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J. Phys. Chem. C 2010, 114, 16408–16415
Green Rust Reduction of Chromium Part 2: Comparison of Heterogeneous and Homogeneous Chromate Reduction Matthew C. F. Wander*,† and Martin A. A. Schoonen‡ Department of Chemistry, Drexel UniVersity, Philadelphia, PennsylVania 19102, and Department of Geosciences, Stony Brook UniVersity, Stony Brook, New York 11794 ReceiVed: March 9, 2010; ReVised Manuscript ReceiVed: August 10, 2010
White and green rusts are the active chemical reagents of buried scrap iron pollutant remediation. In this work, a comparison of the initial electron-transfer step for the reduction of CrO4-2 by Fe2+(aq) and Fe(OH)2(s) is presented. Using hybrid density functional theory and Hartree-Fock cluster calculations for the aqueous reaction, the rate constant for the homogeneous reduction of chromium by ferrous iron was determined to be 5 × 10-2 M-1 s-1 for the initial electron transfer. Using a combination of Hartree-Fock slab and cluster calculations for the heterogeneous reaction, the initial electron transfer for the heterogeneous reduction of chromium by ferrous iron was determined to be 1 × 102 s-1. The difference in rates is driven by the respective free energies of reaction: 33.4 vs -653.2 kJ/mol. This computational result is apparently the opposite of what has been observed experimentally, but further analysis suggests that these results are fully convergent with experiment. The experimental heterogeneous rate is limited by surface passivation from slow intersheet electron transfer, while the aqueous reaction may be an autocatalytic heterogeneous reaction involving the iron oxyhydroxide product. As a result, it is possible to produce a clear model of the pollutant reduction reaction sequence for these two reactants. Introduction In our previous work, a single sheet of Fe(OH)2 was used as an analogue for green rust. Its calculated electronic and chemical properties were compared to experimentally determined properties of green rust.1 The structural deformations associated with oxidation were examined, along with their potential intramolecular electron transfers (ET). An initial oxidizing defect created a polaron consistent with both of the oxidized structures of green rust: chloride green rust (GR-Cl) and carbonate/sulfate green rust (GR-CO3 and GR-SO4).2 A localized electron hole allowed for the testing of cluster approaches in order to characterize the available internal electron-transfer reactions. Electron transfers within the sheet were calculated to be quite fast, but transfers between sheets were slow. With this basis for understanding, what is the reactivity of this material with a pollutant such as chromate? Chromate can oxidize green rust completely, producing a range of mineral assemblages. Wilkin et al. showed that scrap metal iron with a coating of green rust removed chromate completely by reducing it to a 3+ state.3 The solubility, now less than 1ug/L, was determined by precipitating Cr3+ as Cr(OH)3(ppt) or other hydroxide phases.3,4 Secondary iron-bearing minerals, particularly those containing ferrous iron, still participate directly in the reduction of chromate either by coating the metal or as an independent mineral phase.3,5,6 Early laboratory experiments on Fe(OH)2(ppt) and sulfate green rust (Fe(II)6Fe(III)2OH16SO4 · nH2O) showed rapid oxidation in the presence of chromate.7 Due to the immediacy of the reaction, rates were not measured for either of the two conditions explored.7 In the first case, the product was a 3:1, Fe:Cr, poorly * To whom correspondence should be addressed. E-mail: mcfwander@ gmail.com. † Drexel University. ‡ Stony Brook University.
crystalline δ-FeOOH. In the second case, the product was akin to a more poorly crystalline ferrihydrite solid solution of 4.56:1 Fe:Cr. In both cases, the reaction is so favorable that the initial hydroxide decomposes completely. Further work on sulfate green rust and chloride green rust confirmed the complete chromate reaction.8 In both cases, a ferrihydrite-like, solid solution was produced with a solubility approximating that of Cr(OH)3(ppt). While varying the ratio of the iron reactant to the concentration of chromate might produce different Fe products, including goethite and green rusts, the chromate phase was always a 4.5:1 ferrihydrite.8 However, with excess green rust, oxidation occurs slowly and consistently regardless of its interlayer anion.9,10 Williams and Scherer indicated a rate constant on the order of from 10-3 to 10-2 s-1.9 Aging did not alter this result, which was essentially zero order with respect to surface area.9 There was a small deviation from zero order in carbonate green rusts, which indicated the possibility of chromate slipping between the hydroxide sheets.11 However, the measured surface saturation of chromate, even at low concentrations, illustrates the importance of regeneration of electrons from the bulk.12 Chromate armoring, in the form of (Cr,Fe)(OH)3(ppt), appears to reduce the effectiveness of subsequent reductions of additional quantities of chromate.13 These rates were confirmed with chloride green rust, which is the most rapid reducer of chromate at ∼3 × 10-2 s-1. Sulfate green rust and carbonate green rust reduced chromate more slowly, ∼1 × 10-3 s-1.10 This study identified lepidochrosite and magnetite as the primary oxidation products of the reaction.10 Regardless, green rust appears to be many orders of magnitude faster than other potential reductants, such as magnetite.10 The reduced reactivity of green rust, which results from chromate armoring of the surface, has also been examined.14,15 These rates, on the order of from 10-4 to 10-2 s-1, are comparable to other reduction rates observed for green rust and
10.1021/jp1021328 2010 American Chemical Society Published on Web 09/08/2010
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chromate.14,15 In this experiment, a ferric oxyhydroxycarbonate is the armoring agent with a depth of oxidation of nearly 8.3 nm, which corresponds to a precipitation of 2.1 µM CrO4-2 per g of green rust. When there was an excess of surface sites relative to chromium reduction rates were strictly first order, but when that excess was eliminated a second-order mechanism, incorporating available surface sites, was observed.15 Kinetic work demonstrated that ferrous iron is significantly faster at reducing chromate than green rust. Measurements were made of the reaction ratio of chromate oxidation of ferrous iron vs the oxidation of ferrous iron by oxygen.16 The rate for the latter reaction varies over many orders of magnitude depending on pH and has the form d[Fe3+]/dt ) k[OH]2PO2[Fe2+] where k ranges from 1013 to 1016.17 When ferrous iron reacted with chromate or oxygen, the chromate reduction was favored below pH 10, except in phosphate-rich waters where chromate reduction was favored below pH 7.16 While phosphate is known to increase the rate of O2 oxidation, it may also inhibit chromate oxidation by ligating with the iron.18,19 Chromate reduction was still faster than it was possible to measure in all pH regimes.16 Direct measurements of the reaction rates of aqueous ferrous iron with chromate show quadratic behavior with changing pH and a minimum rate of 100-101 M-1 s-1 at pH 4-5.20,21 In alkaline conditions, the rate expression is second order in [OH-], indicating the effect of hydrolysis on ferrous iron oxidation.22 In summary, the reaction rate of aqueous ferrous iron could be as much as 10 orders of magnitude higher than the green rust rate. Marcus Theory As a Framework for Calculating ElectronTransfer Rates. Marcus theory has two different, but related, expressions for electron-transfer reactions, depending on whether the reaction is adiabatic (vibration limited) or diabatic (electronic overlap limited). For the adiabatic reaction, the overall rate expression for the electron-transfer rate constant is
(
kET ) νETe-
(λ + ∆GET)2 4λ
)
-VAB /kbT
where λ ) λI + λE
(1) and νET is the vibrational mode most closely associated with the electron transfer. For this study the longitudinal relaxation frequency of 1.85 × 1013 s-1 was used;23,24 λ is the electronic reorganization energy, which is composed of the sum of the intrinsic reorganization energy (λI) and the external or solvent reorganization energy (λE); ∆GET is the free energy associated with the reaction; VAB is the electronic overlap integral, which represents the stabilization at the crossing point between reactant and product states; kb is Boltzman’s constant; and T is temperature in Kelvin. In this case ∆GET ≈ ∆EET; because the number and type of bonds are not changing, the likelihood is that there would be negligible contributions from the zero-point energy and the thermal corrections to the free energy of this reaction.25 For diabatic electron transfers the expression for the rate constant is
kET )
2 2πVAB
p(4πkbT)0.5
e ( -
(λ + ∆GET)2 4λ
)
/kbT
(2)
where p is Plank’s constant and all other symbols have the same meaning as the adiabatic case. These two expressions represent the limits of large and small VAB of the general electron-transfer rate expression.26 In the diabatic case, VAB , 1 kJ/mol, which
Figure 1. Diagrammatic representation of the Marcus rate expression components. The solid line is the diabatic pathway, the dashed line is the adiabatic path, and the dotted line is the path for the extrapolation used when the charged reversed energy cannot be computed.
means that the barrier height (Ea) has the same expression as the adiabatic case. As a matter of utility, when comparing different rates, Ea is defined as
(
(λ + ∆GET)2 - VAB 4λ Ea ) kbT
)
(3)
where Ea is the barrier height for the reaction after all of the factors in the Marcus rate expression have been considered. There are two ways to calculate λI; these approaches are shown in Figure 1. If the product’s wave function can be evaluated at the reactants’ geometry or vice versa, then the difference between that energy and the corresponding minima is λI. This approach proved successful for the heterogeneous reaction. Otherwise, calculations must be made along the reaction coordinate and the appropriate energy values determined by extrapolation. This was how λI was determined for the aqueous reaction. To determine the contribution of the external solvent to the reorganization energy, the Marcus solvent formula was used27
(
λE ) e2
)(
1 1 1 1 1 + ε∞ ε0 2rD 2rA rDA
)
(4)
where e2 is the fundamental charge of an electron squared, ε0 is the static dielectric constant for water (78.39), ε∞ is the optical dielectric constant for water (1.7689),28 rD and rA are the radii of the atoms donating and accepting the electron, respectively, measured as the average of metal-oxygen bond lengths, and rDA is the distance between the donor and the acceptor centers. A quasi-diabatic method was used to calculate the electronic coupling matrix element VAB in conjunction with the wave function output of NWChem.29 At the crossing-point geometry, VAB was obtained by solving the secular equation representing the electronic overlap, which is represented as29 2 -1 2 VAB ) (1 - SAB ) |HAB - (HAA + HBB)SAB /2|
(5)
where SAB is the overlap density, HAA is the energy of the reactant state at the crossing point, HBB is the energy of the product state at the crossing point, and HAB is the energy of
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Figure 2. Meyer square for slab ET. The clusters shown here are for illustrative purposes only and were not used in any calculations.
overlap between the product and the reactant wave functions evaluated at the crossing point. In addition to the pure electron-transfer (ET) framework described above, an extension of the Marcus theory framework for proton-coupled electron-transfer (PCET) reactions is required.30 This extension requires separate steps for proton and electron transfers in order to determine if the proton transfers (PT) precede (catalyze), are in concert with (hydrogen-atom transfers), or are subsequent to (charge induced pKa change) the electron transfer. Like pure ET reactions, PCET reactions can be adiabatic or diabatic and use the same Marcus rate expressions. Therefore, the intrinsic electronic reorganization energy (λI), solvent contribution to the reorganization energy (λE), and the electronic overlap energy (VAB) are still required. The products and reactants differed by two proton transfers in the heterogeneous reaction, so the two possible nonprotoncoupled electron-transfer intermediates were optimized. These intermediates occur when the electron transferred without the accompanying protons or the protons transferred without the electron. The protons were moved in pairs, because the important question was whether the ET was coupled to the PT. This resulted in two additional intermediates, one where the protons transferred but the electron remained on the iron and another where the electron transferred but left the protons behind. All four optimizations were used to determine the possible reaction paths; see Figure 2 for a schematic. Unfortunately, in Crystal it is only possible to freeze atom positions not bond lengths. This limitation means there might be a small error in the interaction distance and bond angles of the hydrogens for the intermediates. To map the reaction coordinate between the reactants and products, linear synchronous transit (LST) was used.29 In internal, or z-matrix, coordinates, the reaction path to each internal coordinate was mapped as follows
xi ) (1 - ε)xRi + εxPi
(6)
where xi is the value of the z-matrix variable along the reaction path for the ith variable, i ranges from 1 to 3N - 6 (the number of internal coordinates), ε is the relative distance along the path and ranges from 0 to 1 in 0.1 increments, xRi is the ith z-matrix variable in the reactants geometry, and xPi is the ith z-matrix variable in the products geometry. There are many LST pathways connecting reactants and products, one for each z-matrix arrangement. Picking a z-matrix to match to those changes from the electron and proton transfers between reactants and products is important to ensure that the LST resembles the
true pathway. For example, metal-oxygen bond lengths are integral to the ET path, as are the oxygen-hydrogen bond lengths to the PT path, so both were included. The focus of this study is on characterizing the electron transfer between iron and chromium in homogeneous and heterogeneous settings. As with the previous paper on green rust,1 all rate constants were calculated within a Marcus theory framework. Equivalent methods have been used extensively for homogeneous aqueous reactions.31-34 However, a few small modifications were required in order to account for the effect of crystal boundaries and to overcome limitations in the various codes for the heterogeneous reaction. Since this is a pilot study, many important, but secondary, factors were not included such as solvent, either as explicit water molecules or as an implicit dielectric field. In future work, many of these factors can be added back into the approach, as computational limits will allow. Methods All optimizations and calculations for the aqueous reaction between Fe2+ and CrO42- were performed in NWchem.35 Calculations were performed in the gas phase using B3LYP with the same basis set as in Part 1,1 a 6-311++G(d,p) for H and O and the Ahlrichs TZV basis set for Fe augmented by Hay-Wadt diffuse and polarization functions.36-44 The same basis set was used for chromium as for iron.38,42 The optimizations started with an outer-sphere configuration, but a water molecule became displaced during the optimization as the chromate ligated to the iron. This water was discarded, and the optimization continued. The reactants were optimized at a spin s ) 5 (four unpaired alpha spin electrons), which corresponded to the combination of a high-spin Fe2+ and a closed-shell chromate. The products were initially optimized at a spin s ) 7 (six unpaired alpha spin electrons), but the final geometry was refined by further optimization at s ) 5. There were no significant differences in energies or geometries between calculations at s ) 7 and 5. However, since s ) 5 is the correct product state, all reported single-point energy calculations and geometries were performed at s ) 5. Resulting energies and ET rates were then compared against experimental thermodynamics and kinetics. Collision Frequency Calculation. Because of the charges on the iron and the chromate, it is necessary to correct the electrontransfer rate constant to account for changes in the collision frequency that results from these charges. Since the two ions are oppositely charged they will collide more frequently, which is not considered in eqs 1-6. This increase is approximated in this paper as 2 kpre ) 4πrDA e-∆GR /RT W
(7)
where kpre is the collision frequency, which is a multiplicative prefactor of the rate expression, rDA is the distance between the donor (D) and the acceptor (A), ∆GWR is the free energy associated with the work to bring two charged ions together, R is the ideal gas constant, and T is the temperature in Kelvin. ∆GWR has the form
∆GRW ) z1z2e2ε0εs(1 + 0.3281rDA√I)
(8)
where z1 and z2 are the charges on the respective ions, e2 is the fundamental charge squared, ε0 is the permittivity of vacuum, εs is the static dielectric constant for water, rDA is the distance
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TABLE 1: Chromium Basis Seta 24 6 2798.29400 423.137000 92.4388600 120.280600 26.0372700 7.84417200 12.5605383718 2.97947019945 1.00769501233 10.3191157823 2.54249054852 0.613655723487 a
0 0 3 2.0 1.0 0.638240000 × 10-1 0.377084000 0.680989000 0 1 3 8.0 1.0 -0.117779000 0.101431000 0.957198000 0 1 2 8.0 1.0 -1.026025208 -0.0778842558595 0 1 1 0.0 1.0 1.0 0 3 2 4.0 1.0 0.305429874639 1.02515655515 0 3 1 0.0 1.0 1.00000000
0.139878000 0.555983000 0.474818000 -0.0220931803195 0.575671989795 1.0
This basis set has been formatted for input into Crystal03.
between the donor (D) and the acceptor (A), and I is the ionic strength of the solution. The values are presented in Table 4. Heterogeneous Reaction. In the heterogeneous case, a single sheet of Fe(OH)2 was used as an analogue of green rust. For the Fe(OH)2 slab, optimizations were performed using Crystal with a unique basis set.45 The basis set for Fe(OH)2 was the same as that used in Part 1. This basis set was optimized from the Durand-21d41G pseudopotential basis set for Fe3+ developed by Catti et al. using the program LOptCG.46,47 This basis set was optimized in Fe(OH)2, along with the diffuse functions for O, in the Durand-41G basis set of Apra.48 The Cr basis set is an all-electron 3-21G basis set, optimized using LOptCG that was used for both Cr(V) and Cr(VI), see Table 1.49 Two configurations for chromate were examined as possible reactants. The first configuration had one oxygen atom pointing down toward the sheet and three pointing up. The second configuration had three oxygens pointing down, which were hydrogen bound to the surface, with one oxygen atom pointing up. The second configuration had a lower energy, and as a result, it was used as the reactant. Crystal was incapable of controlling the input of spins like NWchem, so all optimizations were treated as spin up, even though the transferring electron was a “beta” electron on the iron. For the product geometry at its minimum, the difference in energies between the two spin states on the chromium was insignificant. This insignificance is an important confirmation that coupling between electrons on the Fe and Cr is not present, which had it been present would have indicated that the structure is not a true minimum. LST and Slab Calculation. Performing a linear synchronous transit calculation on a slab system presents an additional complication. In addition to atom positions being relative to each other, they are also relative to the vectors that determine the periodic boundary conditions. The easiest answer was to define the vectors using dummy atoms. The first atom would be at (0,0) in fractional coordinates, with the other two at (1,0)
and (0,1), respectively. These atoms would be used to define internal coordinates for the first three atoms, and all subsequent atoms would be defined according to those three, as per a typical LST calculation. This system is overdetermined mathematically but fully consistent with the LST cluster approach. In this case, the LST allowed for the determination of the energy of the charge-reversed state: the products’ state at the geometry of the reactants. This is the most accurate way to determine λ for both the ET and the PCET paths, because it does not involve extrapolation. VAB Calculations on the Heterogeneous Slab. Since NWchem was required for all VAB calculations, a representative cluster for the heterogeneous reaction was required. The choice was a Fe7(OH)12(H2O)122+ cluster. This cluster provided a uniform boundary to the defect iron atom, as well as a consistent environment for the ligated OH groups involved in bridging the electron transfer. Because of the large value for the Fe-Cr distances in the optimized slab geometries, a large basis set is required for the VAB calculation. The same large, combined basis set was used as in the homogeneous reaction, and the spin was s ) 5. All results from the slab case, like the homogeneous aqueous reaction, were compared against experimental results. Results In both cases, pH played a significant role in understanding the results of our models. While these effects were not treated exactly, the selection of starting reactants predicated a pH range. In the case of the aqueous reaction the number of protons in the cluster constrains the reactant to between pH 6.5 and 8.5. At this pH, ∆Grxn is 2.4 kJ/mol and the experimental rate is between 102 and 104.20 The heterogeneous reactants were limited to more alkaline waters, pH > ∼8. Benchmarking. The calculations were benchmarked against existing experimental data in the form of standard Gibbs free energies for minerals and ions (see the Discussion section for that data). While those experimental values involve reactants/ products at infinite separations, they are not unrelated, and it is important that the calculated numbers be similar because the single largest error in calculating a rate constant is the error in the free energy of the reaction.50 In particular, there is an assumption when clusters are used to represent interfacial or solid state electron transfers that the cluster accurately represent the true periodic system. As can be observed in Table 2, the B3LYP cluster calculations are fundamentally different from the slab calculations, indicating that this assumption is violated. Furthermore, DFT slab calculations (also in Table 2) for the heterogeneous reaction resulted in unfavorable ET and PCET energies, which is not remotely consistent with experimental energies. Those values, if they had been used in this study, would have produced wildly inaccurate rates. This raises the more general possibility, that cluster representations of surface sites might not be valid in all cases. Overall, the HF slab calculations provided the best match to the experimental predictions of the ∆Grxn heterogeneous reaction. There is a small but significant set of the literature performed with very similar approaches, which supports the view that Hartree-Fock in
TABLE 2: Summary of Energies Used To Benchmark the Reaction calcuated energies/hartrees
reaction energies/kJmol
system representation
reactants
PT
ET
PCET
PT
ET
PCET
cluster-B3lyp slab-B3lyp slab-HF
-12 019.7 -1993.067 -1966.902
-12 019.6 -1992.952 -1966.820
-12 019.6 -1992.898 -1966.924
-12 020.3 -1993.035 -1967.151
258.0 303 213.9
142.0 444 -57.3
-1492.9 84 -653.2
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Figure 3. Fe-Cr aqueous reactants and products.
TABLE 3: Cluster Bond Lengths for Aqueous Reactants (Å)a iron
chromate
ligand
distance
W W W W OH O average (rD) Fe-Cr (rDA)
2.238 2.398 2.272 2.217 1.919 2.178 2.204 3.10197
ligand
distance
O O O OH
1.65221 1.63203 1.56932 1.75537
average (rA)
1.65223
a This table details the metal-oxygen and metal-metal distances for use in the Marcus solvent formula.
conjunction with periodic atomic orbitals provides a reasonable electronic description of the electron-transfer reaction.51-54 Finally, the choice of choosing to model both the aqueous and the interfacial electron-transfer reactions with minimal solvent deserves some justification. The addition of solvent may actually be harmful if variations in the solvation were to overwhelm the calculated reaction energies.55 The fact that we are able to match experimental energies of reaction indicates a fortuitous cancellation of errors, but that is something that is always the case with large electronic structure calculations and should not interfere with an approximate rate constant determination. The quality of the respective matches is the reason that HF slab calculations were used even though the aqueous, or homogeneous, rates were calculated with B3LYP. Aqueous Reaction. The optimization started with two outersphere reactants. One of the waters was dislodged, and a proton was transferred from the Fe to the Cr. A proton transfer for an outer-sphere reaction was unexpected, so it is likely the result of a chromate ligand altering iron’s proton affinity. It could be a gas-phase artifact, as differences in pKa between Fe2+ and chromate are small. The final rate expression would be rate ) k[Fe2+][CrO4-2]. Despite being unfavorable energetically, this reaction is quite fast; its adiabaticity is a result of close proximity and covalent bonding between donor and acceptor. The fact that the result is low relative to experimental values may result from overestimation in the Marcus solvent formula. In this case, the solvating radii for the iron and the chromate overlap significantly, and it is not clear if this case violates the mathematical
assumptions underlying this formula.56 Figure 3 shows the resulting geometries, and Table 3 contains the metal-oxygen and metal-metal bond lengths. Figure 4 shows the results of the energy calculations from both the optimizations and the LST calculations. Table 4 contains the resulting values for ∆Grxn, λI, λE, Ea, and k. See Table 4 for the correction to the collision frequency employed to account for the charges of the reactants. Heterogeneous Reaction. The favored product of the heterogeneous reaction was an H2CrO41- species. This species indicated that two protons may have been coupled to the electron transfer. As a result, in order to determine the intermediates that would define independent reaction paths it was necessary to separate the proton transfers from the transferring electron. Figure 5 shows the four geometries, two constrained and two unconstrained, showing the reactants, intermediates, and products of the slab reaction. Table 3 indicates the thermodynamics resulting from the slab calculations, and Table 6 lists the metal-oxygen bond lengths used to calculate the solvent reorganization energy. Unlike the aqueous case, both the pure ET and the PCET reactions are outer-sphere, weakly coupled diabatic reactions. To emphasize again, the cluster used to calculate VAB did not match the thermodynamics of the slab reaction either in magnitude or in direction of the driving force and was not used for any other calculations. Relevance to More Oxidized Forms. The question arises as to how relevant these results are to more oxidized ferrous hydroxide rusts, i.e., green rusts between 20% and 33% oxidation. The key difference in these rusts is the presence of an interlayer anion, which alters the thermodynamics of the ET and PCET products (see Figure 6 for a diagram with a carbonate anion). When considering the net charges of the slab-ion outersphere complex, the net charges for the green rust ET product would be -2, +1, -3 and electronic repulsion would stabilize the energy of the products by 100-200 kJ/mol. The products’ combination of charges is -2, -1, -2, then the energy of the PCET state should rise by a similar amount. It is possible to estimate the effect of this change using a semiempirical approach. A semiempirical approach exists, which uses the experimental thermodynamics of the reaction in conjunction with the calculated λI, λE, and VAB, to estimate an electron-transfer rate constant.57 As a result, it is possible to use the experimental data (∆Gorxn) to estimate the rate constant for the pure ET. The resulting rate, shown in Table 5, is identified as GR-ET and
TABLE 4: Summary of for ET Rates and Energies (kJ/mol)
a
rate
∆Gorxn
λIntrinsic
λExternal
VAB
Ea
ka
type
aqueous surface-PCET surface-ET GR-ET
33.4 -653.2 -72.3 -143.7
345.5 741.4 160.5 160.5
159.0 241.3 241.3 241.3
45.9 2.6×10-3 5.9×10-3 5.9×10-3
97.6 27.6 67.5 41.2
7.9 × 10-5 2.2 × 10+2 3.5 × 10-5 4.2 × 10+0
adiabatic diabatic diabatic diabatic
Units for the electron transfer are M-1 s-1 for the aqueous reaction and s-1 for the surface reaction.
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Figure 4. Linear synchronous transit energies for the aqueous reaction.
Figure 5. Four different slabs. The reactants are in the upper left and the products in the lower right. The pure ET product is in the upper right and the two proton-transfer intermediates in the lower right.
TABLE 5: Collision Frequency Correction z1
z2
R (Å)
ionic strength
dielectric constant
∆GWR (J/mol)
KPRE (M-1)
log KPRE (M-1)
rate, k (M s-1)
2
-2
3.10197
0.0
78.39
-22 855
5.9 × 102
2.8
4.7 × 10-2
indicates that even with a sizable driving force the overall rate is still limited by the diabatic nature of the reaction. The calculated value lies midway between the calculated pure ET rate and the calculated PCET rate, which is reasonable. However, because the intersheet transfer still limits the ET reaction to chromium, the overall observed rate is not likely to change. Discussion Comparing our calculated results to the thermochemistry of the reactions that are being modeled reveals some apparent discrepancies. Aqueous reactions have a substantially less favorable ∆Grxn58-61
TABLE 6: Slab Bond Lengths: Heterogeneous Reactants (Å)a Fe(OH)2
chromate
ligand
distance
OH OH OH OH OH OH average (rD) Fe-Cr (rDA)
2.23619 2.23376 2.23444 2.13755 2.13956 2.13962 2.18685 4.41949
a
ligand
distance
O O O O
1.60459 1.60286 1.60299 1.60187
average (rA)
1.60308
This table details the metal-oxygen and metal-metal distances used in the Marcus solvent formula.
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+2 -2 +3 -3 Fe(aq) + CrO4(aq) f Fe(aq) + CrO4(aq) ∆Grxn ) 2.4 kJ/mol (r1) +2 -1 +3 -3 + Fe(aq) + HCrO4(aq) f Fe(aq) + CrO4(aq) + H(aq) ∆Grxn ) 39.4 kJ/mol (r2)
than
1 -2 3Fe(OH)2(ppt) + CrO4(aq) + CO3(aq)-2 f 2 1 -3 Fe(II)4Fe(III)2OH12CO3 + CrO4(aq) 2 ∆Grxn ) -143 kJ/mol
(r3)
1 -2 3Fe(OH)2(ppt) + CrO4(aq) + SO4(aq)-2 f 2 1 -3 Fe(II)2Fe(III)1OH6SO4 + CrO4(aq) 2 ∆Grxn ) -143 kJ/mol
(r4)
-2 4Fe(OH)2(ppt) + CrO4(aq) + Cl- f -3 Fe(II)3Fe(III)1OH8Cl + CrO4(aq) ∆Grxn ) -126 kJ/mol
(r5)
Taken together, this would suggest that the green rust reaction is much faster than the aqueous reaction, the opposite of what is observed experimentally. Again, the predicted rates are slower than the experimental rates for the aqueous reaction and much faster than those for the green rust reaction, approximately in line with the free-energy values above (r1-r5). In both cases, the assumption is that the reduction of Cr(VI) to Cr(V) is the slow step in the overall reduction to Cr(III). After the initial reduction, Cr(V) could disproportionate as62
2Cr(V) f Cr(IV) + Cr(VI)
(r6)
2Cr(IV) f Cr(V) + Cr(III)
(r7)
Cr(IV) + Cr(V) f Cr(III) + Cr(VI)
(r8)
The alternative is that further reduction by iron could drive the chromium intermediates to Cr(III).63 Cr(III) and Cr(VI) are the most stable oxidation states in natural waters, which supports the disproportionation hypothesis.4 Precipitation subsequent to reduction could explain the variety of mineral products, indicat-
Figure 6. Image for showing the difference between low-oxidation (20% Fe3+) green rust (right). The interlayer anion with its negative charge stabilizes the pure ET product and raises the energy of the PCET product due to the proximity of the charges of the slab and ions.
ing that understanding the initial electron transfer is the key to the overall transformation. For the aqueous ferrous iron reaction, the results can be reconciled with the experiment by considering the ultimate product of these reactions. The primary products is Fe3+(aq), an ion with a large hydration energy.64 Despite that, the Fe2+/Fe3+ redox couple is in fact actually weak.65 The reason for this is that while Fe3+ is extremely stable in water it is even more stable as an iron oxyhydroxide. For the aqueous discussed here, the product is an impure ferric oxyhydroxide, either ferryhydrite or goethite. In the case of the heterogeneous reaction of white rust it is possible to take the oxidation up to the dissolution process, but the rates discussed here do not proceed that far.9 These minerals are capable of autocatalytic oxidation of ferrous iron by a variety of pollutants including chromate.16,66-71 Altering the product from aqueous ferric iron to a more stable ferric oxyhydroxide can drive the electron transfer of an adsorbed ferrous iron ion. Thus, while the initial oxidation of ferrous iron might be slower than measured, the experimental rate is not determined by the aqueous reaction but by the heterogeneous adsorbed species and its thermodynamically stabilized product. For the heterogeneous reaction of green rust, according to the calculations presented here, the presence of chromate would oxidize the top layer of a white or green rust coating on scrap metal iron quickly. However, the vast majority of the microcrystalline material’s mass is not on the surface. For further reaction to occur, an internal ET must move electrons from the bulk to the surface. In Part 1, the lower limit of the rate of an intersheet ET was estimated to be 1 × 10-8 s-1.1 This estimate is consistent with known estimates for green rust, considering that an interlayer anion or water molecule would mediate the transfer and accelerate the rate. As a result, passivation of the top surface could reasonably explain the experimental ET kinetics. Conclusions As measured experimentally, the reaction rate of aqueous ferrous iron with chromate is not likely to be the aqueous reaction modeled here but actually the result of an autocatalytic heterogeneous reaction. While the initial electron transfer would proceed as modeled, as soon as there was any sort of mineral product the autocatalytic mechanism would dominate. As a result, the value for the initial homogeneous ET, while slow, is still reasonably convergent with the experimental result. Oxidation of a single sheet of ferrous hydroxide provided a good match to the reaction of green rust and chromate. When the calculations for the green rust and the aqueous ferrous iron systems were compared, the heterogeneous initial ET behaved as predicted from the thermodynamics (r3-r5) but not from the experimental rate constant. The experimental rate of oxidation of Fe(OH)2 was likely a result of limited, intersheet transfer and passivation. Even though the calculations used only the Fe(OH)2 sheet, the effect of the interlayer anion was approximated using a semiempirical method to predict a pure ET rate. In this case, the experimental rate is still likely to be limited by the intersheet transfer. Acknowledgment. M.C.F.W. and M.A.A.S. are grateful for financial support from the National Science Foundation (NSF), Chemistry Division, through the Center for Environmental Molecular Science at Stony Brook, Award #CHE-0221934, and a supplement that allowed M.C.F.W. to visit the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL)
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