Equilibrium fractionation

Mass conservation and equilibrium fractionation together require that... 

For Bcarb we take isofractionation at Bhco3"Bco2 to be 10%o, which is close to that of the measured equilibrium fractionation. Embedding this in the complete model gives ... 

As for carbonate measurements, the model confirms that bioapatite-carbonate should exhibit a 8 C value offset from the average-ingested-diet by an amount equal to the isotopic equilibrium fractionation between carbonate and gaseous CO2. 

Kase and Horiuchi (1996) obtained a large number of analytical data on sphalerites from sixteen Besshi-type deposits, mainly at Besshi and its vicinity, Hitachi, and Shimokawa. They revealed that (1) the Mn/Zn and Co/Zn ratios of sphalerite may have markedly increased during contact metamorphism, while the Cd/Zn ratios remained unchanged (2) the Emco/ lwzn (2/n total dissolved concentration in ore fluids) and Emco/S/wzn ratios in the initial ore solutions responsible for the mineralizations at Besshi which was calculated ba.sed on the equilibrium fractionation model between hydrothermal solution and sphalerite and analytical data on sphalerites are quite similar to the ratios of hydrothermal solutions at EPR 21 °N (3) however, these ratios for the Hitachi solutions are very low and different from those of the Besshi-subtype solution. 

If nonstoichiometric amounts of reactants are present in the initial system, the presence of excess reactants tends to increase the equilibrium fractional conversion of the limiting re-... 

The amount of particles determine the quantity of decay products that stay in the air (equilibrium fraction, F) and the fraction of activity associated with the "unattached or ultrafine mode of the size distribution (fDot) These decay products are certainly harmful at high concentrations but we cannot yet detect the effects at normal levels because the vast majority of lung cancer death are due to smoking. Models predict that potentially 9000 lung cancer deaths per year in the United States are due to indoor radon. Methods are currently available and new methods are being developed and tested for lowering the levels of radon in indoor air. 

The possibility (Budd et al., 1995c) that non-equilibrium fractionation of tin isotopes in recycled bronzes may allow the degree of recycling to be established (as discussed above) begins to assume great significance. 

Equilibrium fractionation. A simple fractionation law, called the linear law (e.g., Hofmann, 1971), relates the measured and natural isotopic ratios through a function f (Am/) of the mass difference Am/ = m,—m, between the isotopes defining the ratios... 

Since the denominator falls in the range Dt to 1, concentration in the solid is closer to that of the liquid away from the interface than equilibrium fractionation would require. Again, disequilibrium partitioning during crystal growth decreases solid-liquid chemical fractionation. 

The rate equation is presented in equation (7). It can also be derived from v = k2[MS][H2] when we substitute for [MS] the equilibrium fraction of the catalyst that occurs as MS. Note that we do not fulfil the Bodenstein approximation, but our general formula (6) is still valid. If the free eneigy of... 

Figure 4. Illustration of mass-dependent fractionation of Mg isotopes, cast in terms of 5 values. 5 Mg and 5 Mg values based on Mg/ Mg and Mg/ Mg ratios, respectively. A common equilibrium fractionation model, as defined by exponential relations between a values (fractionation factors) for different isotope ratios, is shown in the gray line. A simple linear relation, where the slope is proportional to the mass difference of the isotope pair, is shown in the black line. Additional mass-dependent fractionation laws may be defined, and all are closely convergent over small ranges (a few per mil) in isotope compositions at 5 values that are close to zero.

Figure 8. Example of apparent closed-system equilibrium fractionation, where Mo in solution is sorbed to Mn oxides (Barling and Anbar 2004). The 6 Mo values of the Mo remaining in solution during sorption follow die linear trends that are consistent widi closed-system equilibrium fractionation where isotopic equilibrium is continuously maintained between Mo in solution and diat sorbed to die Mn oxides. Three aqueous-solid pairs (shown widi tie lines) are consistent with this interpretation. The isotopic data cannot be ejqilained dirough a Rayleigh process, where die product of die reaction (sorbed Mo) is isolated from isotopic exchange widi aqueous Mo.

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). 

Rules (2) and (3) imply that large equilibrium fractionations are most likely to occur at low temperatures between substances with markedly different oxidation states, bond partners, electronic configurations, or coordination numbers. 

A more general expression for the equilibrium fractionation of isotopes X and " yX, applicable to all diatomic and larger molecules is ... 

This relation correctly predicts that most equilibrium stable isotope fractionations are inversely proportional to the square of absolute temperature, and is the basis of equilibrium fractionation rule (1). A detailed derivation of the Bigeleisen and Mayer model has been presented in an earlier review (Criss 1991). 

It should be clear from the preceding discussions that practical application of equilibrium stable isotope fractionation theory often requires a certain amormt of simplification of complex and poorly studied systems. Given this reality, one should not be surprised to find that theoretically determined equilibrium fractionations rarely achieve accuracies approaching the nominal precisions of measurements made with modem analytical techniques. It should... 

Kinetic fractionations can occur when there is incomplete isotopic exchange between the different phases present in a system. A thorough introduction to kinetic stable isotope fractionation theory is unfortunately beyond the scope of the present review. Flowever, it is useful to include a brief discussion of some basic aspects, particularly in comparison to equilibrium fractionation theory. A simple example of kinetic fractionation is the evaporation of a liquid water droplet into a vacuum, in this example FljO molecules entering the gas phase are physically removed from the vicinity of the droplet, so there is no chance for isotopic equilibration between vapor-phase molecules and the residual liquid. Isotopic fractionation in this case is determined by a one-way reaction path, and will not, in general, be the same as the fractionation in a system where vapor-phase molecules are able to equilibrate and exchange with the liquid. In other reactions, isotopic exchange is limited by an energy barrier—an... 

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).

Patel A, Price GD, Mendelssohn MJ (1991) A computer simulation approach to modeling the structure, thermodynamics and oxygen isotope equilibria of silicates. Phys Chem Min 17 690-699 Polyakov VB (1997) Equilibrium fractionation of the iron isotopes estimation from Mossbauer spectroscopy data. Geochim Cosmochim Acta 61 4213-4217... 

APPENDIX ANNOTATED BIBLIOGRAPHY OF THEORETICAL EQUILIBRIUM FRACTIONATIONS... 

This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

In this study the authors develop simplified equations relating equilibrium fractionations to mass-scaling factors and molecular force constants. Equilibrium isotopic fractionations of heavy elements (Si and Sn) are predicted to be small, based on highly simplified, one-parameter empirical force-field models (bond-stretching only) of Sip4, [SiFJ, SnCl4, and [SnCl,] -. 

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