Uncertainties and error propagation in kinetic hard-modelling of spectroscopic data

Abstract

A novel method is presented for the rigorous propagation of uncertainties in initial concentrations and in dosing rates into the errors in the rate constants fitted by multivariate kinetic hard-modelling of spectroscopic data using the Newton-Gauss-Levenberg/Marquardt optimisation algorithm. The method was successfully validated by Monte-Carlo sampling. The impact of the uncertainties in initial concentrations and in the dosing rate was quantified for simulated spectroscopic data based on a second and a formal third order rate law under batch and semi-batch conditions respectively. An important consequence of this study regarding optimum experimental design is the fact that the propagated error in a second order rate constant is minimal under exact stoichiometric conditions or when the reactant with the lowest associated uncertainty in its initial concentration is in a reasonable excess (pseudo first order conditions). As an experimental example, the reaction of benzophenone with phenylhydrazine in THF was investigated repeatedly (17 individual experiments) by UV-vis and mid-IR spectroscopy under the same semi-batch conditions, dosing the catalyst acetic acid. For all experiments and spectroscopic signals, reproducible formal third order rate constants were determined. Applying the proposed method of error propagation to any single experiment, it was possible to predict 80% (UV-vis) and 40% (mid-IR) of the observed standard deviation in the rate constants obtained from all experiments. The largest contribution to this predicted error in the rate constant could be assigned to the dosing rate. The proposed method of error propagation is flexible and can straightforwardly be extended to propagate other possible sources of error.

Kinetic hard-modelling and spectral validation of rank-deficient spectroscopic data: A case study

“…Note that for all experimental conditions (a to e), quick injections were done within 24 s. Experimental conditions (a) are similar to those used in previous papers [21,22]. The experiment performed under these conditions (Experiment #1) was analysed assuming batch and semibatch conditions.…”

“…Products of the reaction are benzophenonephenylhydrazone (BP), water and the regenerated catalyst (Aa). This model reaction has been extensively described for acetic acid being dosed into benzophenone and phenylhydrazine [21,22]. In a recent paper, we have also shown that this condensation reaction and the optimisation of its rate constant are highly reproducible [21], thus being an ideal case study to validate our systematic method to treat kinetic rank deficiencies.…”

“…In the least-squares analysis, the sum of all squared residuals R (nt×nw) is used as the objective function to be minimised by iteratively optimising the rate constants k. For this article, the Newton-Gauss-Levenberg/Marquardt algorithm (NGL/M) was used to solve this non-linear regression [2,10,[24][25][26][27]. This gradientbased method allows estimating the uncertainties in the optimised rate constants from the variance/covariance matrix, including the propagation of errors, such as the uncertainties in initial concentrations and in dosing rate, as discussed previously [21].…”

“…1). The kinetic mechanism of this reaction has been discussed in previous articles [21,22] and is used here as a test reaction to experimentally validate the theoretical concepts presented previously [17]. Strategies (1) to (4) are applied to this reaction in order to break the rank deficiency in the concentration matrix without distorting the calculated rate constants.…”

Extent-based kinetic identification using spectroscopic measurements and multivariate calibration

“…Owing to the decoupling of the various rates and extents, each reaction and each mass transfer can be dealt with individually, unlike in the simultaneous identification, where an overall model including all the reactions and mass transfers has to be postulated [4][5][6]. Hence, the reaction and masstransfer rate parameters estimated by the incremental methods are largely uncorrelated.…”