Uncertainties in kinetic hard-modelling of multivariate spectroscopic data

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.

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

Julien Léo Billeter, Konrad Hungerbühler

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.

Elsevier2008

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

Julien Léo Billeter, Konrad Hungerbühler

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.

2009

Nitrous Oxide Abatement over Fe-Zeolite

Bryan Bromley

The use of zeolites for catalytic reactions is a field in continuous development. Since it was demonstrated that metal-zeolites are efficient catalysts for NOx abatement, decomposition of nitrous oxide (N2O) over Fe-containing zeolites has attracted great interest by the scientific community mainly due: N2O is the third most important gas contributing to global warming, its concentration in atmosphere is still on the rise and direct N2O decomposition to N2 and O2 is a sustainable alternative. N2O interaction with Fe-zeolite leads to the formation of "special" adsorbed oxygen (often called α-oxygen) with great potential for selective oxidation. Although the focus of a number of studies, the nature of the actives sites for the formation of this adsorbed oxygen is still unclear. The main objectives of this work are: 1) study the decomposition mechanism of N2O over isomorpously substituted Fe-ZSM-5 zeolite aiming on catalyst optimization and 2) optimization and further use of structured Fe-zeolite in novel micro-structured reactor. A number of refined transient kinetics methods were intensively used such as transient response, Temperature Programmed Desorption (TPD) and Temporal Analysis of Product (TAP). Density Functional Theory (DFT) and in situ IR spectroscopy studies were conducted in collaboration and our results support the following: In chapter 3, quantitative measurements have been carrier out to analyze the effect of temperature on the activation (under helium flow) and the irreversible deactivations that can happen during the pretreatment. The results obtained indicate that high temperature treatment (1273 K) results in the formation of a high amount of α-sites. The addition of a very low fraction of oxygen (2%) in the feed demonstrates that the catalyst is sensitive to its presence. Certain sites can be irreversibly destroyed, a result ascribed to the oxidation of active Fe(II) to Fe2O3 clusters which contain inactive Fe(III). Catalyst treatment at high temperature (1273 K) in He for up to 3 hours, also reduced irreversibly the concentration of α-sites. TPD measurements have shown desorption of oxygen at 3 different temperatures. Desorption at the lowest temperature (650 K) was assign to the α-oxygen recombination. The oxygen which desorbs at 700 K was associated to the oxygen from the deactivated sites. Oxygen which desorbs at the highest temperature (730 K) is originated from the surface NOx that desorbed as NO and O2. After a comparison of the concentration of the molecules accumulated, we propose that NO2 is the adsorbed specie. The mechanism of N2O decomposition on the binuclear oxo-hydroxo bridged extraframework iron core site [FeII(μ-O)(μ-OH)FeII]+ inside the ZSM-5 zeolite has been studied by combining theoretical and experimental approaches (see chapter 4). Rate parameters computed using standard statistical mechanics and transition state theory reveal that elementary catalytic steps involved into N2O decomposition are strongly dependent on the temperature. This theoretical result was contrasted with the experimentally observed steady state kinetics of the N2O decomposition and TPD experiments. As predicted by the theoretical study, the switch of the reaction order with respect to N2O pressure (from zero to one) occurs at around 800 K which confirms a change of the rate determining step from the α-oxygen recombination to α-oxygen formation. The proposed mechanism was also confirmed by O2-TD which confirmed the viability of the binuclear complex. The α-oxygen recombination at low temperature (

EPFL2010

Nitrous Oxide Abatement over Fe-Zeolite

Bryan Bromley

EPFL2010

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

Julien Léo Billeter, Konrad Hungerbühler

2009