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