental threshold energies (ΔE₀) was treated as a single adjustable parameter for all Sr-backscatterer shells 34 . Least-squares fits were performed on both filtered spectra of individual peaks in the radial structure functions (RSF) and on normalized χ(k) spectra with no significant differences in fit results. Detailed analyses of crystalline and hydrated strontium reference compounds and strontium in aqueous solution at ambient and cryogenic temperatures are given in our previous study 25 . These references allowed us to constrain adjustable EXAFS fitting parameters and to estimate errors in fit parameters based on empirical analysis (rather than using only the statistical errors derived from the least-squares fit). Our previous study 25 , showed that anharmonic vibrational disorder of oxygen-ligated strontium compounds can be neglected because the third cummulate term (C3) of the EXAFS phase shift function is generally not significant above the error in fitted EXAFS distances (i.e., R ± 0.02 Å with and without C3). Also, we showed that rapid quenching of sorption samples for data collection at low temperature does not appear to introduce any new features into EXAFS spectra when compared with room temperature spectra. Based on our previous strontium EXAFS analyses, S₀ ² was fixed at 0.92 and estimated empirical errors in fit parameters for first-shell Sr-O analysis are: R ± 0.02 Å; N ± 1 for N in the range of 6–12; σ ² ± 25% 25 26 .

GEOSURF 35 and FITEQL 36 were used to fit specific surface complexation reactions to the experimental data. GEOSURF is tied to a thermodynamic database that automatically accounts for aqueous speciation and ionic strength during the sorption simulation. We used GEOSURF to fit strontium sorption to amorphous silica and goethite in carbonate-free suspensions and to amorphous silica in the presence of dissolved carbonate because SrCO₃(aq) is negligible and dissolved carbonate is not known to sorb to amorphous silica to the best of our knowledge. We used FITEQL to fit strontium sorption data to goethite in the presence of dissolved carbonate to account for the significant carbonate sorption to the goethite surface because FITEQL allows the input of measured carbonate concentrations at each titration point for the speciation calculation. The pH dependence of measured dissolved carbonate concentrations in goethite suspensions was used to estimate dissolved carbonate concentrations when modeling strontium sorption for those experiments in which dissolved carbonate was not measured. The extended-Debye Hückel equation was used to correct for ionic strength effects. Aqueous equilibrium constants used in the calculations are listed in Table 7 37 .

Absorption spectra were collected for samples of strontium sorbed to amorphous silica from solutions of 10^-3 M SrCl₂ and 0.1 M NaCl with dissolved CO₂ from pH 8 to 10. Normalized χ(k) EXAFS spectra and Fourier transforms of the spectra are shown in Figure 1. Numerical fit results are given in Table 8. For all sorption samples collected at low temperature, there is only a single shell of oxygen backscatterers with a Sr-O distance of 2.60 ± 0.02 Å and a coordination number of 10 ± 1. Compared to a spectrum of strontium sorbed to silica gel collected at room temperature reported in our previous study 26 , the fitted Sr-O distance at room temperature is slightly shorter (2.57 ± 0.02 Å) than that derived for low-temperature spectra. This small distance contraction was also noted for room- and low-temperature spectra of strontium sorbed to kaolinite 26 and probably results from a small anharmonic effect 25 . For strontium sorption on both silica gel and kaolinite, the fitted Sr-O distance is slightly shorter than the Sr-O distance (2.65 ± 0.02 Å) determined for aqueous Sr²⁺ in a 10^-3 M SrCl₂ solution 25 .

Comparison of EXAFS sorption spectra indicates no change in strontium coordination with increasing pH and sorption. There is no evidence for silica backscatterers from the substrate, nor is there evidence for carbon or strontium backscatterers indicative of strontianite precipitation or other multi-nuclear sorption complexes. Evidence for scattering from atoms beyond the first coordination shell would be seen in multiple sine-wave oscillations in normalized spectra (i.e., "beat" patterns or shoulders on primary sine waves). Figure 1 compares the strontium sorption spectra to reference spectra for crystalline strontium carbonate (SrCO₃(s)) and strontium in the calcium zeolite mineral heulandite (≈ 4500 ppm strontium substitution in the calcium site). In the calcium site in heulandite, strontium is eight-coordinated by oxygen, with three ligands of framework oxygen atoms from the mineral surface and five ligands of water extending into the zeolite channel. As shown here and in our previous study of strontium in zeolites 25 , scattering from aluminum or silicon atoms in the zeolite framework is apparent in spectra up to a distance of 4.15 Å from central strontium because of direct bonding to the zeolite framework. Thus, strontium in zeolites is a good analog structure for inner-sphere complexation of strontium on silica gel, if it occurs. Although aluminum and silicon are relatively light backscatterers, they are easily identified as backscatterers in zeolite when strontium is partially dehydrated and bonds to the framework structure. Likewise, precipitation of strontianite is readily observed by backscattering from carbon and strontium atoms at distances of ≈ 4 Å or less, but is not seen in the sorption sample spectra, even for samples in which reacting solutions were supersaturated with respect to strontianite (Figure 2a). In our previous study of strontium reference compounds 25 , we found that backscatterer atoms beyond the first oxygen coordination shell were apparent in normalized spectra when fitted values of σ ² (the Debye-Waller disorder parameter) were below ≈ 0.025 Å ² (for N < 12 and R > 3 Å). We did not collect EXAFS spectra on samples of strontium sorbed to amorphous silica in the absence of CO₂ because the bulk sorption behavior was the same with CO₂ present. Nor did we collect EXAFS spectra on strontium sorption in 5 × 10^-3 M NaCl solutions.

Figure 3 and Table 8 reproduce EXAFS spectra on strontium sorption to goethite with dissolved carbonate 26 to show changes in bonding at the surface as a function of solution pH. These results show that strontium forms a surface precipitate at pH 8.5, but not at higher pH where the solutions were more supersaturated with respect to strontianite for total strontium concentrations of 10^-3 M (Figure 2b). For solutions with pH above 8.5 only Sr-O backscatters were detected indicating that strontium retains all or part of its hydration sphere when sorbed to the surface. This behavior was attributed to a maximum sorption of carbonate on goethite near pH 8.5 that nucleated a SrCO₃-type surface precipitate, and decreasing carbonate sorption at higher pH that resulted in formation of hydrated Sr surface complexes 26 .

The fitted and experimental results are shown in Figure 4 for the sorption of strontium to amorphous silica in solutions with total Sr ranging from ~7 × 10^-7 to 10^-3 M, 0.1 M NaCl, and with and without dissolved carbonate, and for one experiment with total Sr = 10^-4 M, 0.005 M NaCl, and dissolved carbonate. Strontium sorption to amorphous silica in the presence and absence of dissolved carbonate can be described with two tetradentate strontium complexes on the β-plane and one monodentate strontium complex on the diffuse plane (Table 9):

4 >SOH + Sr²⁺ = (>SOH)₂(>SO-)₂_ Sr²⁺ + 2H⁺

4 >SOH + Sr²⁺ +H₂O = (>SOH)₂(>SO-)₂_ SrOH⁺ + 3H⁺

>SOH + Sr²⁺ = >SOH...Sr²⁺

In 0.1 M NaCl solutions, tetradentate Sr²⁺ and SrOH⁺ complexes capture the gradual sorption edge with increasing pH. The tetradentate Sr²⁺ complex, (>SOH)₂(>SO-)₂_Sr²⁺, dominates at near neutral pH where sorption is minimal. As pH and percent sorption increase, the tetradentate hydrolyzed SrOH⁺ complex, (>SOH)₂(>SO-)₂_ SrOH⁺, accounts for most of the strontium sorbed to the surface. In solutions with lower ionic strength (0.005 M NaCl), the sorption edge shifts to lower pH and a monodenate Sr²⁺ complex on the diffuse plane, >SOH...Sr²⁺, is needed to fit the data in addition to the tetradentate strontium complexes. Inclusion of other outersphere complexes (such as >SOH_ Sr²⁺, tetradentate SrCl⁺, and tetradentate SrClOH) failed to capture the enhanced strontium uptake at lower ionic strength. The dominance of the β-plane Sr²⁺ complexes at lower ionic strength over the β-plane SrOH⁺ complexes reflects the interplay between the charge in the β-plane and the charge of the surface complexes. As the ionic strength increases the β-plane SrOH⁺ complex becomes more dominant. The position of the surface complexes on the β- and diffuse planes suggests that strontium retains some or all of its waters of hydration at the amorphous silica surface and is an outer-sphere complex. The designation of sorbed strontium as outer-sphere is supported by the shift in the sorption edge from pH = 7.2 at I = 0.005 M NaCl to pH = 9 at I = 0.1 M NaCl (Figure 4B,F) and by EXAFS data showing only Sr-O bonding and coordination similar to aqueous Sr from pH 8.5 to 9.9 (Figure 1).

The strontium surface complexation model presented here is consistent with Sverjensky's 16 model fit for other alkaline earth sorption data for amorphous silica. Sverjensky 16 described Ca 50 and Mg 47 sorption data over a range of ionic strengths (0.001 to 0.1 N for Ca and 0.005 to 0.05 N for Mg) for a single surface coverage for each cation. The Ca model used the same type of complexes used to describe strontium sorption [>SOH...Ca²⁺, (>SOH)₂(>SO-)₂_ Ca²⁺, and (>SOH)₂(>SO-)₂_ CaOH⁺], and the Mg model used two of the three reactions [>SOH... Mg²⁺ and (>SOH)₂(>SO-)₂_ Mg²⁺].

The fitted and experimental results are shown in Figure 5(D–F) for the sorption of strontium to goethite with total Sr ranging from 10^-6 to 10^-4 M and 0.1 M NaCl in suspensions with dissolved carbonate. The strontium surface complexation model developed here is based on sorption data from experiments with total Sr from 10^-6 to 10^-4 M to avoid possible precipitation of strontium carbonate from experiments with the total Sr ~ 10^-3 M (data not shown). The base model consists of the tetradentate and monodenate SrOH⁺ complexes (Equations 4 and 5) and carbonate surface complexation reactions and constants from Villalobos and Leckie 41 adjusted in accordance with the site-occupancy standard state 14 (Table 9) to account for carbonate sorption to goethite:

>SOH + CO₃ ^2- + H⁺ = >SO^-0.2_COO^-0.8 + H₂O

>SOH + CO₃ ^2- + H⁺ + Na⁺ = >SOCOONa + H₂O

>SOH + CO₃ ^2- + 2H⁺ = >SOCOOH + H₂O

The binuclear tetradentate strontium surface complex (Equation 6) was not included in the calculations because its overall contribution is minimal at lower surface coverages (total Sr from 10^-6 to 10^-4 M) and because it led to convergence problems within FITEQL. Fits to the strontium sorption data require the addition of two monodentate strontium carbonate complexes. In the reactions below we maintain the carbonate stoichiometry and charge distribution between the 0- and β-planes as modeled by Villalobos and Leckie 41 for the strontium carbonate complexes.

>SOH + CO₃ ^2- + Sr²⁺ = >SO^0.2-_COOSrOH^0.2+ + H₂O

>SOH + CO₃ ^2- + H⁺ + Sr²⁺ = >SO^0.2-_COOSr^1.2+ + H₂O

Strontium carbonate surface complexes dominate the goethite surface from pH 6 to 10, with the non-carbonate SrOH⁺ complexes becoming important only at higher pH where less carbonate sorbs to the surface.

The strontium sorption model provides some insight into the pH dependence of SrCO₃ precipitation at the goethite surface observed using EXAFS 26 . Strontium carbonate precipitate has been identified in the presence of goethite at pH 8.5, but not at pH greater than 8.7 in samples with the same surface loading (total Sr = 10^-3 M) and for one sample at pH 9.9 with total strontium = 10^-4 M. (Figure 3). Sahai et al 26 concluded that the absence of a strontium carbonate precipitate at pH above 8.7 was due to insufficient carbonate on the surface to nucleate the precipitate, based on a decrease in carbonate sorption on the goethite surface from pH 6 to 10 (using constants from VanGeen et al. 39 ). Figure 6A is an example of the amount of carbonate predicted to sorb to the goethite surface using our strontium surface complexation model and measured dissolved carbonate. Strontium carbonate complexes comprise at most 3% of the total amount of carbonate on the surface. The amount of carbonate on the surface has a complex dependence on solution pH. Unlike the linear increase of dissolved carbonate concentrations with pH shown in Figure 6B, the amount of carbonate on the surface increases from pH 6 to 8, reaches a maximum from pH 8 to 9, and sharply decreases with increasing pH where the aqueous carbonate complexes dominate. From pH 6 to 9, there is 10 to 4 times more carbonate on the surface than dissolved in solution, illustrating the high affinity of the goethite surface for carbonate in near-neutral pH suspensions, where surface precipitation of strontium carbonate has been observed 26 . The stoichiometry of the carbonate complexes in our model further suggests that the >SO^0.2-_COOSr^1.2+ may be a precursor to surface precipitation, where as the hydrolyzed carbonate complex is not. Thus strontium carbonate precipitation at the goethite surface is inhibited despite having solutions that are supersaturated at higher pH.

Figure 7 compares measured and predicted strontium sorption to goethite in suspensions containing dissolved carbonate, 0.1 M NaNO₃ and total Sr ranging from 10^-6 to 10^-4 M. In this figure we show the total amount of strontium sorbed assuming an uncertainty of ± 0.3 log K for the strontium sorption constants (Equations 4, 5, 10, 11). Dissolved carbonate concentrations were estimated from concentrations measured as a function of pH in the NaCl experiments because they were not measured in the NaNO₃ experiments. The surface complexation model is identical to the strontium carbonate surface complexation model described above with the exception that NO₃ ^- replaces Cl^- for the complexation of the background electrolyte with the surface (Table 9). All constants have been adjusted in accordance with the site-occupancy standard state 14 , where the total number of sites equals 16.4 nm^-2, the solid concentration equals 10 g L^-1 and the BET surface area of 27.7 m²g^-1.

Application of the strontium carbonate surface complexation model developed in 0.1 M NaCl goethite suspensions to experiments conducted in 0.1 M NaNO₃ goethite suspensions is a test of the site-occupancy standard state which allows sorption experiments with varying solid concentrations and specific surface areas to be compared. The solid concentration of the NaNO₃ experiments was 25% of the solid concentration of the NaCl experiments and the goethite was synthesized from nitrate starting materials resulting in goethite with a surface area that was 75% of the goethite surface area synthesized from chloride starting materials. The strontium carbonate model slightly underpredicts the strontium sorption in NaNO₃ goethite suspensions assuming an uncertainty of ± 0.3 log K suggested for surface complexation models 16 .

Sverjensky and colleagues have applied Born solvation and crystal-chemistry theory using a site-occupancy standard state to develop a predictive model for surface protonation, alkali, heavy metal, alkaline earth, and anion sorption 6 7 8 9 10 11 12 13 14 15 16 . In this section we augment and re-calibrate Sverjenksy's 16 model for the prediction of alkaline earth speciation sorbed on oxide surfaces by including the fitted strontium surface complexation reactions for goethite and amorphous silica from this study.

Application of Born solvation and crystal-chemistry theory to metal sorption assumes that the standard Gibbs free energy of sorption (ΔG ^θ _r,m) depends on contributions from ion solvation (ΔG ^θ _s,m), electrostatic interactions between the sorbing ion and the surface sites (ΔG ^θ _ai,m), and contributions specific to the sorbing ion (ΔG ^θ _ii,m):

ΔG ^θ _r,m = ΔG ^θ _s,m + ΔG ^θ _ai,m + ΔG ^θ _ii,m

such that a given surface complexation equilibrium constant (Log K ^θ _r,m) can be expressed as:

Log K ^θ _r,m = -ΔΩ_r,m/RT*(1/ε _s) - B_m(s/r_m) + log K"_ii,m

The first term on the right hand side of equation 13 accounts for ion solvation, where ΔΩ_r,m is the Born solvation coefficient for the r^th reaction containing the metal m and ε _s is the dielectric constant for the solid. The second term accounts for the repulsive interaction between the sorbing ion and near surface species, where s is the Pauling's bond strength of the metal-oxygen bonds in the bulk mineral, r_m is the distance the sorbing ion is repulsed by the underlying metal in the solid due to short-range electrostatic interactions, and B_m is a constant characteristic of the surface reaction. The final term represents interactions intrinsic to the sorbing ion as well as solvation contributions from the interfacial dielectric constant and the electrostatic attractive interactions. Equation 13 can be reduced to:

Log K ^θ _r,m = -ΔΩ_r,m/RT*(1/ε _s)+log K''_ii,m

for a given surface reaction if the repulsive interactions in the electrostatic term are minimal. Linear regressions of log K ^θ _r,m versus 1/ε _s yield a slope equal to -ΔΩ_r,m/RT and a y-intercept equal to log K"_ii,m (or B_m(s/r_m) + log K"_ii,m if repulsion interactions are important). This regression serves as a fundamental calibration for the predictive model, because it allows the equilibrium constant for a given surface reaction to be estimated for solids of varying dielectric constants.

In the absence of enough data to calibrate Equation 14, surface equilibrium constants can be calculated with ΔΩ_r,m and log K"_ii,m derived from two additional regressions (Equations 15 and 17). Unknown values of ΔΩ_r,m can be estimated from the absolute solvation coefficient of a given surface complex, Ω_{abs r}, and the effective electrostatic radius of the sorbing ion, R_e,m. The Ω_{abs r} is estimated from a regression of known ΔΩ_r,m versus R_e,m:

ΔΩ_r,m= η/R_e,m - Ω_{abs r}

where η = 166.027 kcal Å mol^-1 8 . R_e,m is a function of the hydrated radii, r_m,hydr, 55 and an empirical constant specific to the sorbed species, γ _m:

R_e,m = r_m,hydr + γ _m

In the regression analysis γ _m is a variable used to produce a theoretical slope equal to 1 for ΔΩ_r,m versus η/R_e,m. Similarly, unknown log K"_ii,m for a given surface complex can be estimated from a linear regression of known values of log K"_ii,m (the y-intercept in Equation 14 if repulsive interactions between the near surface species and sorbing ion are not important) versus the ion radius, r_x,m, because log K"_ii,m is assumed to be intrinsic to the sorbing ion and independent of differing solid properties. The resulting regression is:

Log K''_ii,m = slope * r_x,m + y-intercept

In this section we analyze the new strontium surface reactions for amorphous silica and goethite together with other alkaline earth surface reactions to re-calibrate Sverjensky's 16 predictive model for alkaline earth sorption. Strontium surface reactions on goethite and amorphous silica from this study are re-written as:

2>SOH + 2>SO^- + Sr²⁺ + OH^- = (>SOH)₂(>SO^-)₂_SrOH⁺ (tet_SrOH⁺)

2>SOH + 2>SO^- + Sr²⁺ = (>SOH)₂(>SO^-)₂_Sr²⁺ (tet_Sr²⁺)

>SO^- + Sr²⁺ + OH^- = >SO^-_SrOH⁺

>SOH + Sr²⁺ = >SOH...Sr²⁺

and the equilibrium constants used in the fits are converted from the 1 M standard state (Log K⁰) to the site-occupancy standard state (Log K ^θ ) using the following equations to be consistent with stoichiometry presented by Sverjensky 16 :

Log K ^θ _{tet_SrOH+} = Log K⁰ _{tet_SrOH+} + 2pH_zpc + ΔpK^o _n + log C_s ³(N_sA_s)⁴/(N^†A^†)² + 14

Log K ^θ _{tet_Sr2+} = Log K⁰ _{tet_Sr2+} + 2pH_zpc + ΔpK^o _n + log C_s ³(N_sA_s)⁴/(N^†A^†)²

Log K ^θ _{>SO-_SrOH+} = Log K⁰ _{>SO-_SrOH+} + pH_zpc + ΔpK^o _n + log (N_sA_s)/(N^†A^†) + 14

Log K ^θ _>SOH...Sr2+ = Log K⁰ _>SOH...Sr2+ + log (N_sA_s)/(N^†A^†)

We cannot evaluate the binuclear and strontium carbonate surface reactions with this model, because data are insufficient for calibration (Equations 6–11). Table 10 lists the surface equilibrium constants (Equation 18a-21a) from fits to Sr sorption to goethite and amorphous silica from this study and from fits for other alkaline earth cations 16 . The regression of log K ^θ _r,m versus 1/ε _s for the formation of Sr, Ca, Mg, and Ba surface reactions are linear with correlation coefficients of 0.938 ≤ R ≤ 0.997 with at least three data points (Figure 8, Table 10). We include regressions for Ca, Mg and Ba in Figure 8 because the slopes and y-intercepts used in the calibration of the model are slightly different than those reported in Sverjensky 16 and to provide the reader with snapshot of the data used to calibrate the model. Table 11 lists ΔΩ_r,m and log K"_ii,m derived for Sr and for Ca, Mg, and Ba from the regressions in Figure 8 (Equation 14), Ω_{abs r} and fitting parameter γ _i from regressions of ΔΩ_r,m and η/R_e,m in Figure 9 (Equation 15 and 16). Our calculations differ from Sverjensky 16 only in our use of alkaline earth hydration radii from Robinson and Stokes 55 for r_m,hydr. In some cases Sverjensky 16 used r_m,hydr as a fitting parameter to achieve reasonable linear regressions. We chose not do this because it introduces a second fitting parameter in addition to γ _i, which is adjusted to yield a theoretical slope equal to one in Equation 15. Additionally, we observe identical values of ΔΩ_r,m for the formation of (>SOH)₂(>SO^-)₂_CaOH⁺ and (>SOH)₂(>SO^-)₂_SrOH⁺ which is consistent with the two species having the same effective hydration radius. There is a greater difference in ΔΩ_r,m for the formation of (>SOH)₂(>SO^-)₂_M²⁺ and >SO-_MOH⁺ for Ca and Sr; however it is only slightly greater than an uncertainty of ± 1.5 ΔΩ_r,m 16 .

Figure 10 is a plot of the log K"_ii for (>SOH)₂(>SO^-)₂_MOH⁺, (>SOH)₂(>SO^-)₂_M²⁺ and >SO^-_MOH⁺ versus ion radii (Equation 17). The linear trends (R > 0.97) support the notion that repulsive interactions between alkaline earths and metals in the underlying substrate are minimal and the y-intercept in Figure 9 represents log K"_ii,m for all three surface species. Sverjensky 16 estimated the repulsive interactions for M²⁺ surface reactions, but assumed they were insignificant for surface reactions involving MOH⁺. Results from EXAFS analyses indicate that strontium sorbs mostly as a hydrated surface complex, which is consistent with a large separation between the sorbed cation and metal cations in the substrate and thus minimal repulsion.

Our strontium surface reaction model fits nicely within the constraints of Sverjensky's 16 larger alkaline earth sorption model based on the regression analysis (Figures 8, 9, 10). The good correlation between Log K ^θ _r,Sr versus 1/ε _s for the formation of >SO^-_SrOH⁺ and (>SOH)₂(>SO^-)₂_SrOH⁺ surface complexes suggests that equilibrium constants for these reactions can be predicted for many solids with varying dielectric constants. It is also possible to estimate equilibrium constants for (>SOH)₂(>SO^-)₂_Sr²⁺ for other solids even though the regression is based only on two solids, because γ-alumina (ε _s = 10.4) and amorphous silica (ε _s = 4.6) span a fairly wide range in dielectric constant. It is not possible to estimate the formation of >SOH...Sr for a wide range of solids, because data are available only for amorphous silica (I = 0.005 N NaCl). Estimates of ΔΩ_r,m for the formation of the diffuse layer species (>SOH...M²⁺) from ΔΩ_r,m and a theoretical slope equal to one did not reproduce the limited data for this reaction. This species appears to be a very important complex for the uptake of alkaline earths on amorphous silica (and perhaps quartz) at low ionic strength, but it doesn't appear to be important for other oxides and hydroxides based on the available data.

Figure 11 compares the difference between equilibrium constants for the formation of (>SOH)₂(>SO^-)₂_MOH⁺, (>SOH)₂(>SO^-)₂_M²⁺ and >SO^-_MOH⁺ fitted to sorption data and predicted directly by the regression of Log K ^θ _r,m vs 1/ε _s (Equation 14) and by substitution of estimated values of ΔΩ_r,m and log K"_ii,m into Equation 14 from regressions in Equations 15 and 17. We compare predicted constants from regression of Equation 14 if the surface reaction is calibrated with three or more different solids. On average the regression over predicts Δlog K ^θ _r,m by 0.1 ± 0.6 (1σ). Prediction of all fitted equilibrium constants using regressed values for ΔΩ_r,m and log K"_ii,m (Equation 15 and 17) substituted into Equation 14 yield an average log K ^θ = 0.2 ± 0.7 (1σ). The overall uncertainty of the predicted model appears to be about twice that of log K ^θ values fitted to sorption data (log K ^θ ± 0.3).