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Abstract
Kinetic hard-modelling of multivariate spectroscopic data [1, 2] is a well-established method for the determination of reaction mechanisms and associated rate constants. Kinetic hard-modelling relies on a hard model defined by kinetic rate laws and requires the numerical integration of these rate laws nested into Beer's law in order to model the spectroscopic signal. The modelled absorbances are compared with the measured ones and the residuals are minimised in the least squares sense by optimising the corresponding rate constants. Uncertainties in experimental conditions, such as initial concentrations or dosing rates, are often neglected and thus errors in the fitted rate constants estimated from the analysis of one single experiment typically tend to be underestimated. In this contribution, we present a rigorous method for the 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 and non-linear optimisation [3].
First, the importance of the uncertainties in initial concentrations is shown for simulated spectroscopic data based on a second order rate law under batch conditions. Our method of error propagation leads to interesting and useful results for the optimum design of kinetic experiments used for the non-linear optimisation of second order rate constants by kinetic hard-modelling.
We then present an experimental validation of the method using the acid-catalysed reaction of benzophenone with phenylhydrazine in THF repeatedly investigated (seventeen times) by UV-vis and mid-IR pectroscopy under semi-batch conditions. By applying our method of error propagation to each single experiment, we have been able to cover an important proportion of the observed standard deviation in the rate constants obtained from all experiments. Differences between results obtained from UV-vis and mid-IR spectroscopy, as well as the importance of the dosing rate in the design of semi-batch experiments are discussed.
[1] M. Maeder, Y.M. Neuhold, Chapter 7 in P. Gemperline (Ed.), Practical Guide to Chemometrics, Taylor and Francis, Boca Raton, USA, 2006.
[2] G. Puxty, U. Fischer, M. Jecklin, K. Hungerbühler, Chimia (2006), 60, 605.
[3] Billeter, J., Neuhold, Y.-M., Simon, L., Puxty, G., Hungerbühler, K., Chemom. Intell. Lab. Syst. 93(2), 120-131, 2008.
First, the importance of the uncertainties in initial concentrations is shown for simulated spectroscopic data based on a second order rate law under batch conditions. Our method of error propagation leads to interesting and useful results for the optimum design of kinetic experiments used for the non-linear optimisation of second order rate constants by kinetic hard-modelling.
We then present an experimental validation of the method using the acid-catalysed reaction of benzophenone with phenylhydrazine in THF repeatedly investigated (seventeen times) by UV-vis and mid-IR pectroscopy under semi-batch conditions. By applying our method of error propagation to each single experiment, we have been able to cover an important proportion of the observed standard deviation in the rate constants obtained from all experiments. Differences between results obtained from UV-vis and mid-IR spectroscopy, as well as the importance of the dosing rate in the design of semi-batch experiments are discussed.
[1] M. Maeder, Y.M. Neuhold, Chapter 7 in P. Gemperline (Ed.), Practical Guide to Chemometrics, Taylor and Francis, Boca Raton, USA, 2006.
[2] G. Puxty, U. Fischer, M. Jecklin, K. Hungerbühler, Chimia (2006), 60, 605.
[3] Billeter, J., Neuhold, Y.-M., Simon, L., Puxty, G., Hungerbühler, K., Chemom. Intell. Lab. Syst. 93(2), 120-131, 2008.