Kinetic hard-modelling and spectral validation of rank-deficient spectroscopic data

In recent years, chemometric methods for the analysis of multivariate kinetic data have considerably progressed. Kinetic hard-modelling is one of these methods that is based on the rate law and used to determine the kinetic parameters (e.g. rate constants) of chemical reactions by non-linear optimisation. Applied to spectroscopy, kinetic hard-modelling relies on Beer’s law to decompose the time and wavelength resolved data into the concentration profiles and the molar spectra of the pure components. In direct implicit kinetic hardmodelling, the concentration profiles are obtained by numerical integration of the rate laws and pure spectra are linearly estimated at each iteration using the pseudo-inverse of the concentration matrix [1]. Direct implicit kinetic hard-modelling of spectroscopic data allows the validation of the kinetic mechanism by comparing the estimated component spectra with independently measured ones. A severe drawback, however, is that this implicit method fails when concentrations profiles are linearly dependent, as the pseudo-inverse and thus the component spectra cannot be computed. Different Strategies have been proposed as a remedy to this rank deficiency problem, such as (1) defining some absorbing species as uncoloured, (2) providing some component spectra for the analysis, (3) dosing one or more species or (4) analysing simultaneously several experiments recorded under different initial concentrations (3-way analysis). In absence of a systematic method, the appropriate species to be included in these four Strategies are selected by experience or trial and error. Spectral validation of the kinetic mechanism can also be difficult when Strategy (1) is employed, as the fitted component spectra are complex linear combinations of the true pure spectra. We have recently proposed a systematic method for the experimental and data analytical design of kinetic data measured by spectroscopy that allows identifying the species to be incorporated in Strategies (1) – (4) and calculating the linear combinations of the true pure spectra when Strategy (1) is used, an important step for spectral validation [2]. This systematic method is based on a time-invariant matrix that avoids the numerical integration of the time-variant concentration profiles and allows the experimental design of chemical reactions, even if the associated rate constants are not yet known, i.e. before optimisation. This time-invariant matrix uses the entire set of kinetic reactions and only requires a reduction to independent reactions if linear combinations of the true pure spectra are desired (Strategy 1). For this, a method has also been developed. In this presentation, the systematic method of species selection is presented using simulated data and, from this, appropriate experimental designs (Strategies) are suggested. The method is also presented using experimental results obtained from the reaction of benzophenone with phenylhydrazine in THF (under catalysis of acetic acid), for which the postulated kinetic mechanism has been successfully validated via the comparison between fitted and measured component spectra in mid-IR and UV-vis [3]. [1] M. Maeder, A.D. Zuberbühler, Anal. Chem., 62 (1990), 2220-2224. [2] J. Billeter, Y.M. Neuhold, K. Hungerbühler, Chemom. Intell. Lab. Syst., 95 (2009), 170-187. [3] J. Billeter, Y.M. Neuhold, K. Hungerbühler, Chemom. Intell. Lab. Syst., 98 (2009), 213-226.