Reduced Partition Function Ratios Research Articles

18O/16O and D/H Isotopic Preference in Hydration Spheres of Alkali Metal Ions

Abstract With the final goal set at theoretical elucidation of experimentally observed isotope salt effects, molecular orbital calculations were performed to estimate the 18O/16O and D/H isotopic reduced partition function ratios (RPFRs) of water molecules around lithium, sodium, and potassium ions. As model water molecules in the ith hydration sphere of the cation in aqueous solutions containing that cation, we considered water molecules in the ith hydration sphere that were surrounded by water molecules in the (i+1)th hydration sphere in clusters, M+(H2O)n (M = Li, Na or K; n up to 100). The calculations indicated that the decreasing order of the 18 O preference over 16 O in the primary hydration sphere is: Li+ > (bulk water) ≥ Na+ > K+. That is, water molecules in the primary hydration spheres of the Li+, Na+, and K+ ions are, respectively, enriched, slightly depleted, and depleted in the heavier isotope of oxygen relative to water molecules in bulk. No such preference was observed for hydrogen isotopes in any hydration sphere or for oxygen isotopes in the secondary and outer hydration spheres.

D/H and 18O/16O Isotopic Reduced Partition Function Ratios ofWater Molecules around a Sodium Ion

With the final goal set at theoretical elucidation of experimentally observed isotope salt effects, molecular orbital calculations were performed to estimate the D/H and 18O/16O isotopic reduced partition function ratios (RPFRs) of water molecules around a sodium ion. As model water molecules in the ith hydration sphere of the sodium ion in sodium ion-bearing aqueous solution, we considered water molecules in the ith hydration sphere that were surrounded by water molecules in the (i+1)th hydration sphere in clusters, Na+(H2O)n (n up to 100). The calculations indicated that the 18O/16O RPFR in the primary hydration sphere is slightly smaller than that of bulk water while the D/H RPFR is practically the same as that of bulk water, and that the influence of the existence of the sodium ion is limited to the primary hydration sphere.

Reduced partition function ratios for isotopes of hydrated alkali and alkaline earth ions calculated by a simple electrostatic model

We describe a simple electrostatic model of hydrated ions [M(H2O)n]+ (n = 3–18, M = Li, Mg, Ca, K) which enables to calculate ion vibration frequency of the ground state. In this model the considered ion with a reduced mass vibrates in quasispherical well formed by the ion-dipole attractive potential and repulsive valence potential, these simplifications allowed to solve one dimensional Schrödinger equation, whilst the calculated ground state was considered as one of triply degenerated state of the three dimensional motion of the ion vs. hydration shell. The reduced partition function ratios were calculated from the vibration frequencies using Urey's (1947) harmonic approximation formula. The results obtained in this way are in good agreement with those obtained by much more laborious ab initio molecular orbital methods, like SCF Hartree-Fock, DFT, MP2, etc. Moreover, we were able to extend calculations to hydrated Li and K ions surrounded with two shells of water molecules. These results are the first estimations of the upper limit of isotope fractionation in water solutions, which are 99.3‰ for Li and only 2.5‰ for K isotopes.

New data on equilibrium iron isotope fractionation among sulfides: Constraints on mechanisms of sulfide formation in hydrothermal and igneous systems

Fe, S, and Cu reduced partition function ratios (β-factors) allow calculation of equilibrium isotope fractionation factors. β-Factors for chalcopyrite are calculated from experimental and theoretical partial phonon densities of state states ( Kobayashi et al., 2007). The Fe β-factors for mackinawite are calculated from Mössbauer spectroscopy data ( Bertaut et al., 1965). Excellent agreement exists between Fe β-factors for chalcopyrite calculated from the experimental and theoretical 57Fe phonon densities of states, supporting the reliability of the Fe β-factors for chalcopyrite. The 34S β-factor for chalcopyrite is consistent with experimental data on equilibrium sulfur isotope fractionation factors among sulfides and theoretical 34S β-factors, except those recently calculated by a DFT approach. Up-to-date experimental isotope-exchange data on equilibrium Fe isotope fractionation factors between minerals and aqueous Fe were critically reevaluated in conjunction with Fe β-factors for minerals, and the following expressions for β-factors for aqueous Fe 2+ and Fe 3+ were obtained: 10 3 ln β Fe aq 3 + = 1.0063 10 6 T 2 ; T > 273 K 10 3 ln β Fe aq 2 + = 0.6537 10 6 T 2 ; T > 273 K. .Application of these new β-factors to Fe and S isotope fractionation at seafloor hydrothermal conditions shows that chalcopyrite may precipitate in equilibrium with Fe in hydrothermal fluids. In contrast, Fe and S isotope compositions of pyrite are may not be controlled by isotope equilibration with Fe in the hydrothermal fluid. Previously reported data on Fe isotope fractionation between pyrite and chalcopyrite in intrusions in the Grasberg Igneous Complex seem to be reasonable, and are in general agreement with previous “non-isotopic” estimates. Fe isotope fractionation between pyrite and chalcopyrite appear to reflect equilibrium in the intrusions of the Grasberg Igneous Complex, and Fe isotopes may therefore be considered as a potential geothermometer. The pyrite–chalcopyrite Fe isotope geothermometer, however, cannot be applied to the skarn facies of the Grasberg complex, where the initial Fe isotope compositions may have been disturbed by subsequent processes related to skarn formation.

D/H and18o/16O Isotopic Reduced Partition Function Ratios of Water Molecules around a Sodium Ion

D/H and¹⁸o/¹⁶O Isotopic Reduced Partition Function Ratios of Water Molecules around a Sodium Ion

Calcium Isotope Effects Accompanying Electrochemical Insertion of Calcium into Graphite from Electrolyte Solution

Calcium has been electrochemically inserted from a 1:2(v/v) mixed solution of ethylene carbonate and methylethyl carbonate containing 0.5mol dm−3 Ca(ClO4)2 into graphite, and the calcium isotope fractionation accompanying the insertion was observed. The lighter isotope, 40Ca, was preferentially fractionated into graphite. The single-stage separation coefficient for the 48Ca/40Ca isotopic pair ranged from 1.0×10−3 to 7.2×10−3 at 50°C. These experimental results were qualitatively consistent with the estimation of the equilibrium constant of the calcium isotope exchange reaction between the calcium ion in the electrolyte solution and the calcium atom in graphite based on molecular orbital calculations of the reduced partition function ratios of the calcium species involved in the equilibrium.

Calculation of equilibrium stable isotope partition function ratios for aqueous zinc complexes and metallic zinc

The goal of this study is to determine reduced partition function ratios for a variety of species of zinc, both as a metal and in aqueous solutions in order to calculate equilibrium stable isotope partitioning. We present calculations of the magnitude of Zn stable-isotope fractionation ( 66,67,68Zn/ 64Zn) between aqueous species and metallic zinc using measured vibrational spectra (fit from neutron scattering studies of metallic zinc) and a variety of electronic structure models. The results show that the reduced metal, Zn(0), will be light in equilibrium with oxidized Zn(II) aqueous species, with the best estimates for the Zn(II)–Zn(0) fractionation between hexaquo species and metallic zinc being Δ 66/64Zn aq-metal ∼ 1.6‰ at 25 °C, and Δ 66/64Zn aq-metal ∼ 0.8‰ between the tetrachloro zinc complex and metallic zinc at 25 °C using B3LYP/aug-cc-pVDZ level of theory and basis set. To examine the behavior of zinc in various aqueous solution chemistries, models for Zn(II) complex speciation were used to determine which species are thermodynamically favorable and abundant under a variety of different conditions relevant to natural waters, experimental and industrial solutions. The optimal molecular geometries for [Zn(H 2O) 6] 2+, [Zn(H 2O) 6]·SO 4, [ZnCl 4] 2− and [Zn(H 2O) 3(C 3H 5O(COO) 3)] − complexes in various states of solvation, protonation and coordination were calculated at various levels of electronic structure theory and basis set size. Isotopic reduced partition function ratios were calculated from frequency analyses of these optimized structures. Increasing the basis set size typically led to a decrease in the calculated reduced partition function ratios of ∼0.5‰ with values approaching a plateau using the aug-cc-pVDZ basis set or larger. The widest range of species were studied at the B3LYP/LAN2DZ/6-31G ∗ level of theory and basis-set size for comparison. Aqueous zinc complexes where oxygen is bound to the metal center tended to have the largest reduced partition function ratios, with estimated fractionations ranging from 2.2 to 2.9‰ ( 66Zn/ 64Zn) at 25 °C relative to metallic zinc. The tetrahedrally coordinated tetrachloro zinc complex, where zinc is bound exclusively to chloride, had the lowest reduced partition function ratio for a Zn(II) species (Δ 66/64Zn aq-metal ∼ 1–1.3‰ at 25 °C). Increasing the number of waters in the second shell of solvation of the above complexes led to variable results, most commonly leading to a decrease of ∼0.2 to 0.3‰ in calculated Δ 66/64Zn aq-metal at 25 °C. These estimates are useful in the interpretation of observed fractionations during the electrochemical deposition of zinc, where aqueous-metal fractionations of up to 5.5‰ are observed. The models show these are not caused by an equilibrium fractionation process. These results suggest that the redox cycle of zinc during industrial processing may be responsible for isotopically distinct reservoirs of zinc observed in polluted environments. The leaching of metallic zinc or zinc tailings from industrial sites could lead to the observed heavy signature in river systems, the magnitude of which will be reliant on the source material and the aqueous species that form.

Effects of changing solution chemistry on Fe 3+/Fe 2+ isotope fractionation in aqueous Fe–Cl solutions

The range in 56Fe/ 54Fe isotopic compositions measured in naturally occurring iron-bearing species is greater than 5‰. Both theoretical modeling and experimental studies of equilibrium isotopic fractionation among iron-bearing species have shown that significant fractionations can be caused by differences in oxidation state (i.e., redox effects in the environment) as well as by bond partner and coordination number (i.e., nonredox effects due to speciation). To test the relative effects of redox vs. nonredox attributes on total Fe equilibrium isotopic fractionation, we measured changes, both experimentally and theoretically, in the isotopic composition of an Fe 2+–Fe 3+–Cl–H 2O solution as the chlorinity was varied. We made use of the unique solubility of FeCl 4 − in immiscible diethyl ether to create a separate spectator phase against which changes in the aqueous phase could be quantified. Our experiments showed a reduction in the redox isotopic fractionation between Fe 2+- and Fe 3+-bearing species from 3.4‰ at [Cl −] = 1.5 M to 2.4‰ at [Cl −] = 5.0 M, due to changes in speciation in the Fe–Cl solution. This experimental design was also used to demonstrate the attainment of isotopic equilibrium between the two phases, using a 54Fe spike. To better understand speciation effects on redox fractionation, we created four new sets of ab initio models of the ferrous chloride complexes used in the experiments. These were combined with corresponding ab initio models for the ferric chloride complexes from previous work. At 20 °C, 1000 ln β ( β = 56Fe/ 54Fe reduced partition function ratio relative to a dissociated Fe atom) values range from 6.39‰ to 5.42‰ for Fe(H 2O) 6 2+, 5.98‰ to 5.34‰ for FeCl(H 2O) 5 +, and 5.91‰ to 4.86‰ for FeCl 2(H 2O) 4, depending on the model. The theoretical models predict ferric–ferrous fractionation about half as large (depending on model) as the experimental results. Our results show (1) oxidation state is likely to be the dominant factor controlling equilibrium Fe isotope fractionation in solution and (2) nonredox attributes (such as ligands present in the aqueous solution, speciation and relative abundances, and ionic strength of the solution) can also have significant effects. Changes in the isotopic composition of an Fe-bearing solution will influence the resultant Fe isotopic signature of any precipitates.

Isotopic fractionation of Mg 2+(aq), Ca 2+(aq), and Fe 2+(aq) with carbonate minerals

Density-functional electronic structure calculations are used to compute the equilibrium constants for 26Mg/ 24Mg and 44Ca/ 40Ca isotope exchange between carbonate minerals and uncomplexed divalent aquo ions. The most reliable calculations at the B3LYP/6-311++G(2d,2p) level predict equilibrium constants K, reported as 10 3ln ( K) at 25 °C, of −5.3, −1.1, and +1.2 for 26Mg/ 24Mg exchange between calcite (CaCO 3), magnesite (MgCO 3), and dolomite (Ca 0.5Mg 0.5CO 3), respectively, and Mg 2+(aq), with positive values indicating enrichment of the heavy isotope in the mineral phase. For 44Ca/ 40Ca exchange between calcite and Ca 2+(aq) at 25 °C, the calculations predict values of +1.5 for Ca 2+(aq) in 6-fold coordination and +4.1 for Ca 2+(aq) in 7-fold coordination. We find that the reduced partition function ratios can be reliably computed from systems as small as M ( CO 3 ) 6 10 - and M ( H 2 O ) 6 2 + embedded in a set of fixed atoms representing the second-shell (and greater) coordination environment. We find that the aqueous cluster representing the aquo ion is much more sensitive to improvements in the basis set than the calculations on the mineral systems, and that fractionation factors should be computed using the best possible basis set for the aquo complex, even if the reduced partition function ratio calculated with the same basis set is not available for the mineral system. The new calculations show that the previous discrepancies between theory and experiment for Fe 3+–hematite and Fe 2+–siderite fractionations arise from an insufficiently accurate reduced partition function ratio for the Fe 3+(aq) and Fe 2+(aq) species.

Revisit to Vapor Pressure Isotope Effects of Water Studied by Molecular Orbital Calculations

H/D and 16O/18O vapor pressure isotope effects (vpies) of water between liquid water and water vapor were reinvestigated by estimating the reduced partition function ratios, rpfrs, of water clusters, (H2O)n, with n up to 100 at the HF/6-31G(d) level of molecular orbital theory. Large water clusters with n = 80−100 resulted in a reasonable estimate of the 16O/18O vpie, which was in contrast to the one for which only small water clusters had been considered. Contrary to this, no appreciable improvement in the H/D vpie reproduction by using large water clusters was observed.

Prediction of Iron-Isotope Fractionation Between Hematite (α-Fe2O3) and Ferric and Ferrous Iron in Aqueous Solution from Density Functional Theory

Density functional theory electronic structure calculations are used to compute equilibrium constants for iron-isotope exchange among Fe2+(aq), Fe3+(aq), and hematite (alpha-Fe2O3). The hematite is represented in both bulk and surface environments. The iron-isotope fractionation between Fe2+(aq) and Fe3+(aq), determined using a range of exchange-correlation functionals and basis sets, is in good agreement with experimental measurements. The calculated reduced partition function ratio for bulk hematite is very close to previous estimates based on Mossbauer and inelastic nuclear resonance X-ray spectroscopy. However, the calculated fractionation between hematite bulk and the aqueous species Fe3+(aq) and Fe2+(aq) differs from experimental measurements carried out at the aqueous-hematite interface. We find a heavy iron enrichment trend in the order Fe2+(aq) < hematite bulk approximately hematite surface < Fe3+(aq). In contrast to experimental studies, we find a significant positive fractionation (heavy enrichment) for Fe(3+)(aq) relative to hematite, regardless of whether the hematite is represented by a bulk or a surface model. Our calculations indicate that it is unlikely that the aqueous interfacial structure of hematite is a simple termination of the bulk structure.

D/H isotopic fractionation between brucite Mg(OH) 2 and water from first-principles vibrational modeling

Isotopic fractionation between two phases can be calculated if the vibrational properties of isotopic end-members are fully characterized. We assessed a theoretical approach based on first-principles density-functional density (DFT) prediction of vibrational frequencies by comparing with spectroscopic data on isotopically substituted brucite, a model mineral for which detailed experimental data on isotopic effects and brucite–water D/H partitioning exist. The deviation from experimental values averages to less than 1% for lattice modes, and 4% for OH stretching modes. The reduced partition function ratio (RPFR) for D/H substitution in brucite was combined with experimental RPFR for water to calculate D/H fractionation factor between water and brucite. Results within the harmonic approximation systematically deviate from experimental data by + 20–25‰. RPFR is very sensitive to the frequency and isotopic shift of high frequency OH stretching modes, for which anharmonic effects are important and usually not explicitly taken introduced in calculations for minerals. The asymmetry of the O–H potential was calculated by DFT and accounts for spectroscopic measurements of the anharmonic shifts of OH stretching mode overtones and isotopic frequency ratio. Calculations of D/H fractionation introducing anharmonic corrections to the RPFR of brucite are fitted with the expression 1000ln α D/H(brucite/H 2O) = − 23.3 10 3 / T + 2.55 10 6 / T 2 − 1.51 10 9 / T 3, that yields an improved agreement with experiments at high temperature, but the deviation at 25 °C is − 30‰. Uncertainties in calculated fractionation factors can arise from dispersion effects or from DFT errors. For instance, a 1% change of the stretching mode frequencies or a 3% change of lattice mode frequencies could account for the discrepancy between model and experimental fractionation factors. The pressure dependence of brucite RPRF was calculated, and is given by 1000ln Г Pbrucite = P (− 1.005 10 3 / T + 2.18 10 6 / T 2 − 0.213 10 9 / T 3), with P in GPa. The large calculated temperature dependence of the fractionation factor suggests that a single experimental fractionation curve can be fitted to a partial consistent set of experimental data as 1000ln α D/H(brucite/H 2O) = − 27.9(30) 10 3 / T + 8.8(29) 10 6 / T 2 − 2.24(63) 10 9 / T 3. These equations allow calculating the temperature and pressure dependence of the D/H fractionation factor between brucite and water in the 300–900 K and 0.1–100 MPa range when combined with the pressure correction for water RPFR [Polyakov, V.B., Horita, J. and Cole, D.R., 2006. Pressure effects on the reduced partition function ratio for hydrogen isotopes in water. Geochimica Cosmochimica Acta, 70: 1904–1913.] for geochemical applications. The DFT model used here for estimating RPFR of the model mineral brucite can be extended to other hydrous minerals using vibrational spectroscopy as a test for the accuracy of the DFT prediction, and the brucite–water equations proposed here as a reference for determining mineral–water fractionation factors and geochemical applications.

The iron-isotope fractionation dictated by the carboxylic functional: An ab-initio investigation

The ground-state geometries, electronic energies and vibrational properties of carboxylic complexes of iron were investigated both in vacuo and under the effect of a reaction field, to determine thermodynamic properties of iron–acetates and the role of the carboxylic functional on the isotopic imprinting of this metal in metalorganic complexation. The electronic energy, zero point corrections and thermal corrections of these substances at variational state were investigated at the DFT/B3LYP level of theory with different basis set expansions and the effect of the reaction field on the variational structures was investigated through the Polarized Continuun Model. Thermochemical cycle calculations, combined with solvation energy calculations and appropriate scaling from absolute to conventional properties allowed to compute the Gibbs free energy of formation from the elements of the investigated aqueous species and to select the best procedure to be applied in the successive vibrational analysis. The best compliance with the few existing thermodynamic data for these substances was obtained by coupling the gas phase calculations at DFT/B3LYP level with the [6-31G(d,p)]–[6-31G+(d,p)] (for cations and neutral molecules – anions; respectively) with solvation calculations adopting atomic radii optimized for the HF/6-31G(d) level of theory (UAHF). A vibrational analysis conducted on 54Fe, 56Fe, 57Fe and 58Fe gaseous isotopomers yielded reduced partition function ratios which increased not only with the nominal valence of the central cation, as expected, but, more importantly, with the extent of the complexation operated by the organic functional. Coupling thermodynamic data with separative effects it was shown that this last is controlled, as expected, by the relative bond strength of the complex in both aggregation states. Through the Integral Equation Formalism of the Polarized Continuum Model (IEFPCM) the effect of the ionic strength of the solution and of a T-dependent permittivity on the energy and separative effects of the solvated metalorganic complexes were analyzed in detail. The solvent effect in the standard state (hypothetical one-molal solution referred to infinite dilution; T = 298.15 K, P = 1 bar) is a limited reduction of the separative effects of all the isotopomeric couples. With an increase in T (and the concomitant decrease in the dielectric constant of the solvent) this effect diminishes progressively.

Isotope Fractionation of Strontium in a Precipitation Reaction of SrO2

Isotope fractionation of strontium in a solid-liquid equilibrium (at room temperature) was investigated, in which possible species in the liquid phase and solid phase are assumed to be Sr(H2O)82+ and SrO2, respectively. Isotopic analysis was performed by the multi-collector thermal ionization spectrometry (MC-TIMS). Isotope enrichment factor ε86,m was observed to be, 0.35 ± 0.20‱(m = 84), 0.00 ± 0.04‱(m = 87), or −0.61 ± 0.23‱(m = 88). Reduced partition function ratios due to intermolecular vibrational frequency shifts were estimated by a quantum mechanics simulation. The estimated ε86,84 and ε86,88 showed similar values with those experimentally obtained, while calculated ε86,87 disagreed with the experimental result. These results suggest an existence of mass-independent isotope fractionation which may be due to the nuclear field shift.

Calculation of site-specific carbon-isotope fractionation in pedogenic oxide minerals

Ab initio molecular dynamics and quantum chemistry techniques are used to calculate the structure, vibrational frequencies, and carbon-isotope fractionation factors of the carbon dioxide component [CO(2)(m)] of soil (oxy)hydroxide minerals goethite, diaspore, and gibbsite. We have identified two possible pathways of incorporation of CO(2)(m) into (oxy)hydroxide crystal structures: one in which the C(4+) substitutes for four H(+) [CO(2)(m)(A)] and another in which C(4+) substitutes for (Al(3+),Fe(3+)) + H(+) [CO(2)(m)(B)]. Calculations of isotope fractionation factors give large differences between the two structures, with the CO(2)(m)(A) being isotopically lighter than CO(2)(m)(B) by approximately 10 per mil in the case of gibbsite and nearly 20 per mil in the case of goethite. The reduced partition function ratio of CO(2)(m)(B) structure in goethite differs from CO(2)(g) by <1 per mil. The predicted fractionation for gibbsite is >10 per mil higher, close to those measured for calcite and aragonite. The surprisingly large difference in the carbon-isotope fractionation factor between the CO(2)(m)(A) and CO(2)(m)(B) structures within a given mineral suggests that the isotopic signatures of soil (oxy)hydroxide could be heterogeneous.

Theoretical calculation of oxygen isotope fractionation factors in carbonate systems

Using established methods of statistical mechanical calculation and a recent compilation of vibrational frequency data, we have computed oxygen isotope reduced partition function ratios ( β values) for a large number of carbonate minerals. The oxygen isotope β values of carbonates are inversely correlated to both the mass and radius of the cation bonded to the carbonate anion but neither correlation is good enough to be used as a precise and accurate predictor of β values. There is an approximately 0.6% relative increase in the β values of aragonite per 10 kbar increase in pressure. These estimates of the pressure effect on β values are broadly similar to those deduced previously for calcite using the methods of mineral physics. In comparing the β values of our study with those derived recently from first-principles lattice dynamics calculations, we find near-perfect agreement for calcite and witherite (<0.3% deviation), reasonable agreement for dolomite (<0.9% deviation) and somewhat poorer agreement for aragonite and magnesite (1.5–2% deviation). In the system for which we have the most robust constraints, CO 2–calcite, there is excellent agreement between our calculations and experimental data over a broad range of temperatures (0–900 °C). Similarly, there is good to excellent correspondence between calculation and experiment for most other low to moderate atomic mass carbonate minerals (aragonite to strontianite). The agreement is not as good for high atomic mass carbonates (witherite, cerussite, otavite). In the case of witherite and cerussite, the discrepancy may be due, in part, to our calculation methodology, which does not account for the effect of cation mass on the magnitude of vibrational frequency shifts associated with heavy isotope substitution. However, the calculations also reveal an incompatibility between the high- and low-temperature experimental datasets for witherite and cerussite. Specifically, the shapes of fractionation factor versus 1/ T 2 curves in the calcite–witherite and calcite–cerussite systems do not conform to the robust constraints on the basic shape of these curves provided by theory. This suggests that either the high- or low-temperature datasets for both minerals is in error. Dolomite–calcite fractionation factors derived from our calculations fall within the wide range of fractionations for this system given by previous experimental and natural sample studies. However, our compilation of available low-temperature (25–80 °C) experimental data reveal an unusual temperature dependence of fractionations in this system; namely, the data indicate an increase in the magnitude of fractionations between dolomite (or proto-dolomite) and calcite with increasing temperature. Such a trend is incompatible with theory, which stipulates that fractionations between carbonate minerals must decrease monotonically with increasing temperature. We propose that the anomalous temperature dependence seen in the low-temperature experimental data reflect changes in the crystallinity and degree of cation ordering of the dolomite phase over this temperature interval and the effect these changes have on the vibrational frequencies of dolomite. Similar effects may be present in natural systems at low-temperature and must be considered in applying experimental or theoretical fractionation data to these systems. In nearly all cases, carbonate mineral–calcite fractionation factors given by the present calculations are in as good or better agreement with experimental data than are fractionations derived from semi-empirical bond strength methods.

Counterion effects on the isotopic properties of the Cl− and Li+ ions in aqueous solutions

The reduced partition function ratios between isotopic forms (β-factors) were calculated by the ab initio RHF/6-311++G**(3df, 3p) and MP2/6-311++G**(3df, 3p) quantum-chemical methods for hydrated chloride ion and ion pair hydrates NaCl·nH2O and LiOH · nH2O. The influence of the Na+ cation on the β-factor value and the chlorine isotope separation factor in the precipitation of NaCl from concentrated aqueous solutions was found to be substantial. At the same time, the presence of OH− counterions had no noticeable effect on the β-factor of the hydrated Li+ cation.

Ab initio calculations for equilibrium fractionations in multiple sulfur isotope systems

Ab initio calculations of vibrational frequencies ( ν), reduced partition function ratios ( β) of various gaseous and aqueous sulfur-bearing compounds, and isotopic fractionation factors ( α) between them were performed on the multiple isotope systems using the Hartree-Fock (HF) and the Becke three-parameter Lee-Yang-Parr (B3LYP) density functional methods. Based on the comparison of the geometry, vibrational frequencies, and fractionation factors with those determined experimentally, the HF/6-31G(d) was chosen as the optimal method for the sulfur isotope systems. The results of computation for multiple sulfur fractionation factors indicate that the mass-dependent relationships, i.e., ( 33 α − 1) / ( 34 α − 1) and ( 36 α − 1) / ( 34 α − 1) values, converge only at high temperature (e.g., T > 500 °C) to 0.515 and 1.89, respectively, which are the values previously derived from the Bigeleisen-Mayer approximations. However, they significantly deviate from these values with decreasing temperature, and depending on the sulfur species pairs that are involved in equilibrium isotope exchange reactions. The ( 33 α − 1) / ( 34 α − 1) values and the ( 36 α − 1) / ( 34 α − 1) values range from 0.505 to 0.517 and from 1.88 to 1.96, respectively, at 0 °C. An isotopic crossover relationship, in which the fractionation factor changes from > 1.0 to < 1.0 (or from < 1.0 to > 1.0) about a specific temperature (i.e., crossover temperature), was observed in several species couples (e.g., S 2–H 2S, S 8–H 2S, CS 2–H 2S, SO 3–SO 4 2− (aq)). Since the crossover temperatures vary among different isotope molecules, large changes in the ( 33 α − 1) / ( 34 α − 1) and ( 36 α − 1) / ( 34 α − 1) values (e.g., > 1 to < 0) also occur near the crossover temperatures. The effects of solvation on the fractionation factors were investigated using the PCM models. The dielectric constant ( ε) of solvent, which is the dominant solvent characteristic, affects the fractionation factors between gaseous and aqueous species (e.g., SO 2(g) and SO 2(aq)), but the effect levels out for ε > 20. Among the solvent models, the IEF-PCM model yields the greatest fractionation factors between gaseous and aqueous species. The effects of anharmonicities were also investigated for H 2S and HS −. When anharmonicity is corrected without scaling, the DFT methods, rather than the HF methods, yield frequency values closer to those determined spectroscopically. Our results show that anharmonicity can significantly affect the reduced partition function ratios (0.5‰ for the 1000( 34 β H 2S − 1) value and 0.1‰ for the 1000( 34 β HS − − 1) value) and fractionation factors; however, it has little effect on the multiple isotope fractionation factor ratios. Overall, temperature, harmonic frequency, and mass of atoms bonded to sulfur ( M) are the three important parameters affecting the reduced partition function ratios, fractionation factors, and mass-dependent relationships. By decreasing temperature and increasing frequency and mass, the 1000( 34 β − 1) values increase and the ( 33 β − 1) / ( 34 β − 1) values decrease (i.e., deviate from “expected” 0.515). The ( 33 β − 1) / ( 34 β − 1) values converge to 0.516 only at higher temperature (e.g., T > 600 °C), lower frequency ( ω e < 500 cm − 1 ), and smaller mass ( M < 7 amu) and become as low as 0.503 at T = 50 °C, ω e = 2500 cm − 1 , and M = 56 amu.

Modeling the effects of bond environment on equilibrium iron isotope fractionation in ferric aquo-chloro complexes

Four or five sets of ab initio models, including Unrestricted Hartree Fock (UHF) and hybrid Density Functional Theory (DFT) are calculated for each species in a series of aqueous ferric aquo-chloro complexes: Fe ( H 2 O ) 6 3 + , FeCl ( H 2 O ) 5 2 + , FeCl 2 ( H 2 O ) 4 + , FeCl 3(H 2O) 3, FeCl 3(H 2O) 2, FeCl 4 - , FeCl 5H 2O 2−, FeCl 5 2 - , FeCl 6 3 - ) in order to determine the relative isotopic fractionation among the complexes, to compare the results of different models for the same complexes, to examine factors that influence the magnitude of the isotopic fractionation, and to compare bond-partner-driven fractionation with redox-driven fractionation. Relative to Fe ( H 2 O ) 6 3 + , all models show a nearly linear decrease in 56Fe/ 54Fe as the number of Cl − ions per Fe 3+ ion increases, with slopes of −0.8‰ to −1.0‰ per Cl − at 20 °C. At 20 °C, 1000 ln β ( β = 56Fe/ 54Fe reduced partition function ratio relative to a dissociated Fe atom) values range from 8.93‰ to 9.73‰ for Fe ( H 2 O ) 6 3 + , 8.04–9.12‰ for FeCl ( H 2 O ) 5 2 + , 7.61–8.73‰ for FeCl 2 ( H 2 O ) 4 + , 7.14–8.25‰ for FeCl 4 - , and 3.09–4.41‰ for FeCl 6 3 - . The fractionation between Fe ( H 2 O ) 6 3 + and FeCl 4 - ranges from 1.5‰ to 2.6‰, depending on the model; this is comparable in magnitude to fractionation effects due to Fe 3+/Fe 2+ redox reactions. β values from the UHF models are consistently higher than those from the hybrid DFT models. Isotopic fractionation is shown to be sensitive to differences in ligand bond stiffness (above), coordination number, bond length, and the frequency of the asymmetric Fe–X stretching vibrational mode, as predicted by previous theoretical studies. Complexes with smaller coordination numbers have higher 1000 ln β (7.46‰, 5.25‰, and 3.48‰ for FeCl 4 - , FeCl 5 2 - , FeCl 6 3 - , respectively, from the B3LYP/6-31G(d) model). Species with the same number of chlorides but fewer waters also show the effect of coordination number on 1000 ln β: (7.46‰ vs. 7.05‰ for FeCl 3(H 2O) 2 vs. FeCl 3(H 2O) 3 and 5.25‰ vs. 4.94‰ for FeCl 5 2 - vs. FeCl 5H 2O 2− with the B3LYP/6-31G(d) model). As more Fe–Cl bonds substitute for Fe–OH 2 bonds (with a resulting decrease in β), the lengths of the Fe–Cl bonds and the Fe–O bonds increase. Preliminary modeling of Fe ( H 2 O ) 6 2 + shows an Fe 3+/Fe 2+ fractionation of 3.2‰ for the B3LYP/6-31G(d) model, in agreement with previous studies. The addition of an explicit outer hydration sphere of 12 H 2O molecules to models of Fe ( H 2 O ) 6 3 + improves agreement with measured vibrational frequencies and bond lengths; 1000 ln β increases by 0.8–1.0‰. An additional hydration sphere around FeCl 4 - increases 1000 ln β by only 0.1‰. Isotopic fractionations predicted for this simple system imply that ligands present in an aqueous iron environment are potentially important drivers of fractionation, and suggest that significant fractionation effects are likely in other aqueous systems containing sulfides or organic ligands. Fractionation effects due to both speciation and redox must be considered when interpreting iron isotope fractionations in the geological record.

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.