Reduced Partition Function Ratios Research Articles

Quantum-Chemical Calculations of Carbon-Isotope Fractionation in CO2(g), Aqueous Carbonate Species, and Carbonate Minerals

Quantum chemical calculations on large supermolecular carbonate-water and carbonate mineral clusters are used to predict equilibrium constants for 13,12C-isotope-exchange reactions between CO2(g), aqueous carbonate species, and the common carbonate minerals. For the aqueous species, we evaluate the influence of the size and conformational variability of the solvation shell, the exchange-correlation functional, and the basis set. The choice of exchange-correlation functional (PBE vs B3LYP), the basis set (6-31G* vs aug-cc-pVDZ), and solvation shell size (first shell only vs first shell and a partial second shell) each produce changes of approximately 5-10 per mil in the reduced partition function ratio. Conformational variability gives rise to a standard error of approximately 0.5 per mil using approximately 10 solute-solvent conformations. The best results are obtained with the B3LYP/aug-cc-pVDZ combination, but because the improvements in the basis set and exchange correlation functional drive the reduced partition function ratios in opposite directions, reasonably good results are also obtained with the PBE/6-31G* combination. To construct molecular clusters representative of mineral environments, a new method is introduced on the basis of conservation of Pauling bond strength. Using these clusters as models for minerals, calculations of mineral-gas and mineral-aqueous carbon-isotope fractionation factors, are in good agreement with experimental measurements. Carbon-isotope fractionation factors for gas, aqueous, and mineral phases are thus integrated into a single theoretical/computational framework.

Quantum chemical calculations of equilibrium copper (I) isotope fractionations in ore-forming fluids

We calculated the equilibrium isotope fractionation of Cu (I) complexes in hydrothermal ore-forming fluids using quantum chemical calculations (based on the density functional theory and Hartree–Fock approximations) of molecular structures and vibrational frequencies to gain insights into the Cu isotope ( 63Cu, 65Cu) composition in natural systems. The calculated molecular structures of liquid — (copper chlorides and copper hydrosulfides) and vapor-phase (CuCl(H 2O) and Cu 3Cl 3) Cu complexes are largely consistent with the experimental data. The predicted vibrational frequencies are dependent on the energy levels of theory and basis sets used in the calculations. The vibrational frequency shift due to isotopic substitution is most prominent in the stretching (for linear molecules such as CuCl) and bending (for [CuCl 2] 1− and [CuCl 3] 2−) modes of vibration. The calculated reduced partition function ratio (i.e. 10 3 · ln( β 65–63)) of each copper isotopomer depends on the Cu coordination environments and types of ligands. The Cu complexes with a longer bond length (e.g., Cu–Cl or Cu–S) and larger coordination number apparently have greater isotope fractionation among the monomer complexes. A significant copper isotope fractionation has been predicted for each Cu complex: Cu 3Cl 3 (vapor phases) contain the most enriched 65Cu isotopes, whereas [CuCl 3] 2− (liquid phase) is the most depleted. The calculated δ 65Cu range [maximum δ 65Cu for Cu 3Cl 3–minimum δ 65Cu for [CuCl 3] 2−] increases with a decrease in the temperature; the ranges are 0.76–0.89‰ at 500 °C, 1.00–1.17‰ at 400 °C, 1.37–1.61‰ at 300 °C, 2.01–2.35‰ at 200 °C, and 2.50–2.92‰ at 150 °C. These ranges around 150–500 °C are somewhat larger than the observed isotopic compositions in sea-floor hydrothermal vents (approximately 0.8–1.3‰) and porphyry copper deposits (0.7‰). While detailed information about phase equilibria (stability fields) among both vapor and liquid copper complexes as function of composition, temperature and pressure should be known to better discuss the origin of the Cu isotope composition in natural systems, current theoretical prediction shows the effect of temperature could contribute to a variation in isotopic composition in the natural system. Our calculation indicates that temperature and types of ligands affect the copper isotope fractionation in the ore-forming fluids, and also imply that the liquid-vapor equilibrium (e.g., volcanic degassing) could lead to significant copper isotope fractionation. On the other hands, Cu isotope fractionations of [CuCl 2] 1− and [Cu(HS) 2] 1−, the major copper-bearing species in hydrothermal conditions, are rather similar; thus, the fractionation without a change in the oxidation states may not fully account for the natural isotopic variation unless the concentrations of both Cu 3Cl 3 and for [CuCl 3] 2− are significant in the hydrothermal fluids.

Vapor pressure isotope effects in methyl fluoride studied by density functional theory

H/D and 12C/ 13C vapor pressure isotope effects in methyl fluoride (CH 3F) were studied at the MPW1PW91/6-31++G(d) level of theory. The CH 3F molecule and small CH 3F clusters were used as models of vapor and liquid methyl fluoride, respectively, and their reduced partition function ratios were calculated at their optimized structures. The equilibrium constants of the isotope exchange reactions between the two phases were evaluated as 1.0263 and 0.9991 for the H/D and 12C/ 13C isotope exchanges at 150 K, respectively, which reproduced reasonably well the experimental results. The effect of CH⋯F interaction between neighboring molecules on the vpies was discussed.

Comment on the studies of oxygen isotope fractionation between calcium carbonates and water at low temperatures by Zhou and Zheng (2003; 2005)

Experimental and theoretical aspects of oxygen isotope fractionation in the system calcite–water at low temperatures were critically examined. Contrary to the claim made by Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2003) An experimental study of oxygen isotope fractionation between inorganically precipitated aragonite and water at low temperatures. Geochim. Cosmochim. Acta 67, 387–399], there is excellent agreement between fractionation factors that were experimentally determined by means of slow, inorganic precipitation of calcite from solutions and those obtained largely from theoretical, statistical–mechanical calculations of the reduced partition function ratios. This agreement strongly suggests that calcite was precipitated from a solution very close to isotopic equilibrium. However, recently Zhou and Zheng [Zhou G.-T., and Zheng Y.-F. (2005) Effect of polymorphic transition on oxygen isotope fractionation between aragonite, calcite and water: a low-temperature experimental study. Am. Miner. 90, 1121–1130] presented, without any explanation, conclusions on these major aspects that contradict the previous statements of Zhou and Zheng (2003). The apparent discrepancy in calcite–water oxygen isotope fractionation between experimental and theoretical studies discussed by Zhou and Zheng (2003) originates from the “mineral–water interaction” term in the modified increment method, which was developed by one of the authors (Y.-F. Zheng). We call for evidence for the theoretical nature of the modified increment method, which has never been presented in any of Zheng’s papers. Without such evidence, great caution must be exercised in using fractionation factors derived from the modified increment method.

Boron isotope fractionation accompanying formation of potassium, sodium and lithium borates from boron-bearing solutions

A series of experiments was conducted in which boron minerals were precipitated by water evaporation from solutions containing boron and potassium, sodium or lithium at 25°C, and boron isotope fractionation accompanying such mineral precipitation was investigated. In the boron-potassium ion system, K2[B4O5(OH)4]·2H2O, santite (K[B5O6(OH)4]·2H2O), KBO2·1.33H2O, KBO2·1.25H2O and sassolite (B(OH)3) were found deposited as boron minerals. Borax (Na2[B4O5(OH)4·8H2O) was found deposited in the boron-sodium ion system, and Li2B2O4·16H2O, Li2B4O7·5H2O, Li2B10O16·10H2O, LiB2O3(OH)·H2O and sassolite in the boron-lithium ion system. The boron isotopic analysis was conducted for santite, K2[B4O5(OH)4]·2H2O, borax and Li2B2O4·16H2O. The separation factor, S, defined as the 11B/10B isotopic ratio of the precipitate divided by that of the solution, ranged from 0.991 to 1.012. Computer simulations for modeling boron mineral formations, in which polyborates were decomposed into three coordinated BO3 unit and four coordinated BO4 unit for the purpose of calculation of their boron isotopic reduced partition function ratios, were attempted to estimate the equilibrium constant, KB, of the boron isotope exchange between the boric acid molecule (B(OH)3) and the monoborate anion (B(OH)4-). As a result, the KB value of 1.015 to 1.029 was obtained. The simulations indicated that the KB value might be dependent on the kind of boron minerals, which qualitatively agreed with molecular orbital calculations independently carried out.

Isotope Fractionation of Iron(III) in Chemical Exchange Reactions Using Solvent Extraction with Crown Ether

This work reports on the chemical isotope fractionation of Fe(III) by a solvent extraction method with a crown ether of dicyclohexano-18-crown-6. The (56)Fe/(54)Fe and (57)Fe/(54)Fe ratios were analyzed by multiple-collector inductively coupled plasma mass spectrometry. We determined the dependence of the isotope enrichment factors (epsilon) on the strength of HCl. The relative deviation of the (56)Fe/(54)Fe ratios relative to the unprocessed material (10(4) epsilon(56)) increases from -15.3 to -6.3 with [HCl] increasing from 1.6 to 3.5 mol/L. Likewise, 10(4) epsilon(57) increases from -22.8 to -9.6 under the same conditions. The correlation between epsilon(56) and epsilon(57) is mass dependent within the errors. The observed fractionation was broken down into the effects of competing extraction reactions and of a reaction between Fe(III) species (FeCl(2)(+) and FeCl(3)) in the aqueous phase. We found that the isotope fractionation between the Fe(III) species is mass dependent, which we confirmed by calculating the reduced partition function ratios.

Preferential formation of 13C– 18O bonds in carbonate minerals, estimated using first-principles lattice dynamics

Equilibrium constants for internal isotopic exchange reactions of the type: Ca 12 C 18 O 16 O 2 + Ca 13 C 16 O 3 ↔ Ca 13 C 18 O 16 O 2 + Ca 12 C 16 O 3 for individual CO 3 2− groups in the carbonate minerals calcite (CaCO 3), aragonite (CaCO 3), dolomite (CaMg(CO 3) 2), magnesite (MgCO 3), witherite (BaCO 3), and nahcolite (NaHCO 3) are calculated using first-principles lattice dynamics. Calculations rely on density functional perturbation theory (DFPT) with norm-conserving planewave pseudopotentials to determine the vibrational frequencies of isotopically substituted crystals. Our results predict an ∼0.4‰ excess of 13 C 18 O 16 O 2 2 − groups in all studied carbonate minerals at room-temperature equilibrium, relative to what would be expected in a stochastic mixture of carbonate isotopologues with the same bulk 13C/ 12C, 18O/ 16O, and 17O/ 16O ratios. The amount of excess 13 C 18 O 16 O 2 2 − decreases with increasing temperature of equilibration, from 0.5‰ at 0 °C to <0.1‰ at 300 °C, suggesting that measurements of multiply substituted isotopologues of carbonate could be used to infer temperatures of ancient carbonate mineral precipitation and alteration events, even where the δ 18O of coexisting fluids is uncertain. The predicted temperature sensitivity of the equilibrium constant is ∼0.003‰/°C at 25 °C. Estimated equilibrium constants for the formation of 13 C 18 O 16 O 2 2 − are remarkably uniform for the variety of minerals studied, suggesting that temperature calibrations will also be applicable to carbonate minerals not studied here without greatly compromising accuracy. A related equilibrium constant for the reaction: Ca 12 C 18 O 16 O 2 + Ca 12 C 17 O 16 O 2 ↔ Ca 12 C 18 O 17 O 16 O + Ca 12 C 16 O 3 in calcite indicates formation of 0.1‰ excess 12C 18O 17O 16O 2− at 25 °C. In a conventional phosphoric acid reaction of carbonate to form CO 2 for mass-spectrometric analysis, molecules derived from 13 C 18 O 16 O 2 2 − dominate (∼96%) the mass 47 signal, and 12C 18O 17O 16O 2− contributes most of the remainder (3%). This suggests that carbonate internal equilibration temperatures can be recovered from acid-generated CO 2 if abundances of isotopologues with mass 44–47 can be measured to sufficient precision. We have also calculated 18O/ 16O and 13C/ 12C reduced partition function ratios for carbonate minerals, and find them to be in good agreement with experiments and empirical calibrations. Carbon and oxygen isotope fractionation factors in hypothetical 40Mg—magnesite and 40Ba—witherite indicate that M 2+-cation mass does not contribute significantly to equilibrium isotopic fractionations between carbonate minerals.

Pressure effects on the reduced partition function ratio for hydrogen isotopes in water

We have developed a simple, yet accurate theoretical method for calculating the reduced isotope partition function ratio (RIPFR) for hydrogen of water at elevated pressures. This approach requires only accurate equations of state (EOS) for pure isotopic end-members (H 2O and D 2O), which are available in the literature. The effect of pressure or density on the RIPFR of water was calculated relative to that of ideal-gas water at infinitely low pressure for the temperature range from 0 to 527 °C. For gaseous and low-pressure (ca. ⩽15 MPa) supercritical phases of water, the RIPFR increases slightly (1–1.3‰) with pressure or density in a fashion similar to those of many other geologic materials. However, in liquid and high-pressure (>20 MPa) supercritical phases, the RIPFR of water decreases (0.5–6‰) with increasing pressure (or density) to 100 MPa. This rather unique phenomenon is ascribed to the inverse molar volume isotope effects (MVIE) of liquid and high-density supercritical waters, V (D 2O) > V (H 2O), while other substances including minerals show the normal MVIE. These theoretical predictions were experimentally confirmed by Horita et al. [Horita, J., Cole, D.R., Polyakov, V.B., Driesner, T., 2002. Experimental and theoretical study of pressure effects on hydrogen isotope fractionation in the system brucite–water at elevated temperatures. Geochim. Cosmochim. Acta 66, 3769 – 3788.] for the system brucite–water. Although the P–T ranges for the EOS of normal and heavy waters are rather limited, our modeling indicates that the RIPFR of water continues to decrease with pressure above 100 MPa. The method developed here can be applied to any other geologic fluids, if accurate EOS for their isotopic end-members is available. These results have important implications for the interpretation of high-pressure isotopic partitioning in the Earth, the outer planets, and their moons.

Calculation of sulfur isotope fractionation in sulfides

The increment method has been successfully applied to calculate thermodynamic isotope fractionation factors of oxygen in silicates, oxides, carbonates, and sulfates. In this paper, we modified the increment method to calculate thermodynamic isotope fractionation factors of sulfur in sulfides, based on chemical features of sulfur–metal bonds and crystal features of sulfide minerals. To approximate the bond strength of sulfides, a new constant, known as the Madelung constant, was introduced. The increment method was then extended to calculate the reduced partition function ratios of sphalerite, chalcopyrite, galena, pyrrhotite, greenockite, bornite, cubanite, sulvanite, and violarite, as well as the isotope fractionation factors between them over the temperature range from 0 to 1000 °C. The order of 34S enrichment in these nine minerals is pyrrhotite > greenockite > sphalerite > chalcopyrite > cubanite > sulvanite > bornite > violarite > galena. Our improved method constitutes another model for calculating the thermodynamic isotope fractionation factors of sulfur in sulfides of geochemical interest.

Boron isotopic fractionation between minerals and fluids: New insights from in situ high pressure-high temperature vibrational spectroscopic data

Equilibrium boron isotopic fractionations between trigonal B(OH) 3 and tetragonal B(OH) 4 − aqueous species have been calculated at high P- T conditions using measured vibrational spectra (Raman and IR) and force-field modeling to compute reduced partition function ratios for B-isotopic exchange following Urey’s theory. The calculated isotopic fractionation factor at 300 K, α 3/4 = 1.0176(2), is slightly lower than the formerly calculated value of α 3/4 = 1.0193 ( Kakihana and Kotaka, 1977), due to differences in the determined vibrational frequencies. The effect of pressure on α 3/4 up to 10 GPa and 723 K is shown to be negligible relative to temperature or speciation (pH) effects. Implications for the interpretation of boron fractionation in experimental and natural systems are discussed. We also show that the relationship between seawater-mineral B isotope fractionation and pH can be expressed using two variables, α 3/4 on one hand, and the pK a of the boric acid-borate equilibrium on the other hand. This latter value is given by the equilibrium of boron species in water for the carbonate-water exchange, but could be governed by mineral surface properties in the case of clays. This may allow defining intrinsic paleo-pHmeters from B isotope fractionation between carbonate and authigenic minerals. Finally, it is shown that fractionation of boron isotopes can be rationalized in terms of the changes in 1) coordination of B from trigonal to tetrahedral in both fluids and minerals; and 2) the ligand nature around B from OH − in the fluid and some hydrous minerals to non-hydrogenated O in many minerals. Relationships are established that allow predicting the isotopic fractionation factor of B between minerals and fluid.

Tritium Separation by Electrolysis Using Solid Polymer Electrolyte

Hydrogen isotope separation effect by electrolysis of water was theoretically investigated and was compared with experimental results. The separation mechanism was analyzed as the hydrogen isotope exchange reaction between water and diatomic hydride that consists of hydrogen and cathode material. The equilibrium constants of the isotope exchange reaction were calculated from reduced partition function ratio. Using the constants, the separation factor (SF) of the isotopes was calculated according to the two-phase distribution theory for isotopes. Experimentally, light or heavy water spiked with tritiated water was electrolyzed by a device with a solid polymer electrolyte, which equipped with SUS, Ni, or carbon cathode. Thus, the SFs were experimentally obtained. Calculated SFs were well agreed with the experimentally values for SUS and Ni cathodes, and that for carbon cathode was somewhat small then the experimental value.

Periodicity in 6Li-to-7Li Isotopic Reduced Partition Function Ratios of Diatomic Lithium Compounds

The UB3LYP/LanL2DZ level calculations were carried out for sixty-seven Li-M diatomic molecules of lithium between LiH and LiBi to explore the periodic properties of their 6Li-to-7Li isotopic reduced partition function ratios (rpfrs) and some other quantities. The Li-M bond distance, the frequency shift upon 7Li/6Li isotope substitution and the rpfr all show plots typical of any chemical and physical property which is a periodic function of the atomic number. The rpfr is strongly correlated with the Li-M bond stretching force constant except for LiH. The present calculations show that the rpfrs of diatomic molecules of group 2, group 12 and group 18 elements are relatively small, which reflects the filled electron shell and subshell characters of those elements. The results of the calculations and the consideration on chemical properties of elements suggest that zinc is a possible substitute for mercury in lithium isotope separation processes utilizing the redox reaction of lithium.

Oxygen isotope fractionation factors involving cassiterite (SnO 2): I. calculation of reduced partition function ratios from heat capacity and X-ray resonant studies

Oxygen isotope equilibrium fractionation constants (β 18O-factors) of cassiterite were evaluated on the basis of heat capacity and X-ray resonant (Mössbauer spectroscopy and X-ray inelastic scattering) data. The low-temperature heat capacity of cassiterite was measured in the range from 13 to 340 K using an adiabatic calorimeter. Results of measurements of two samples agree very closely but deviate more than 5% from previous heat capacity data used for calculation of thermodynamic functions. The temperature dependence of heat capacity was treated using the modern version of the Thirring expansion, and the appropriate temperature dependence of the vibrational kinetic energy was found. Measurements of temperature-dependent Mössbauer parameters of cassiterite were conducted in the range from 300 to 900 K. The attempt to describe Mössbauer fraction and the second order Doppler (SOD) shift on the basis of the Debye model failed. The first term of the Thirring expansion of the Mössbauer SOD shift agrees with that calculated from the Sn sublattice vibration density of states (VDOS) obtained via synchrotron X-ray scattering. Based on this agreement we calculated the kinetic energy of the cassiterite Sn sublattice from VDOS. From the kinetic energy of the total cassiterite crystalline lattice and its Sn sublattice, β 18O-factors of cassiterite were computed in the temperature range 300–1500 K by the method of Polyakov and Mineev (2000). Appropriate polynomials, which are valid at temperatures above 400 K, are the following: 1 0 3 ln ⁡ β SnO 2 = ( 7.176 ± 0.252 ) x − ( 0.07369 ± 0.00089 ) x 2 + ( 0.0008026 ± 0.0001022 ) x 3 , x = 10 6 / T 2 ; 10 3 ln ⁡ α CaCO 3 − S nO 2 = 4.607 x − 0.3463 x 2 + 0.01500 x 3 , x = 10 6 / T 2 ; 10 3 ln ⁡ α SiO 2 − S nO 2 = 4.942 x − 0.2963 x 2 + 0.01150 x 3 , x = 10 6 / T 2 . The evaluated cassiterite isotope fractionation factors are significantly different from those obtained by synthesis, increment and empirical methods. To resolve the differences, laboratory direct exchange experiments are needed.

Oxygen isotope fractionation factors involving cassiterite (SnO 2): II. determination by direct isotope exchange between cassiterite and calcite

Direct oxygen isotope fractionation between cassiterite and calcite has been investigated experimentally at 15 kbar with temperature ranging from 800 to 1000°C. Combined with the quartz-calcite fractionation measured with the same technique ( Clayton et al., 1989), the calcite-cassiterite and quartz-cassiterite oxygen isotope fractionations can be expressed as: Δ CaCO 3 − SnO 2 = 4.14 × 10 6 T − 2 Δ SiO 2 − SnO 2 = 4.52 × 10 6 T − 2 at temperatures higher than 600°C. These calibrations are in good agreement with those obtained from heat capacity and X-ray resonant data. The theory of using X-ray resonant data to calculate the reduced partition function ratios of transition elements that have at least one Mössbauer-sensitive isotope is evaluated with the current experimental result.

Theoretical investigation of iron isotope fractionation between Fe(H 2O) 63+ and Fe(H 2O) 62+: Implications for iron stable isotope geochemistry

The magnitude of equilibrium iron isotope fractionation between Fe(H 2O) 6 3+ and Fe(H 2O) 6 2+ is calculated using density functional theory (DFT) and compared to prior theoretical and experimental results. DFT is a quantum chemical approach that permits a priori estimation of all vibrational modes and frequencies of these complexes and the effects of isotopic substitution. This information is used to calculate reduced partition function ratios of the complexes (10 3 · ln(β)), and hence, the equilibrium isotope fractionation factor (10 3 · ln(α)). Solvent effects are considered using the polarization continuum model (PCM). DFT calculations predict fractionations of several per mil in 56Fe/ 54Fe favoring partitioning of heavy isotopes in the ferric complex. Quantitatively, 10 3 · ln(α) predicted at 22°C, ∼ 3 ‰, agrees with experimental determinations but is roughly half the size predicted by prior theoretical results using the Modified Urey-Bradley Force Field (MUBFF) model. Similar comparisons are seen at other temperatures. MUBFF makes a number of simplifying assumptions about molecular geometry and requires as input IR spectroscopic data. The difference between DFT and MUBFF results is primarily due to the difference between the DFT-predicted frequency for the ν 4 mode (O-Fe-O deformation) of Fe(H 2O) 6 3+ and spectroscopic determinations of this frequency used as input for MUBFF models (185–190 cm −1 vs. 304 cm −1, respectively). Hence, DFT-PCM estimates of 10 3 · ln(β) for this complex are ∼ 20% smaller than MUBFF estimates. The DFT derived values can be used to refine predictions of equilibrium fractionation between ferric minerals and dissolved ferric iron, important for the interpretation of Fe isotope variations in ancient sediments. Our findings increase confidence in experimental determinations of the Fe(H 2O) 6 3+ − Fe(H 2O) 6 2+ fractionation factor and demonstrate the utility of DFT for applications in “heavy” stable isotope geochemistry.

Periodicity in 6Li-to-7Li Isotopic Reduced Partition Function Ratios of Diatomic Lithium Compounds

Periodicity in 6Li-to-7Li Isotopic Reduced Partition Function Ratios of Diatomic Lithium Compounds

Calculations of Reduced Partition Function Ratios of Hydrated Boric Acid Molecule by the ab initio Molecular Orbital Theory

Based on the ab initio molecular orbital theory at the HF/6-31G(d) level, the effect of hydration on the reduced partition function ratio (RPFR) of the boric acid molecule (B(OH)3) was evaluated in order to better understand boron isotope fractionation observed in aqueous systems. The B(OH)3(H2O) n species with n up to 12 were considered and their geometry optimization and RPFR calculations were carried out. It was induced that hydration decreased the RPFR of B(OH)3 and the degree of decrease in ln(RPFR) was ca. 1.5%, which was nearly equivalent to that of B(OH)4 -.

A density functional theory study applied for carbon and oxygen isotope effects in the system Ni(CO) 4/CO

DFT calculations were carried out on the carbon and oxygen isotopomers of CO and Ni(CO) 4 and results compared with (Spindel's) experimental results on the carbon and oxygen isotope exchange equilibria between CO and Ni(CO) 4. The isotopic equilibrium constants, K B, of carbon and oxygen exchange reactions between CO and Ni(CO) 4 complexes were theoretically calculated as the ratio of the reduced partition function ratios (RPFRs) of the 13C/ 12C and 18O/ 16O isotopic pairs for CO and Ni(CO) 4 on the assumption of the internal harmonic vibrations. The agreement between calculated separation factor and Spindel's experimental data are surprisingly good, including the temperature dependence.

Theoretical estimates of equilibrium chromium-isotope fractionations

Equilibrium Cr-isotope ( 53Cr/ 52Cr) fractionations are calculated using published vibrational spectra and both empirical and ab initio force-field models. Reduced partition function ratios for chromium isotope exchange, in terms of 1000×ln( β 53–52), are calculated for a number of simple complexes, crystals, and the Cr(CO) 6 molecule. Large (>1‰) fractionations are predicted between coexisting species with different oxidation states or bond partners. The highly oxidized [Cr 6+O 4] 2− anion will tend to have higher 53Cr/ 52Cr than coexisting compounds containing Cr 3+ or Cr 0 at equilibrium. Substances containing chromium bonded to strongly bonding ligands like CO will have higher 53Cr/ 52Cr than compounds with weaker bonds, like [CrCl 6] 3−. Substances with short Cr-ligand bonds (Cr–C in Cr(CO) 6, Cr–O in [Cr(H 2O) 6] 3+ or [CrO 4] 2−) will also tend have higher 53Cr/ 52Cr than substances with longer Cr-ligand bonds ([Cr(NH 3) 6] 3+, [CrCl 6] 3−, and Cr-metal). These systematics are similar to those found in an earlier study on Fe-isotope fractionation (Geochim. Cosmochim. Acta 65 (2001) 2487). The calculated equilibrium fractionation between Cr 6+ in [CrO 4] 2− and Cr 3+ in either [Cr(H 2O) 6] 3+ or Cr 2O 3 agrees qualitatively with the fractionation observed during experimental (probably kinetic) reduction of [CrO 4] 2− in solution (Science 295 (2002) 2060), although the calculated fractionation (∼6–7‰ at 298 K) does appear to be significantly larger than the experimental fractionation (3.3–3.5‰). Our model results suggest that natural inorganic Cr-isotope fractionation at the earth's surface may be driven largely by reduction and oxidation processes.

Calculations of Reduced Partition Function Ratios of Hydrated Boric Acid Molecule by the ab initio Molecular Orbital Theory

Calculations of Reduced Partition Function Ratios of Hydrated Boric Acid Molecule by the ab initio Molecular Orbital Theory