Multiple reservoir isotopic systems

Dividing by the total amount of species N in the whole system and introducing the mass fraction (allotment) XjN of species N hosted by the th reservoir, we get 

For the radioactive species, the mass balance equation contains a radioactive decay term 

Since M and N are different chemical species, we make fractionation factors apparent in the form 

Summarizing, each set of radioactive-radiogenic pair+stable isotope gives a non-linear set of three equations per reservoir 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

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Figure 2. Overview of the hydrogen isotopic compositions of different objects, materials, and reservoirs in the solar system and beyond. The data are discnssed in different portions of this review paper, and most are snmmarized in tabnlar form in the recent review of Robert et al. (2000). Points with error bars represent single valnes with a quoted imcertainty. Solid bars reflect the range of multiple measurements ( 10 or more of analyses in most cases). The two astronomical settings are cold, dense interstellar clouds (T 10 K) and so-called hot cores, such as found in Orion, where temperatures are 70K and above. Behind the data for solar system objects lies much complexity which cannot be represented on a single summary figure. For example, even though the range of D/H in water from bulk chondrites is relatively large, most of the samples fall in a very narrow range, as described in the text.

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