Rayleigh fractionation model

Figure 4.5 Cr as a function of the amount of Cr(VI) remaining in a batch slurry experiments with estuarine sediment. Line gives a Rayleigh fractionation model, with s = 3.4%o. Data from Ellis et al. (2002).

Hollister L. S. (1966) Garnet zoning an interpretation based on the Rayleigh fractionation model. Science 154, 1647-1651. Hollister L. S. (1969) Contact metamorhpism in the Kwoiek Area of British Columbia an end member of the metamorphic process. Geol. Soc. Am. Bull. 80, 2465 - 2494. 

Hollister L. S. (1966) Garnet zoning an interpretation based on the Rayleigh fractionation model. Science 154, 1647-1651. 

Fractionating effects of the escape process can now be calculated analytically if specific assumptions are made about the time dependence of the major (constituent 1) inventory Ni— that it is either replenished as fast as it escapes (constant inventory model), or is lost without replenishment along with the minor atmospheric species (Rayleigh fractionation model). In both cases the inventories N2 of minor species, here the noble gases, are assumed to be in the atmosphere at to and are lost without replenishment during the escape episode. For Rayleigh fractionation, adopted for this discussion. Equations (2), (3), and (7) and the definitions Fi = -dNi/dt and F2 = -dN2/dt may be combined and integrated, in the limit of Xi = 1, mc > m2 > mi, and mc > m2, to yield... 

Figure 10. Fe isotope compositions for total aqueous Fe (Fe,(,T) and ferrihydrite (FH) precipitate and aqueous Fe-ferrihydrite fractionations from the batch oxidation and precipitation experiment of Bullen et al. (2001). (A) Measured S Fe values from Bullen et al. (2001), compared to simple Rayleigh fractionation (short-dashed lines, noted with R ) using 10 1naFe.,-FH = 0.9%o, as well as the two-step re-equilibration model discussed in the text (i.e., Eqn. 12), shown in solid gray lines for the aqueous Fe and ferrihydrite components the predicted 5 Fe value for Fe(III), is shown in the heavy dashed line, which reflects continual isotopic equilibrium between Fe(II), and Fe(III),(. Note that in the experiment of Bullen et al. (2001), aqueous Fe existed almost entirely as Fe(II),(,. (B) Measured fractionation between total aqueous Fe and ferrihydrite precipitate, as measured, and as predicted from simple Rayleigh fractionation (black dashed line) and the two-step model where isotopic equilibrium is maintained between aqueous Fe(II),q and Fe(III),q (solid gray line).

The data of Croal et al. (2004) may also be interpreted to reflect a two-step proeess, where a -2.9%o fractionation occurs between Fe(ll)aq and Fe(lll)aq, accompanied by a +1.4%o fractionation between Fe(lll)aq and ferrihydrite upon precipitation, produces a net fractionation of-1.5%0. When cast in terms of common mechanistic models for separation of solid and liquid phases such as Rayleigh fractionation, it becomes clear that the two-step model produces essentially the same fractionation trend as a single -1.5%o fractionation step between Fe(ll)aq and ferrihydrite if the Fe(lll)aq/Fe(ll)aq ratio is low (Fig. 14). As the Fe(lll)aq/Fe(ll)aq ratio inereases, however, the calculated net Fe(ll)aq-ferrihydrite fractionation in the two-step model deviates from that of simple Rayleigh fractionation (Fig. 14). Unfortunately, the scatter in the data reported by Croal et al. (2004), which likely reflects minor contamination of Fe(ll)aq in the ferrihydrite precipitate, prevents distinguishing between these various models without eonsideration of additional factors. 

As discussed in Section 4.08.3.3, Rayleigh-type fractionation models cannot account for the complexity of large convective systems, such as those occurring in the tropics, for which 8p depends on precipitation amount rather than temperature. Despite such limitations, they are able to reproduce the basic behavior of 5D and 5 0 in precipitation, at least in mid- and high latimdes, where large convective systems do not dominate precipitation production. 

In a Rayleigh distillation model the isotopic ratio of a chemical species (R) is related to its isotopic ratio before any is transformed (Ro), to the fractionation factor associated with the transformation (aA-n), and to the fraction of the original chemical species remaining at any particular extent of evolution of the system (f) (Eqn. 15) ... 

Figure 10. This graph shows how the isotopic composition of sulfate and sulfide evolve in a closed system where sulfide is produced by sulfate reduction with an a of 1.040 and an initial isotopic composition of sulfate of 20 %o. Sulfide is assumed to accumulate quantitatively as sulfate becomes depleted. The parameter /so4 expresses the remaining fraction of the original sulfate in the system. A Rayleigh distillation model was used to calculate these results. See text for details.

Figure 11. This graph shows how both the concentrations and isotopic compositions of snlfate and snffide evolve with sediment depth in the sapropelic sediments of Mangrove Lake, Bermnda. Whereas the isotopic compositions of snlfate and sulfide appeared to evolve as in a Rayleigh distillation model (see Fig. 10), snch a model is inappropriate for determining fractionations. This is because marine sediments are open with respect to the exchange of chemical species. See text for details. Data are replotted from Canfield et al. (1998b).

Figure 8. (A) A water column is divided into fifty equal unit cells and it is assumed there is no liquid or dissolved gas between cells. Each cell originally has the noble gas content of air-equilibrated water and all calculated Ne/Ar ratios are normalized to this value to obtain a fractionation factor F. The column temperature is taken to be 325 K, which for pure water gives Knc = 133245 atm and Kaf= 55389 atm. A gas bubble of constant volume is passed sequentially through the column, equilibrium assumed to occur in each water cell and the Ne and Ar partitioned into the respective gas and water phases (Eqn. 16). The evolution of the Ne/Ar ratio in the gas bubble (bold) and each water phase increment (Faint) is shown for different gas/water volume ratios, Vg/Vi. The gas bubble Ne/Ar ratio approaches the maximum fractionation value predicted for a gas/water phase equilibrium where as Vg/Vi -> 0, F Knc/Kat. The cell Vg/Vi ratio only determines the rate at which this hmit is approached. (B) The same water column with a fixed cell Vg/Vi ratio of 0.01. n subsequent bubbles are passed through the column and the He/Ne distribution between phases calculated at each stage. The gas bubble Ne/Ar ratio evolution for n = 1, 10, 20 and 30 is shown in bold, together with the residual Ne/Ar in the water colunm cells (faint lines). All gas bubbles approach the limit imposed by the phase equilibrium model. The water phase is fractioned in the opposite sense and is fractionated in proportion to the magnitude of gas loss following the Rayleigh fractionation law (Eqn. 24).

Figure 1 Rayleigh distillation model showing the effects of evaporation and precipitation on the <5 0 values in the vapor and liquid phases. The initial conditions are a temperature of 25°C and (5 Owater value of 0%o. This model also assumes that it is a closed system, meaning that water vapor is not added once the cloud moves away from the source regions. As clouds lose moisture, fractionation during the condensation further lowers the <5 Owater Value in the water vapor.

Mq is the initial mass of the magma chamber and Mi is the mass of liquid so that the ratio Mi /Mq is equivalent to the term F — the fraction of melt remaining and used in equilibrium and fractional crystallization models /is the fraction of magma allocated to the solidification zone which is returned to the magma chamber. Equation [4.21] is similar in form to the Rayleigh fractionation equation but with a more complex exponent. 

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