Errors in the solutions of a set of equations representing an experimental system: a case study for the simultaneous determination of 13C/ 12C, 17O/ 16O and 18O/ 16O abundance ratios as CO 2+

Abstract

The relationships between experimental-cum-input errors (∂ R i 's) and the errors (∂ E m 's) of solutions ( E m 's), corresponding to a somewhat generalized and representative experimental case of any given three simultaneous equations ( f i ({ E m }) = R i ( i, m = 1, 2, 3)), are worked out. It is shown that the ∂ E m 's are, however, solutions of the corresponding characteristic differential relationships, f i ({∂ E m }) = ∂ R i ( i, m = 1, 2, 3). Verification of this finding, and/or its implication that derived parameters might turn out better or even worse representative of the investigating system than corresponding experimental quantities, is provided in terms of simultaneous determination of carbon and oxygen isotopic abundance ratios ( E m 's) via isotopic CO 2 + abundance ratios ( R i 's). The CO 2 +-system offers not just (required) 3 but 15 different R i = f i ({ E m }) = f i ( E 13/12, E 17/16, E 18/16) relationships. In addition, the well-known oxygen-systems based formula, E 17 / 16 = β E 18 / 16 R α (with R α and β as chosen constants), is often used to represent the CO 2 +-system. All such relevant relationships are discussed, and it is shown how can different typical sets of their combinations of required three be solved for the desired E m 's. It is demonstrated how the desired results (solutions: E m 's) vary with input errors, and examined whether the rate of such variations are governed by the choice of the required three representative equations ( monitors) and/or by the (magnitudes of) E m 's themselves. Essentially, more than providing a hitherto unexamined but required background for accurate simultaneous determination of 13C/ 12C, 17O/ 16O and 18O/ 16O abundance ratios as CO 2 +, it is clarified that acceptable accuracy in measurements alone cannot ensure the conclusions of indirect experimental studies to be always valid. The fundamental requirement is shown to be the ‘a priori’ examining of the role of computational step involved in defining the errors (∂ E m 's) in desired results ( E m 's), and hence to accordingly design the experiment to be carried out. It is thus in the case of working CO 2 +-system pointed out that “ E 17 / 16 = β E 18 / 16 R α ”, if employed as a monitor, acts as however a purely theoretical tool, and shown how the process of evaluation of corresponding results itself sets the guideline for correctly choosing its parameters ( R α and β) and yielding the desired results equally as accurate as, or even more accurate than, the measurements involved.