Reaction of hydroquinone with hematite

Journal of Colloid and Interface Science 274 (2004) 442–450 www.elsevier.com/locate/jcis

Reaction of hydroquinone with hematite II. Calculated electron-transfer rates and comparison to the reductive dissolution rate Andrew G. Stack,a,∗ Kevin M. Rosso,b Dayle M.A. Smith,c and Carrick M. Eggleston a a Department of Geology and Geophysics, University of Wyoming, Laramie, WY 82071-3006, USA b William R. Wiley Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, MSIN K8-96, Richland,

WA 99352, USA c Department of Physics, Whitman College, Walla Walla, WA 99362, USA

Received 29 August 2003; accepted 7 January 2004 Available online 7 February 2004

Abstract The rate of reaction of hematite with quinones and the quinone moieties of larger molecules may be an important factor in limiting the rate of reductive dissolution of hematite, especially by iron-reducing bacteria. It is possible that the rate of reductive dissolution of hematite in the presence of excess hydroquinone at pH 2.5 may be limited by the electron-transfer rate. Here, a reductive dissolution rate was measured and compared to electron-transfer rates calculated using Marcus theory. An experimental rate constant was measured at 9.5 × 10−6 s−1 and the reaction order with respect to the hematite concentration was found to be 1.1. Both the dissolution rate and the reaction order of hematite concentration compare well with previous measurements. Of the Marcus theory calculations, the inner-sphere part of the reorganization energy and the electronic coupling matrix element for hydroquinone self-exchange electron transfer are calculated using ab initio methods. The second order self-exchange rate constant was calculated to be 1.3 × 107 M−1 s−1 , which compares well with experimental measurements. Using previously published data calculated for hexaquairon(III)/(II), the calculated electron-transfer rate for the cross reaction with hydroquinone also compares well to experimental measurements. A hypothetical reductive dissolution rate is calculated using the first-order electron-transfer rate constant and the concentration of total adsorbed quinone. Three different models of the hematite surface are used as well as multiple estimates for the reduction potential, the surface charge, and the adsorption density of hydroquinone. No calculated dissolution rate is less than five orders of magnitude faster than the experimentally measured one.  2004 Elsevier Inc. All rights reserved. Keywords: Hydroquinone; Hematite; α-Fe2 O3 ; Biological electron transfer; Marcus theory

1. Introduction The rate of iron oxide reduction in soils and groundwaters may play an important role in a variety of geochemical processes, especially with regard to dissimilatory ironreducing bacteria. It has been shown that there may exist a correlation between iron oxide concentration and the rate of benzene and toluene degradation by microorganisms [1]. Thus, one of the (many) controls on the rate of contaminant degradation may be the rate of iron oxide reduction and sub* Corresponding author. Present address: Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USA. E-mail address: [email protected] (A.G. Stack).

0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.01.001

sequent dissolution. Quinone moieties have been shown to be produced by facultative dissimilatory iron-reducing bacteria (DIRB) [2,3] and used by obligate DIRB [4] as an intermediate between a cell surface and an iron oxide phase. Understanding the rate of reaction between hydroquinone (as an analogue for microbial electron-transfer agents) and hematite, particularly the rate-determining step(s) may contribute to an understanding of the mechanism of biologically stimulated dissolution. Electron transfer may be limiting the rate of reductive dissolution of hematite by hydroquinone [5], but to our knowledge this hypothesis has not been tested. In the first paper of this two-part series on the reaction of hydroquinone with hematite, we investigated the adsorption behavior of hydroquinone on hematite

A.G. Stack et al. / Journal of Colloid and Interface Science 274 (2004) 442–450

basal surfaces using electrochemical scanning tunneling microscopy (EC-STM), cyclic voltammetry, and X-ray photoelectron spectroscopy (XPS). In this second paper, we focus on the reaction rate and the possibility that electron transfer is the rate-limiting step. The oxidation of hydroquinone to p-benzoquinone occurs in four steps: two electron transfers and two proton losses (see, e.g., Fig. 2 in [6]). There are four likely reaction pathways that pass through semi-quinone intermediate forms at this pH [7]. From electrochemical studies on dropping mercury electrodes and quinhydrone electrodes (a 1:1 crystalline mixture of hydroquinone and benzoquinone), the oxidation pathway has been found to be eHeH below pH 5 (i.e., an electron transfer followed by a proton loss to form semiquinone, then repeated to form benzoquinone) and HeHe in the pH range 5–10. The electron-transfer steps are suggested to be of similar magnitude and to be ratelimiting [8,9]. While the exact pH of the transition from one pathway to another may vary with electrode material, we have assumed that the general pH dependence of the oxidation pathway is valid for an iron oxide. Since reductive dissolution of hematite occurs primarily at low pH [5], we are interested in the eHeH pathway. At pH 7, it has been shown that hydroquinone adsorbs onto iron oxides through a bridging oxygen [10] (Fig. 1a in [6]). Due to the nature of the sorption, the hydroquinone must lose a proton prior to electron transfer. At low pH, the oxidation pathway prohibits this mechanism because electron transfer occurs prior to proton loss. It is argued in the first paper in this series [6] that an outer-sphere adsorption occurs. Additional evidence for an outer-sphere complex comes from Abdul Rahman et al. [11] in the form of energydispersive X-ray absorption spectroscopy fine structure; they find that, at pH 1.9, no inner-sphere adduct forms as an intermediate between hexaquairon(III) and hydroquinone in aqueous solution. Therefore, in this paper we assume that all electron transfers between hexaquairon(III) and hematite surfaces are outer-sphere. Since the electron transfer is outer-sphere, Marcus theory is used to calculate an electron-transfer rate between an iron surface and a hydroquinone using the cross-relation and self-exchange rates of each of the reactants. Once the electron-transfer rate is calculated, a hypothetical dissolution rate is then estimated and compared to the actual dissolution rate. A brief review of the Marcus theory and the implementation used here is given in Appendix A, but the specific method has been developed previously and is explained in Refs. [12–14].

2. Methods 2.1. Wet chemistry Batch experiments were performed with 18 M cm water (Milli-Q+, Millipore Corp.). Reagent grade hydroquinone

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and benzoquinone (Aldrich) were used without modification. All experiments were performed in anaerobic environments (<1 ppm O2 , Forma Scientific Model 1025 Anaerobic System Glove Box; Coy Laboratories Inc. Gas Analyzer). Because hydroquinone can be oxidized by molecular oxygen [5], all DDI water used in this research was purged with nitrogen gas for at least 30 min prior to use and all experiments were performed in a nitrogen atmosphere. Pyrex glass beakers with neoprene rubber stoppers were washed in a dilute HNO3 solution. An overhead propellor-type stirrer was used. Commercial hematite (SPEX Industries, less than 7 ppm impurities) with a BET surface area of 1.53 m2 /g (S. Samson, personal communication) was used. Solution samples were drawn into a syringe and filtered using a 0.2-µm filter. Reverse-phase high-performance liquid chromatography (HPLC) analyses were performed for benzoquinone (Waters Corp. Nova-Pak-C18 column, Model 978 Photo Diode Array Detector). A 10% methanol, 90% water eluent was found to be the optimum for sufficient peak separation. Sample volume was 20 µl. Maximum absorbance was 246 nm. Peak integrations were done with a flat-baseline method. In most cases the default integration routine used by the software was sufficient, but occasionally a manually determined baseline was used. A five-point calibration was performed with concentrations from 0 to 10−3 M. Dissolved Fe was determined using a ferrozine assay [15]. Reagent was created by adding 1 ml of 10−2 M ferrozine (reagent grade, Eastman–Kodak) in a buffer solution of ammonium acetate (APHA, LabChem Inc.). Initially, some samples were exposed to 1 ml 10% w/v hydroxylamine hydrochloride reductant (APHA, LabChem Inc.), while others had only water added. No difference in dissolved iron concentration was observed, indicating that all dissolved iron was Fe(II), within error. Maximum absorbance of the iron–ferrozine complex was at 562 nm. A 15-point calibration was performed with Fe(II) concentrations from 0 M Fe(II) to 9 × 10−5 M. 2.2. Ab initio calculations Gas-phase energy calculations were performed using Gaussian98 [16] with the B3LYP method, which is a hybrid density functional theory/Hartree–Fock method) [17,18] with a 6-311G(d,p) basis set [19,20]. This methodology has been used before for reorganization energy calculations (see, e.g., [13]) and provides accurate relative energies while retaining economy of computation time [21]. To verify that the energy-minimized molecules are realistic, comparisons of bond lengths and angles between solid-phase hydroquinone and the energy-minimized hydroquinone were made. HAB estimates for the hydroquinone self-exchange electron-transfer reaction were calculated at the unrestricted Hartree–Fock (UHF) level with a 3-21 + G basis set and NWChem [22] by two of us (see [14,23] and references therein). Briefly, the calculation consists of evaluating the

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overlap between the wavefunctions of the reactant (i.e., + + QH+ 2 + QH2 ) and product (i.e., QH2 + QH2 ) electronic states at a transition state configuration known as the crossing point. Because nuclear displacements accompanying electron transfer in this system are small, the transition state configuration is well approximated using a linearized reaction coordinate [22]. The orientations of the two hydroquinone molecules in the encounter complex were based on orientations found in α-hydroquinone.

3. Results and discussion 3.1. Experimental dissolution rate Results from the reductive dissolution of hematite are shown in Fig. 1a, performed in excess hydroquinone (1 mM), 0.01 M NaCl background electrolyte at pH 2.5 HCl. The equivalents of dissolved Fe and benzoquinone produced as a function of time are initially very close, but after 4 h the equivalents of benzoquinone lag behind expectation. However, the uncertainty in our HPLC measurements is quite large and it is possible that during the experiment the calibration of the HPLC drifted. The appearance of benzoquinone is roughly half that of the appearance of dissolved Fe, indicating that the following reaction stoichiometry is preserved [5,7]: α − Fe2 O3(s) + QH2 + 4H+ → 2Fe2+ (aq) + Q + 3H2 O.

(1)

The exact stoichiometry of water and protons is not known (without detailed knowledge of the surface site composition), but in acid solution it is reasonable to write the reaction as consuming protons. When the concentration of hematite is varied (data not shown), a rate constant can be calculated with the expression (see, e.g., [5]) d[Fe(II)]/dt = kS , n

(2)

where k is the rate constant, S is the surface area (we assume surface area is proportional to total hematite concentration, [Fe2 O3 ]total, and n is the reaction order with respect to hematite. The rate constant measured from Fig. 1b was 5.75 × 10−4 h−1 (much faster than nonreductive dissolution [24]), and the reaction order with respect to hematite surface area was found to be 1.11 ± 0.19. In calculating d[Fe(II)/dt only the linear portions of dissolution data were used (the line in Fig. 1a). A studentized t-statistic yields a 63% probability that the reaction order is 1, the expected value for a reaction in which the reaction rate is proportional to the surface area. This is in slight contrast to the reductive dissolution of goethite, where the reaction order was found to be 0.8 [5], but in agreement with reductive dissolution of hematite by 2,6-anthroquinone 9,10-disulfonate (AH2 DS) [25]. If n = 1 is applied to LaKind and Stone’s [5] dissolution data for hematite, the calculated rate constant is approximately 25% greater than what is measured here,

Fig. 1. Wet-chemical data, pH 2.5 acetic acid, 0.01 M NaCl. (a) Equivalents of benzoquinone and dissolved iron produced as a function of time with 3.77 m2 /L of hematite. The linear portion of the dissolved iron data was used to calculate d[Fe]/dt, as indicated by the fit line. (b) Log–log plot of concentration of dissolved iron versus initial hematite concentration. The slope is the log of the reaction order (n) with respect to hematite and the y-intercept (k) is the log of the rate constant (see Eq. (2)).

indicating a fairly close match, given experimental and analytical uncertainties. 3.2. Hydroquinone reorganization and interaction energies calculations Calculated versus experimental average bond lengths and selected angles are shown in Table 1 for hydroquinone and benzoquinone. All C–O and C–C bond lengths are within 1%, whereas C–H and O–H bonds, where available, are overestimated by 15%. All bond angles are within 5%. The inconsistency between calculated and experimental bond

A.G. Stack et al. / Journal of Colloid and Interface Science 274 (2004) 442–450

Table 1 Calculated versus experimental bond lengths and angles Calculated C–C (Å) C–H (Å) C–O (Å) O–H (Å) C–C–O  C–C–C  C–C–H  C–C (Å) C=C (Å) C–H (Å) C=O (Å) C–C–O  C–C–C  C–C–H 

(◦ ) (◦ ) (◦ )

(◦ ) (◦ ) (◦ )

Hydroquinone 1.381 1.071 1.381 0.964 122.9 120.0 119.0 Benzoquinone 1.483 1.319 1.070 1.214 121.9 121.9 122.3

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Table 3 Marcus theory parameters Experimentala 1.381 0.945 1.377 0.842 117.4 120.3 120.2 1.477 1.322 Not reported 1.222 121.1 121.1 Not reported

a QH data from [42], Q data from [43]. 2

Table 2 Calculated inner-sphere reorganization energiesa Couple QH2 /QH+ 2 QH− /QH + QH/QH Q− /Q

λIS (eV) 0.516 0.292 0.487 0.531

a For pathway, see Fig. 2 of [6].

lengths involving protons is not expected to affect the reorganization energy calculations significantly. Inner-sphere reorganization energies for each electrontransfer step are shown in Table 2, with an average value of ∼0.46 eV. Oxidation reactions calculated for 2,6-anthroquinone disulfonate show an average value of ∼0.4 eV [12]. The difference between the two is entirely consistent with similar calculations for benzene versus anthracene [25] and the notion that increased π -electron density increases delocalization, and hence decreases the change in bond length with oxidation or reduction and reduces the λIS . The calculation of the HAB and λOS of hydroquinone self-exchange are listed in the first column of Table 3 for a hydroquinone self-exchange reaction (i.e., QH2 /• QH+ 2 , see, e.g., Fig. 2 of [6]). Axial radii used for the λOS calculations were calculated using center-to-center bond distances from the hydroquinone molecule optimized at the density functional theory level, plus the van der Waals radii for its constituent atoms (carbon = 1.77 Å, oxygen = 1.52 Å, and hydrogen = 1.17 Å [26,27]). Axial radii used were a = 4.29 Å, b = 3.64 Å and c = 1.77 Å. 3.3. Hydroquinone self-exchange and homogeneous reaction with hexaquairon(III) Using the parameters in Table 1 for hydroquinone, the calculated first-order electron-transfer rate constant (Eq. (A.4) in Appendix A) for the QH2 /QH+ 2 couple treated

3+ 2+ b QH2 /QH+ Fe2 O3 surfacec 2 Fe(aq) /Fe(aq) λIS (eV) 0.516 0.611 0.474 0.818 1.502 1.686 λOS (eV) 0.027 0.007 0.034 HAB (eV) R (Å) 5.095 5.32 3.55 0.369 0.771 0.37a E0 (VSHE ) a b c d

Fe2 O3 bulkd 1.03 0.17 0.20 2.99 0.37a

Corrected for pH [32]. [13]. [33]. [14].

Table 4 Rate constants for hydroquinone self-exchange and reaction with hexaquairon(III) Reaction

ket (exp.) (M−1 s−1 )

ket (calc.) (M−1 s−1 )

log(calc./exp.)

+ QH2 + QH+ 2 → QH2 + QH2 3+ QH2 + Fe (H2 O)6 2+ → QH+ 2 + Fe (H2 O)6

6.2 × 107a

1.3 × 107

0.7

0.52b

2.5

0.7

a [26]. b [28].

adiabatically using the 5.095-Å encounter complex model, we arrive at a self-exchange rate constant (ket ) of 8.2 × 107 s−1 . Since one of the hydroquinone molecules in the electron transfer pair is uncharged, there is no electrostatic work and Kpre is 0.2 M−1 . The second-order rate constant, kobs , is thus 1.3 × 107 M−1 s−1 (roughly three orders of magnitude slower than diffusion; see, e.g., [28]). This value is in good agreement with the published experimental value for benzoquinone self-exchange at pH 7, 6.2 × 107 M−1 s−1 [31] and a calculated self-exchange rate of benzoquinone (2.4 × 107 M−1 s−1 ) [30]. For comparison, the calculated self-exchange rate for the equivalent step in the AH2 DS redox pathway (AH2 DS−2 /AH2 DS−1 ), kobs , is 4.9 × 106 M−1 s−1 [12]. Rosso and Rustad [13] have calculated reorganization energies for aqueous ferric and ferrous iron and compared the corresponding rate constants to experimental data. The results of the calculation of the cross relation (Eqs. (A.14) and (A.15) of Appendix A with hydroquinone using their data are shown in Table 4. The estimation of the free energy of reaction for the cross-relation is somewhat difficult. Due to uncertainties that often render the values useless, the reduction potentials for the one-electron oxidation steps using ab initio-methods were not calculated. Rather, an experimental estimate of E 0 = +0.369 VSHE [9] is used for the hydroquinone half-cell reaction, along with E 0 = +0.771 VSHE for the iron half-cell reaction. Using these two standard reduction potentials and an electrostatic work term (Ws ) of 4.3 kJ/mol, the cross-relation first-order rate constant is 14.8 s−1 and the second-order rate constant is 2.5 M−1 s−1 . This compares favorably with the experimental value of Baxendale et al. [32], who measured a second-order rate

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constant of 0.52 M−1 s−1 , less than an order of magnitude smaller than the rate constant calculated here. From the results of the calculations of the self-exchange rate of hydroquinone as well as the reaction with hexaquairon(III), we conclude that our parameters for hydroquinone electron transfer are fairly accurate (i.e., the error is less than an order of magnitude).

Stack et al. [6] measured an adsorption density of ∼1.1 quinones/nm2 with EC-STM. While EC-STM is not reliable as an exact measurement, the Stack et al. result is the same as measured on palladium electrodes with the same technique [38]. Therefore, 1.1 will be used as the “best” estimate for the total quinone adsorption density, and 0.2 QH2 /nm2 is used as a secondary estimate. If both electron transfers are of equal rate and are ratelimiting, the predicted dissolution rate is written as

3.4. Estimation of dissolution rate of hematite

ket d[Fe] ∼ k1 k2 [Qads ], [Qads ] = = dt k1 + k2 2

The next step is to calculate the cross-relation electrontransfer rate constant between hydroquinone and a hematite surface. Without detailed knowledge of the specific surface structure of the hematite a composition and structure must be assumed. Here, the parameters for an A-type site from an iron-terminated surface from Eggleston et al. [33] are used: a tetrahedral iron site, with three bonds to the surface and one adsorbed water (zero net formal charge in the oxidized state). In addition to the surface site model, two extreme cases (in terms of electron-transfer rate) are examined: the aqueous hexaquairon(III) parameters used above for the homogeneous case and a calculation done purely on the bulk hematite (from [14]). The relevant Marcus theory parameters used are listed in Table 3. A further problem is finding a one-electron reduction potential for hematite. The standard reduction potential (E 0 = 1.05 VSHE ) is not suitable because it includes the free energy for dissolution as well as electron transfer. Since hematite is usually an n-type semiconductor and because the bottom of the conduction band is mostly Fe3d in character (see, e.g., [34]), the Fermi level measurement (EF ) of 0.37 VSHE from Quinn et al. [35] (single crystal, corrected for pH 2.5) is used as a lower bound and 0.53 VSHE [36] as an upper bound (polycrystalline, corrected for pH 2.5). The Kpre of the homogeneous case in Eq. (A.5) of Appendix A is unrealistic to use in the heterogeneous case, so the adsorption density of hydroquinone on hematite is used instead. However, experimental data on this system are lacking. McBride and Wesselink [37] measured an adsorption density on boehmite (AlOOH); a Langmuirian fit to their data yields an adsorption density of ∼0.2 QH2 /nm2 .

(3)

where k1 and k2 indicate the first and second electron transfers and [Qads ] is the total concentration of adsorbed quinones (this value includes the concentration of hematite involved in the reaction, since that determines the total surface area of adsorbed quinone). The assumption that the electron transfers are of equal rate is consistent with experimental evidence [9] and the calculated inner-sphere reorganization energies of the present study in Table 2 (for the low-pH pathway only). It is further assumed that the total sorbed quinone is mostly hydroquinone and semiquinone (i.e., concentrations of other intermediates are presumably insignificant because they react quickly). At pH 2.5, with 1.1 [Qads ]/nm2, 3.77 m2 /L hematite, and a Fe-termination A-type iron site for the hematite, a predicted dissolution rate if electron transfer is limiting is 13.3 M s−1 , which is 8.1 orders of magnitude larger from the empirically derived result for the same conditions (9.7 × 10−8 M s−1 ; Table 5). Using the bulk hematite data gives a faster rate, 6.6 × 104 M s−1 (11.8 orders of magnitude too fast), and using the HAB and λ characteristics of hexaquairon, but with the reduction potential of hematite and a charge neutral surface site, gives a rate of 2.9 M s−1 (7.5 orders of magnitude too fast). Obviously, none of the surface models predicts the rate of dissolution using the “best” estimates. However, it is necessary to examine the secondary estimates of some of the parameters prior to making any conclusions about electron transfer as a rate-limiting step. Since the homogeneous electron exchange rate constant between hydroquinone and hexaquairon(III) was predicted

Table 5 Experimental and predicted reductive dissolution rates for hematite by hydroquinone Dissolution rates

QH2ads (nm2 )

EF (Fe2 O3 ) (VSHE )

d[Fe]/dt (M s−1 )a

log(calc./exp.)

9.7 × 10−8

Experimental Fe-terminated Fe-terminated Fe-terminated Fe-terminated Bulk Aqueous Aqueous

Z

1.1 1.1 0.2 1.1 1.1 1.1 0.2

0.37 0.37 0.37 0.53 0.37 0.37 0.53

a pH 2.5, 0.01 M NaCl, 1 µM QH , 1.5 × 10−2 M Fe O . 2 2 3

Calculated λ and HAB model 0 13.3 +1 6.5 0 2.4 0 5.4 × 10−1 0 6.6 × 104 0 2.9 +1 6.1 × 10−3

k (s−1 ) 9.5 × 10−6

8.1 7.8 7.4 6.8 11.8 7.5 4.8

3.9 × 106 1.9 × 106 3.9 × 106 1.6 × 105 1.9 × 1010 8.5 × 105 9.7 × 103

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by our model to within an order of magnitude, it is not likely that there is a large error in the way hydroquinone is treated (i.e., λ and HAB for the self-exchange reaction). The hematite surface structure can have an affect not only on λ and HAB , but on the electrostatic work to bring the reaction forward (Eq. (A.16)). For the bulk and iron-terminated surface models it is assumed that the formal charge on the oxidized, unreacted site is neutral (this is what was calculated in [33]). Numerous proton affinity measurements have shown that hematite surfaces have a net positive charge at low pH (the pH of the point of zero charge of hematite is usually 8–9; see, e.g., [39]). Furthermore, since estimates of the site density reactive to tritium exchange can be as high as 22 sites/nm2 [40], it may be better to use a +1 charge for an unreacted site because this might be more reflective of the “average” site. Unfortunately, we do not have the data to parameterize λ or HAB for such a site precisely, but these parameters are not expected to change substantially. Taking into account the difference in electrostatic work alone gives only 0.3 orders of magnitude decrease in the predicted rate. Clearly if there is an error, it is not due to modeling the formal charge on the surface site properly. Similarly, decreasing the adsorption density to the smaller estimate, 0.2 QH2 /nm2 , does not produce a predicted rate close to the experimental one. With still 7.4 orders of magnitude difference in rates, the adsorption density is probably not the primary source of error either. A third source of error is the reduction potential of the hematite. Using a larger estimate of the reduction potential (0.53 VSHE ) gives an approximately 1.3 orders of magnitude decrease in the predicted rate, but still far too fast to be rate limiting. If we use all the secondary estimates together, the surface site is treated as having the λ and HAB of a homogeneous hexaquairon(III), a +1 formal charge (Z) on the unreacted species, and a hydroquinone adsorption density of 0.2 [Qads ]/nm2 all at once, the rate comes down to 6.1 × 10−3 M s−1 (4.8 orders of magnitude error). Since this is still too fast, we can conclude that: as written, electron transfer is probably not rate-limiting, but clearly we need verification that the model of the interface is accurate (i.e., adsorption behavior of hydroquinone and structure of the surface sites) before any unequivocal conclusion is made. Alternately, if this calculation is correct a discussion of the implications is warranted. As discussed above and in Stack et al. [6], the exact mechanism of the reaction between hydroquinone and hematite is very complex. The overall reaction can be generalized as adsorption of hydroquinone to an iron-surface site, electron transfer (forming a semiquinone radical and a reduced iron), desorption of the semiquinone, and dissolution of the reduced iron. This process is repeated by the semiquinone, generating a benzoquinone and another dissolved ferrous iron. If electron transfer is not the rate-limiting step the possibilities remaining are either adsorption of hydroquinone or semiquinone, desorption of benzoquinone and semiquinone, or dissolution of ferrous iron. Previous studies of reductive dissolution (by hydro-

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quinone) of manganese oxides and cobalt oxyhydroxides indicated that reduced metal detachment could be limiting, especially at high pH [41], because of a lag between appearance of dissolved reduced metal and the appearance of benzoquinone. A study of goethite at low pH [5] did not find this behavior, indicating that ferrous iron detachment is probably not rate-limiting. Our results suggest not only the latter, that reduced metal detachment is probably not rate-limiting, but also that benzoquinone desorption could be limiting. In Fig. 1a, equivalents of benzoquinone produced lag behind dissolved reduced iron concentration, indicating that ferrous iron detachment is quicker than benzoquinone detachment. Additional evidence for this hypothesis is from Stack et al. [6] who did not detect reduced iron on the hematite surface using XPS. Since benzoquinone was detected, this would suggest that release of benzoquinone is slower than release of ferrous iron. However, given the uncertainty, it is clear that more work needs to be done to resolve or explain these findings.

4. Summary The hypothesis that electron transfer is the rate-limiting step in the reaction of hematite and hydroquinone has been tested by calculating electron-transfer rates and comparing them to reductive dissolution data. The self-exchange electron-transfer rate of hydroquinone was calculated and compares well with experimental estimates. The predicted homogeneous cross-reaction rate between hydroquinone and aqueous ferric iron also compares well to experimental estimates. However, when the cross-relation is calculated for a hematite surface and a hypothetical dissolution rate derived, it overestimates the experimental dissolution rate by many orders of magnitude. Three different models for the characteristics of the hematite surface site were used, as well two estimates each of the Fermi level, adsorption density, and surface charge. No alternate model predicts the dissolution rate closer than within five orders of magnitude difference between the hypothetical rate and the experimental one. The implication is that electron transfer is not rate-limiting, as least in the way it has been modeled here.

Acknowledgments This research was supported by NSF Career Grant EAR9875830 to CME and DOE-PNNL EMSL User Grant 2554. A portion of this research was performed at the W.R. Wiley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the U.S. Department of Energy’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated for the DOE by Battelle Memorial Institute under Contract DE-AC0676RLO 1830.

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Appendix A A.1. The Marcus theory, outer-sphere electron transfer Marcus [44] presented an electron-transfer theory that relates, among other things, the electron-transfer rate constant to a reorganization energy which is a function of both nuclear rearrangements in the reactants and polarization of the surrounding solvent molecules. This theory has been developed further to consider semiconductor–liquid interfaces [45] and biological systems such as organic, π -electron system molecules such as benzene and anthracene (see, e.g., [46–48]). The theory deals primarily with outer-sphere electron transfer: it is described as a series of bond-preserving reaction steps consisting of the reactants coming together to form a precursor complex, electron transfer within the precursor complex to form the successor complex, and dissociation of the successor complex, Kpre

O + R ←→ [O · · · R], ket

(A.1)

[O · · · R] ←→ [R · · · O],

(A.2)

[R · · · O] → R + O,

(A.3)

where O and R stand for the reduced and oxidized molecules (O becomes reduced and R becomes oxidized). For a self-exchange reaction, O is an oxidized form of the molecule and R is the reduced form. When the electron-transfer rate is small relative to diffusion in homogeneous phase reactions, the second-order rate constant for reactions 1 and 2 is described as the product of the equilibrium constant for encounter complex formation (Kpre ) and the first-order electron-transfer rate (ket ), kobs = Kpre ket ,

(A.4)

when Kpre is calculated for a aqueous encounter complex and kobs has units of M−1 s−1 and is directly comparable to a self-exchange rate such as measured in a pulse radiolysis experiment (e.g., [29]). In the homogeneous-phase case the Kpre can be described as Kpre = 4πNA R 2 dR exp(−w/RT ),

(A.5)

where NA is Avogadro’s number, R is the distance of closest approach of the reactants, dR is the effective reaction zone thickness (1/1.2 Å−1 , the reciprocal of an electron transfer decay parameter, is commonly used) and w is the electrostatic work term [49], w=

(1/4πε0)ZO ZR e2 NA , √ εs Rm (1 + BRÅ µ)

(A.6)

where ZO and ZR are the charges on the reactants, Rm and RÅ are the closest approach distance in meters and angstroms, respectively, B is a Debye–Hückel parameter (0.328 when R is in angstroms) [50], and µ is the ionic

strength. When one or both of the reactants are uncharged the work term is zero. The rate constant for electron transfer (ket ) is formulated in part in terms of the frequency with which electronic states of the reactants and products are made equal by thermal distortion of nuclear configurations of the encounter complex. For small distortions, the potential energy surfaces of the product and reactant states can be approximated as parabolic energy surfaces with respect to a nuclear configuration coordinate. Electron transfer occurs at the crossing-point between the parabolas, when the reactants and products states have the same energy. The electron-transfer rate constant is described as
 ket = νκ exp (−G∗ )/RT , (A.7) where ν is the frequency with which an electron can move back and forth between reactants (approximately 1013 s−1 ), κ is the transmission coefficient (the probability of electron transfer when the transition state is achieved), and G∗ is the Gibbs free energy of activation. In general, electron-transfer reactions are classified as either adiabatic or nonadiabatic. The difference between the two types is the degree of electronic interaction (or molecular orbital overlap) between the electron donor and acceptor. The interaction is quantified by the electronic coupling matrix element, HAB . Strong electronic interaction (large HAB ) couples the reactant and product electronic states near the crossing-point configuration and leads to the formation of two new adiabatic potential energy surfaces. The system then evolves on the new lower surface. The treatment of this type of system is termed adiabatic and it assumes by convention that every incidence of the reactants reaching the crossing-point nuclear configuration leads to the creation of products. When the electronic interaction is weak (small HAB ), the reaction is treated nonadiabatically and the probability of electron transfer at the transition state is much lower than unity. As an approximate cutoff, when HAB is greater than 0.026 eV (= kT at 298 K), the electron transfer will be considered adiabatic here [23,29], although more rigorous methods can be employed. Here HAB has been calculated directly using the ab initio methods of Faradzel et al. [14,29] from the overlap of the reactant and product wavefunctions. For an adiabatic system, by definition the transmission coefficient (κ) is unity and the frequency factor is [51] ν = 2HAB / h.

(A.8)

This is rationalized as the frequency with which an electron can move back and forth between reactants in the encounter complex and will be dependent on the degree of electronic coupling. The greater the coupling, the faster an electron can hop between the reactants. For nonadiabatic reactions, the product of the frequency factor and the transmission coefficient is 2 2π  νκ = (A.9) HAB  (4πλkT )−1/2 . h¯

A.G. Stack et al. / Journal of Colloid and Interface Science 274 (2004) 442–450

The free energy of activation for the adiabatic self-exchange case is described as [47] 2 )1/2 2λ − (λ2 − 4HAB (A.10) − HAB , 4 where λ is the Marcus reorganization energy. When HAB is small enough to have a negligible effect (i.e., nonadiabatic), this simplifies to the well-known expression used for the nonadiabatic case:

G∗ =

G∗ = λ/4.

(A.11)

Marcus showed that the reorganization energy, λ, is typically separable into the sum of the inner-sphere and outersphere components, λ = λIS + λOS ,

(A.12)

where λIS and λOS are the inner- and outer-sphere reorganization energies, respectively. The outer-sphere reorganization energy (λOS ) reflects changes in the solvent structure surrounding the donor and acceptor complexes due to the movement of charge from one reactant to the other. Because of the nonspherical shapes of quinone species, outer-sphere reorganization energies were calculated from the method of German and Kuznetsov [52] for three-dimensional ellipsoids. This technique is believed to be much more accurate for a planar organic π -system such as hydroquinone. The inner-sphere term (λIS ) reflects changes in nuclear configuration of atoms in the encounter complex (e.g., bond length changes). Following Rosso et al. [12] we have calculated λIS using a four-point method proposed by Nelsen et al. [53]. In this method, ab initio geometry optimizations of the two reactant species are performed and the energies evaluated. The single-point energy of each reactant species frozen in the equilibrium geometry of its electron transfer partners is also evaluated (four total energies). The total inner-sphere reorganization energy is the sum of the relaxation energies for the oxidized and reduced species. Once the self-exchange rate constants for two reactions have been calculated, the cross-reaction rate constant, k12 , can be calculated,  k12 = k11 k22 K12 f , (A.13) where k11 and k22 refer to the first order self-exchange rate constants, and f is f = 10∧

(log K12 )2 , 4 log(k11k22 /ν02 )

449

which is corrected for the electrostatic work removing the precursor complex (Wp ) and forming the successor complex (Ws ). The electrostatic work terms are calculated using Eq. (A.6). A.2. Encounter complex Without exact knowledge of the encounter complex configuration, an estimate must be made before λOS and HAB can be calculated. In solutions at room temperature, thermal motion will lead to the formation of many different encounter complexes, which complicates the measurement of an encounter complex configuration. For our purposes, we are interested in the encounter complex with the highest probability of formation; this is likely to be the orientation with the minimum energy of interaction between reactants. In the case of quinone self-exchange, the most common encounter complex is a face-to-face orientation where π -orbital overlap is maximized [54]. We have modeled the most likely encounter complex by examining X-ray diffractometry-derived structures of condensed phases. We have assumed that in the absence of intermolecular covalent or ionic bonding nearest-neighbor hydroquinone moieties in a crystal composed of hydroquinone should serve as a proxy for an encounter complex in solution. Of importance are not only the distance between quinones in the structure, but the orientations and staggering of the quinones relative to each other in the model encounter complex; all of which may affect HAB significantly [48,55]. The α-hydroquinone structure can be rationalized as rings of hydrogen-bonded hydroxyls (of the hydroquinones) repeating in the c-axis direction to form cages [42]. The intermolecular free energy of interaction for this compound is described [56] as mostly dispersion forces (63%) with significant hydrogen bonding (37%). The lack of significant ionic or covalent bonding between hydroquinone molecules makes this a suitable compound to consider for extracting a model for the encounter complex. The α-hydroquinone structure contains two pairs of face-to-face hydroquinones, one along the c-axis (5.650 Å) and another on the second layer outside of the ring (5.095 Å). We have used the latter of the two since this is presumably the distance of closest approach.

References (A.14)

where ν0 is the geometric mean of the frequency factors of the self-exchange reactions. K12 is the equilibrium constant for the cross reaction,   K12 = exp −G0et /RT , (A.15) where G0et , the Gibbs free energy of the reaction, is  0  0 G0et = −nF Ered (A.16) − Eox − Wp + Ws ,

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