The precision of isotopic analyses is typically calculated by two methods. Pooled standard deviations of raw data are typically
computed from sets of duplicate or triplicate measurements (12). From those pooled standard deviations, standard deviations
of mean values pertaining to specific substances are calculated. More specifically, the standard deviation of a mean value
is the pooled standard deviation divided by n
^{
½
}, in which n is the number of measurements performed on a given sample (13). For carbon, nitrogen, oxygen, and sulfur, the resulting 95%
confidence intervals for a result are typically in the range of ±0.1 to ±0.4‰. For hydrogen, the 95% confidence interval is
typically ±3‰.
Discussion
The precise quantitation of stableisotopic compositions in pharmaceutical intermediates and products requires both mass balance
and isotopic fractionation equations that are applicable to both single and multistep reaction sequences. We start from the
most basic requirement of mass balance and then consider isotopic fractionations in a single reaction. Finally, we discuss
applications to the protection of process patents.
Mass balance.
For A + B → C, in which reactants A and B are quantitatively converted to product C, two mass balances can be written:
in which, m
_{A}, m
_{B}, and m
_{C} are molar amounts of carbon (or any other element) in A, B, and C and the isotopic compositions of that carbon (or any other
element) in A, B, and C are given by δ_{A}, δ_{B}, and δ_{C}. Equation 1 is a mass balance (i.e., carbon in = carbon out) whereas Equation 2 is an isotopic mass balance (^{13}C in = ^{13}C out). Under the conditions postulated (quantitative conversion), the isotopic composition of C can be computed from those
of A and B (10, 11).
Isotopic fractionation.
For isotopic fractionations, calculations must take into account factors such as reaction completeness and isotope effects.
These effects will cause the isotopic composition of C to differ from that computed using the mass balance equation and assuming
quantitative conversion of reactants to products.
To provide a concrete example, assume that A is present in excess whereas B, the limiting reactant, is quantitatively converted
to product. In that case:
in which n
_{A}, n
_{B}, and n
_{C} represent the numbers of carbon atoms (or any other element of interest) in A, B, and C. Because A is not quantitatively
converted to product, the isotopic compositions of the Aderived positions in C can differ from those in the initial reactant.
In this case, the isotopic offset is expressed as ΔA. As Figure 1 shows, its value depends on the isotope effects and on the
fraction of A that remains unconsumed. If the reaction conditions, particularly the magnitude of the excess of A, are consistent,
ΔA will be constant. Because the n values are known exactly, ΔA can be determined from Equation 3 after isotopic analysis of the reactants and product (i.e., determination of δ_{A}, δ_{B}, and δ_{C}).
Values of δ_{A}, δ_{B}, and δ_{C} do not affect the values of ΔA. Accordingly, once ΔA is known for a given reaction and a set of conditions, it is necessary
only to know two of the δ values to compute the third. Thus, for example, when ΔA, δ_{A}, and δ_{B}, are known, the isotopic value of the product (δ_{C}) can be calculated.
If neither A nor B is consumed completely during the course of the reaction, and if the rate of the chemical reaction
(or position of the chemical equilibrium) is sensitive to isotopic substitution on both reactants, it is necessary to consider
values of ΔA and ΔB:
If reaction conditions cannot be manipulated so that f
_{A} and f
_{B} (and thus Δ_{A} and Δ_{B}) can be independently driven to completion (i.e., zero), it will be possible to determine only the sum, n
_{A}Δ_{A} + n
_{B}Δ_{B}. From theoretical considerations (14), Δ_{A} and Δ_{B} can be evaluated separately for all values of f
_{A} and f
_{B} if the isotope effects are known.
Of course, isotopic fractionations such as those previously described accumulate during the different steps of a multistep
synthesis scheme. They can, however, be individually and systematically differentiated, not only for multiple reactants but
also for multiple isotopes. To provide an example, consider carbonisotopic fractionations in a hypothetical fourstep sequence:
