major isotope in the isotopically enriched analyte. For elements with only two stable isotopes, such as boron, chlorine, or silver, the above single dilution equation simplifies to the following: In a typical gas chromatography analysis, isotopic dilution can decrease the uncertainty of the measurement results from 5% to 1%. It can also be used in mass spectrometry (commonly referred to as isotopic dilution mass spectrometry or IDMS), in which the isotopic ratio can be determined with precision typically better than 0.25%. Optimum composition of the blend In a simplified manner, the uncertainty of the measurement results is largely determined from the measurement of RAB: From here, we obtain the relative uncertainty of nA, ur(nA) = u(nA)/nA: The lowest relative uncertainty of nA corresponds to the condition when the first derivative with respect to RAB equals zero. In addition, it is common in mass spectrometry that u(RAB)/RAB is constant and therefore we can replace u(RAB) with RAB. These ideas combine to give Solving this equation leads to the optimum composition of the blend AB, i.e., the geometric mean between the isotopic compositions of standard (A) and spike (B): This simplified equation was first proposed by De Bievre and Debus numerically and later by Komori et al. and by Riepe and Kaiser analytically. It has been noted that this simple expression is only a general approximation and it does not hold, for example, in the presence of Poisson statistics or in the presence of strong isotope signal ratio correlation. Double dilution method The single dilution method requires the knowledge of the isotopic composition of the isotopically enriched analyte (RB) and the amount of the enriched analyte added (nB). Both of these variables are hard to establish since isotopically enriched substances are generally available in small quantities of questionable purity. As a result, before isotope dilution is performed on the sample, the amount of the enriched analyte is ascertained beforehand using isotope dilution. This preparatory step is called the reverse isotope dilution and it involves a standard of natural isotopic-composition analyte (denoted as A*). First proposed in the 1940s and further developed in the 1950s, reverse isotope dilution remains an effective means of characterizing a labeled material. Reverse isotope dilution analysis of the enriched analyte: Isotope dilution analysis of the analyte: Since isotopic composition of A and A* are identical, combining these two expressions eliminates the need to measure the amount of the