Kinetic thermodynamic isotope fractionation

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). 

Figure 1.7 Variation of the potential energy as a function of the reaction coordinate. The potential energy curves for the reagent, activated complex, and reaction product containing the element considered are indicated, together with the vibrational energy levels, which vary according to the mass of the isotope present. Differences in AC govern thermodynamic isotope fractionation, whereas differences in govern kinetic isotope effects. The latter occur when equilibrium cannot be reached.

When isotopes are fractionated kinetically during chemical reactions, the isotope ratio shift of the reaction products relative to the reactants often depends on reaction mechanisms and rates. This contrasts with isotopic fractionations between phases in isotopic equilibrium, where the isotopic differences are thermodynamic quantities and thus do not depend on reaction mechanisms or rates. In this section, we briefly review the well-developed theory for kinetic isotope effects that appears in the S isotope literature. This background serves as a guide for interpreting and predicting Se and Cr isotope systematics. 

Elemental and isotopic fractionations by evaporation of silicate liquids, in particular limiting circumstances, can be simulated by equilibrium calculations, provided that an adequate thermodynamic model of the melt is available. In this approach, a particular starting temperature, pressure, and initial composition of condensed material are chosen and the gas in equilibrium with the melt is calculated from thermodynamic data. The gas is then removed from the system and equilibrium is recalculated. Repeated small steps of this sort can simulate the kinetic behavior during vacuum evaporation (i.e., the limit of fast removal of the gas relative to the rate it is generated by evaporation). This approach has been taken by Grossman et al. (2000, 2002) and Alexander (2001, 2002). 

In a strict sense, the term Rayleigh fractionation should only be used for chemically open systems where the isotopic species removed at every instant are in thermodynamic and isotopic equilibrium with those remaining in the system at the moment of removal. Furthermore, such an ideal Rayleigh distillation is one where the reactant reservoir is finite and well mixed, and does not re-react with the product (Clark and Fritz, 1997). However, the term Rayleigh fractionation is commonly applied to equilibrium closed systems and kinetic fractionations as well because the situations may be computationally identical. Isotopic fractionations are strongly affected by whether a system is open or closed. 

The thermodynamic and kinetic isotope effects which are always associated with a transformation (i.e. distillation, chemical reaction...), induce varying degrees of isotopic fractionation of the product with respect its substrate. However, these differences in isotopic composition may be very small and accurate and precise quantitation methods are required if NMR Spectroscopy or Mass Spectrometry are to be used for isotopic analysis. 

Isotopic fractionation can occur during chemical, physical, and biological processes. The two main mechanisms that cause isotopic fractionation are the kinetic isotope effect, which is produced by differences in reaction rates, and the thermodynamic isotope effect, which relates to the energy state of a system [1,9,13]. Galimov [13] describes equations for calculating kinetic and thermodynamic isotopic effects. 

Calculation of the mass fractionation factor p from experimental data provides insight into whether thermodynamic or equilibrium as opposed to kinetic effects are at the origin of the mass-dependent isotope fractionation, although often the fractionation is shown to be of mixed origin. For example, Wombacher et al. 

The understanding of isotope effects on chemical equilibria, condensed phase equilibria, isotope separation, rates of reaction, and geochemical and meteorological phenomena, share a common foundation, which is the statistical thermodynamic treatment of isotopic differences on the properties of equilibrating species. For that reason the theory of isotope effects on equilibrium constants will be explored in considerable detail in this chapter. The results will carry over to later chapters which treat kinetic isotope effects, condensed phase phenomena, isotope separation, geochemical and biological fractionation, etc. 

Similarly the argument from the er-deuterium isotope effect depends on the value one assumes for the fractionation factor for the transition state at r = 1. The simple geometric mean shown in Fig. 20 can only be a plausible guess. Again, for the Hammett relations, one has to make assumptions about the values of p for the transition state at r = 1 (Fig. 23). Furthermore there are also assumptions in separating the kinetic and thermodynamic contributions. 

Volatility fractionation of elemental abundances and isotopic composition has traditionally been considered in terms of condensation and evaporation, under the assumption that condensation is a slow process controlled by thermodynamic equilibrium, and evaporation is a rapid process controlled by kinetic factors. We see no compelling reason to accept this distinction between evaporation and condensation. However, distinguishing between equilibrium and nonequilibrium conditions is crucial, and thus we have chosen to organize this section in terms of equilibrium and kinetically controlled processes. 

Most common is the process of mass-dependent fractionation, in which the stable isotope ratio is altered as the consequence of physical processes differentially affecting atoms or molecules of different mass. Isotopes are fractionated relative to one another according to thermodynamic, kinetic, and diffusion processes. A simple example is the way in which oxygen isotopes in water molecules are fractionated during the process of evaporation. Water molecules containing the lower mass isotope leO are more likely to become water vapor than those containing the higher mass isotope lsO. Hence the water vapor is enriched in isotope leO and the liquid water is enriched in isotope lsO. 

Fractionations can also occur between two chemical species at equilibrium. The basis for equilibrium fractionations is thermodynamic and, as with kinetic fractionations, is related to mass-dependent differences in bond energies between light and heavy isotopes. The generalized isotope equilibrium between two chemical species is presented in Equation (3). 

The distribution of the minor isotopes relative to is governed by thermodynamic and kinetic fractionation processes, in addition to the radioactive decay associated with... 

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