How to Calculate Isotope Ratios

Equations for the rates of change of individual isotopes in a reservoir are not essentially different from the equations for the rates of change of chemical species. Isotopic abundances, however, are generally expressed as ratios of one isotope to another and, moreover, not just as the ratio but also as the departure of the ratio from a standard. This circumstance introduces some algebra into the derivation of an isotopic conservation equation. It is convenient to pursue this algebra just once, as I shall in this section, after which all isotope simulations can be formulated in the same way. I shall use the carbon isotopes to illustrate this derivation, but the 

The most abundant isotope of carbon has a mass of 12 atomic mass units, 12C. A less abundant stable isotope is 13C. And much less abundant is the radioactive isotope t4C, also called radiocarbon. It is convenient to express the abundances of these rare isotopes in terms of ratios of the number of atoms of the rare isotope in a sample to the number of atoms of the abundant isotope. We call this ratio r, generally a very small number. To arrive at numbers of convenient magnitude, it is conventional to express the ratio in terms of the departure of r from the ratio in a standard, which I call. v, and to express this departure in parts per thousand of s. Thus the standard delta notation is 

813C or 14C = (r — s)/s 1000 per mil This equation can be rearranged to derive an expression for the ratio r = s (1 + dell 1000) where del denotes 813C or 14C. 

Suppose that a given reservoir contains m atoms of the abundant carbon isotope and has an isotopic composition of del. Suppose that carbon of composition deli is supplied to the reservoir at rate fi while carbon of composition delo is removed from the reservoir at rate fo. The equation for the rate of change of the number of atoms of the abundant isotope in the reservoir is 

The number of atoms of the rare isotope in the reservoir is r m, and the rate of change of this number is 

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Let us consider the two isotopes 148Nd and 150Nd and measure the isotope composition of the spiked mixture. Using the subscripts sa, sp, and m for sample, spike, and mixture, it was shown in Section 1.3 how to calculate the spiking ratio r... 

Isotope effects on equilibria have been formulated earlier in this chapter in terms of ratios of (s2/si)f values, referred to as reduced isotopic partition function ratios. From Equation 4.80, we recognize that the true value of the isotope effect is found by multiplying the ratio of reduced isotopic partition function ratios by ratios of s2/si values. Using Equation 4.116 one now knows how to calculate s2/si from ratios of factorials. Note well that symmetry numbers only enter when a molecule contains two or more identical atoms. Also note that at high temperature (s2/si)f approaches unity so that the high temperature equilibrium constant is the symmetry number factor. 

In this chapter I explained how isotope ratios may be calculated from equations that are closely related, but not identical, to the equations for the bulk species. Extra terms arise in the isotope equations because isotopic composition is most conveniently expressed in terms of ratios of concentrations. I illustrated the use of these equations in a calculation of the carbon isotopic composition of atmosphere, surface ocean, and deep ocean and in the response of isotope ratios to the combustion of fossil fuels. As an alternative application, I simulated the response of the carbon system in an evaporating lagoon to seasonal changes in biological productivity, temperature, and evaporation rate. With a simulation like the one presented here it is quite easy to explore the effects of various perturbations. Although not done here, it would be easy also to examine the sensitivity of the results to such parameters as water depth and salinity. 

Now I shall show how the nearly diagonal system can easily be modified to incorporate additional interacting species. In this illustration I shall add the calculation of the stable carbon isotope ratio specified by 813C. All of the parameters that affect the concentrations of carbon and calcium are left as in program SEDS03, so that the concentrations remain those that were plotted in Section 8.4. I shall not repeat the plots of the concentrations but present just the results for the isotope ratio. 

Like the climate system of Chapter 7, this system yields nonzero elements of the sleq array only close to the diagonal. Much computation can be eliminated by modifying the solution subroutines, SLOPER and GAUSS. I presented the modified subroutines SLOPERND and GAUSSND, which differ from the equivalent routines of Chapter 7 in that they can accommodate an arbitrary number of interacting species. To illustrate how the computational method can be applied to more species, I added to the system a calculation of the stable carbon isotope ratio, solving finally for the three... 

Element losses together with element bioavailability determine how much of an element has to be provided through the diet to remain in element balance. For assessment of losses, a label needs to be administered once either orally or intravenously. Compartmental modeling techniques permit to calculate when the label has equilibrated with the natural element in all body compartments. When isotopic labeling has been achieved, continuous replacement of lost isotopic label with the natural element from the diet results in a continuous decline of the body s isotopic enrichment. The change in tracer to tracee ratio in blood corresponds directly to the fraction of the body s element inventory that has been lost and replaced. 

More recent theoretical work has raised questions about these conclusions, how-ever. Particularly extensive calculational treatment of the rearrangement of 54 to vinyl chloride by several research groups failed to duplicate the predictions of an atypical kinetic isotope effect. These later studies indicate that tunneling effects should indeed be greater for H-shift than for the heavier D rearrangement. Consequently, the k /ko ratio should actually decrease at higher temperatures. The discrepancy in predicted results was eventually traced to an error in the earlier calculations. Nevertheless, it... 

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