Equilibrium isotopic fractionation, calculation

Equilibrium stable isotope fractionation is a quantum-mechanical phenomenon, driven mainly by differences in the vibrational energies of molecules and crystals containing atoms of differing masses (Urey 1947). In fact, a list of vibrational frequencies for two isotopic forms of each substance of interest—along with a few fundamental constants—is sufficient to calculate an equilibrium isotope fractionation with reasonable accuracy. A succinct derivation of Urey s formulation follows. This theory has been reviewed many times in the geochemical... 

Isotope (H (deuterium), discovered by Urey et al. (1932), is usually denoted by symbol D. The large relative mass difference between H and D induces significant fractionation ascribable to equilibrium, kinetic, and diffusional effects. The main difference in the calculation of equilibrium isotopic fractionation effects in hydrogen molecules with respect to oxygen arises from the fact that the rotational partition function of hydrogen is nonclassical. Rotational contributions to the isotopic fractionation do not cancel out at high T, as in the classical approximation, and must be accounted for in the estimates of the partition function ratio /. 

Calculations of equilibrium isotope fractionation factors have been particularly successful for gases. Richet et al. (1977) calculated the partition function ratios for a large number of gaseous molecules. They demonstrated that the main source of error in the calculation is the uncertainty in the vibrational molecular constants. 

Additionally, in order to accurately calculate F values from exchange data we need to account for the salt effect on the oxygen and hydrogen equilibrium isotope fractionation between minerals and fluids, based on results reported by Horita et al. (1993a,b 1995) and Chacko et al. (this volume). These results indicate the mineral - salt fractionation may be 0.6 to 1 per mil smaller than the mineral-pure water fractionation at 300°C for a 5 m NaCl solution (Horita et al. 1995). This type of data will play an important role in the ultimate accuracy of the rate constants calculated from partial exchange experiments involving minerals and salt solutions. 

A small amount of theoretical work has been published on the fractionation of uranium isotopes. As mentioned above, Schauble [64] demonstrated that equilibrium isotope fractionation between species of the heaviest elements is not dominated by differences in bond vibrational frequencies, as they are for lighter elements, but by the nuclear field shift effect. This effect is due to interactions between electron shehs, espedahy s shells, that have high electron density near the very large nuclei of heavy atoms. The heavier isotopes partition into those species with fewer s electrons or in which s electrons are shielded by more p, d, or f electrons. Schauble [64] presented calculations for various ojddation states and species of T1 and Hg and the same general conclusions apply to U. Calculations for uranium species were presented at a conference by Schauble ]73]. The largest fractionations are predicted to occur when U(IV) and U(VI) species equilibrate, with values of au(iv) u(vi) as large as 0.0012 at 273 K [Au(i iu(vi) l-2%o at 0°C]. U(IV) has two 5f electrons that apparently shield s electrons from the isotopically... 

This review will introduce basic techniques for calculating equilibrium and kinetic stable isotope fractionations in molecules, aqueous complexes, and solid phases, with a particular focus on the thermodynamic approach that has been most commonly applied to studies of equilibrium fractionations of well-studied elements (H, C, N, O, and S) (Urey 1947). Less direct methods for calculating equilibrium fractionations will be discussed briefly, including techniques based on Mossbauer spectroscopy (Polyakov 1997 Polyakov and Mineev 2000). 

Fractionations are typically very small, on the order of parts per thousand or parts per ten thousand, so it is common to see expressions like 1000 ln(a) or 1000 (a-l) that magnify the difference between a and 1. a =1.001(1000 [a-l] = 1) is equivalent to a 1 per mil (%o) fractionation. Readers of the primary theoretical literature on stable isotope fractionations will frequently encounter results tabulated in terms of P-factors or equilibrium constants. For present purposes, we can think of Pjjh as simply a theoretical fractionation calculated between some substance JiR containing the elementX, and dissociated, non-interacting atoms ofX. In the present review the synonymous term Uxr-x is used. This type of fractionation factor is a convenient way to tabulate theoretical fractionations relative to a common exchange partner (dissociated, isolated atoms), and can easily be converted into fractionation factors for any exchange reaction ... 

Figure 7. Using a theoretically determined equilibrium fractionation to interpret measured isotopic fractionations in a hypothetical mineral-solution system. Three sets of data are shown. The theoretical equilibrium fractionation for this system is indicated by the gray arrow. The first set of data, indicated by circles, closely follow the calculated fractionation, suggesting a batch equilibrium fractionation mechanism. The second set of data (stars) is displaced from the theoretical curve. This may either indicate a temperature-independent kinetic fractionation superimposed on an equilibrium-like fractionation, or that the theoretical calculation is somewhat inaccurate. The third set of data (crosses) shows much greater temperature sensitivity than the equilibrium calculation this provides evidence for a dominantly non-equilibrium fractionation mechanism. For the first data set, the theoretical fractionation curve can be used to extrapolate beyond the measured temperature range. The second data set can also be extrapolated along a scaled theoretical curve (Clayton and Kieffer 1991).

Figure 15. Illustration of possible variations in isotopic fractionation between Fe(III),q and ferric oxide/ hydroxide precipitate (Aje(,n),q.Fenicppt) and precipitation rate. Skulan et al. (2002) noted that the kinetic AF (ni)aq-Feiricppt fractionation produced during precipitation of hematite from Fe(III), was linearly related to precipitation rate, which is shown in the dashed curve (precipitation rate plotted on log scale). The most rapid precipitation rate measured by Skulan et al. (2002) is shown in the black circle. The equilibrium Fe(III),-hematite fractionation is near zero at 98°C, and this is plotted (black square) to the left of the break in scale for precipitation rate. Also shown for comparison is the calculated Fe(III),q-ferrihydrite fractionation from the experiments of Bullen et al. (2001) (grey diamond), as discussed in the previous chapter (Chapter lOA Beard and Johnson 2004). The average oxidation-precipitation rates for the APIO experiments of Croal et al. (2004) are also noted, where the overall process is limited by the rate constant ki. As discussed in the text, if the proportion of Fe(III),q is small relative to total aqueous Fe, the rate constant for the precipitation of ferrihydrite from Fe(III), (Ai) will be higher, assuming first-order rate laws, although its value is unknown.

Seo, J.H., Lee, S.K., Lee, I. 2007. Quantum chemical calculations of equilibrium copper (I) isotope fractionations in ore-forming fluids. Chemicai Geoiogy, 243, 225-237. Wall, A., Heaney, P., Mathur, R. 2007. Insights into copper isotope fractionation during the oxidative phase transition of chalcocite, using time-resolved synchrontron X-ray diffraction. Geochimica Cosmochimica Acta, 77, A1081. 

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