On the number of observable species, observable reactions and observable fluxes in chemometric studies and the role of multichannel integration

It is assumed that preliminary experiments are performed to measure the absorbance of a liquid in a chemically reactive system. Further, given the exploratory nature of the study it is assumed that no further information is available concerning the species present nor the reactions occurring. The principle chemometric goals of such an exploratory experimental study can be stated as follows; 1. (I) determine the number of statistically significant factors associated with the observable species in solution and 2. (II) determine the number of statistically significant factors associated with the observable reactions, consistent with the set of spectroscopic measurements. In the case of infrared (IR) spectroscopy, numerous difficulties are encountered due to the sensitivity of band positions and band shapes to changes in the liquid phase composition during reaction. Although the above mathematical problem and its subsequent goals are typically formulated in terms of the Lambert-Beer-Bouguer Law (LBBL) followed by factor analysis, the merits of utilizing the integral absorption law (IAL) for infrared spectroscopy in order to achieve goals I and II are examined in detail. First, the possibility of transport involving species into and out of the liquid phase is considered, and the associated fluxes are defined. Second, terms representing the model errors for goals I and II arising from the use of both the LBBL and the IAL are specified. Third, well-posed regions RxTP∗ for each model are defined in which the contribution of model error is less than the experimental error associated with the measurements. Fourth, the existence of well-posed solutions for goals I and II for both the LBBL and the IAL representations are presented in terms of the singular value decomposition (SVD). Based on literature data for the sensitivity of LBBL and IAL absorptivities, it appears that goals I and II are considerably more difficult to achieve in the case of a LBBL formulation compared to a IAL formulation, i.e. RxTP∗(LBBL) ⊂ RxTP∗(IAL). Furthermore, and perhaps most important, it seems that only the IAL representation is suitable for systems where changes in band positions and shapes occur. As a generalization, multichannel integration for goals I and II involving other absorption spectroscopies are also considered. The consequences of both non-destructive and destructive absorption spectroscopies on goals I and II are addressed.