Systematic prediction of linear dependencies in the concentration profiles and implications on the kinetic hard-modelling of spectroscopic data

A novel method is presented for the systematic identification of the minimum requirements regarding mathematical pre-treatment, a priori information, or experimental design, in order to allow optimising rate constants and pure component spectra associated with a kinetic model via multivariate kinetic hard-modelling of spectroscopic data. Rank deficiencies in the kinetic concentration matrix represent a major problem for the calibration free method developed by Maeder and Zuberbühler, as its pseudo-inverse, required for the optimisation process, is not defined. In this contribution, the underlying linear dependencies in the concentration profiles are systematically elucidated and appropriate strategies are discussed in order to break them. Also, conditions are predicted for which full spectral resolution can be expected. The method is based on the kernel of a time invariant augmented matrix covering potential rank deficiency due to stoichiometry and rate laws, also relevant for the concentration matrix. Compared to employing the full concentration matrix, this augmented matrix does not require a numerical integration of the differential equations describing the kinetic model and thus can easily be set up. The kernel can be calculated numerically by Singular Value Decomposition (SVD) or determined in a symbolical way, the latter allowing the detection of particular stoichiometric conditions leading to spectral resolution of species. The capabilities of the method are demonstrated analysing three kinetic mechanisms of increasing complexity covering consecutive and parallel reactions.