On One Method of Parameter Estimation in Chemical Kinetics Using Spectra with Unknown Spectral Components

The method of successive estimation of regression parameters, which is widely used in nonlinear regression analysis, is applied to obtain kinetic information from spectral data for the case when the spectra of individual components are unknown. Using a model example with a two-step successive reaction, the reliability of the algorithm is demonstrated. To compare the proposed method with other known methods for estimating kinetic parameters literature data are used. All simulations were done using a new software for nonlinear regression analysis: FITTER. The proposed approach is especially useful when the spectra of reaction components are unknown and when formal calibration methods do not provide desirable accuracy.     

The effect of influential data, model and method on the precision of univariate calibration

Building a calibration model with detection and quantification capabilities is identical to the task of building a regression model. Although commonly used by analysts, an application of the calibration model requires at first careful attention to the three components of the regression triplet (data, model, method), examining (a) the data quality of the proposed model; (b) the model quality; (c) the LS method to be used or a fulfillment of all least-squares assumptions. This paper summarizes these components, describes the effects of deviations from assumptions and considers the correction of such deviations: identifying influential points is the first step in least-squares model building, the calibration task depends on the regression model used, and finally the least squares LS method is based on assumptions of normality of errors, homoscedasticity, independence of errors, overly influential data points and independent variables being subject to error. When some assumptions are violated, the ordinary LS is inconvenient and robust M-estimates with the iterative method of reweighted least-squares must be used. The effects of influential points, heteroscedasticity and non-normality on the calibration precision limits are also elucidated. This paper also considers the proper construction of the statistical uncertainty expressed as confidence limits predicting an unknown concentration (or amount) value, and its dependence on the regression triplet. The authors' objectives were to provide a thorough treatment that includes pertinent references, consistent nomeclature, and related mathematical formulae to show by theory and illustrative examples those approaches best suited to typical problems in analytical chemistry. Two new algorithms, calibration and linear regression written in s-plus and enabling regression triplet analysis, the estimation of calibration precision limits, critical levels, detection limits and quantification limits with the statistical uncertainty of unknown concentrations, form the goal of this paper.     

TOF-SIMS: Accurate mass scale calibration

A study is presented of the factors affecting the calibration of the mass scale in time-of-flight secondary ion mass spectrometry (TOF-SIMS). At the present time, TOF-SIMS analysts using local calibration procedures achieve a rather poor relative mass accuracy of only 150 ppm for large molecules (647 u) whereas for smaller fragments of <200 u this figure only improves to 60 ppm. The instrumental stability is 1 ppm and better than 10 ppm is necessary for unique identification of species. The above experimental uncertainty can lead to unnecessary confusion where peaks are wrongly identified or peaks are ambiguously assigned. Here we study, in detail, the instrumental parameters of a popular single stage reflection TOF-SIMS instrument with ion trajectory calculations using SIMION. The effect of the ion kinetic energy, emission angle, and other instrumental operating parameters on the measured peak position are determined. This shows clearly why molecular and atomic ions have different relative peak positions and the need for an aperture to restrict ions at large emission angles. These data provide the basis for a coherent procedure for optimizing the settings for accurate mass calibration and rules by which calibrations for inorganics and organics may be incorporated. This leads to a new generic set of ions for mass calibration that improves the mass accuracy in our interlaboratory study by a factor of 5. A calibration protocol is developed, which gives a relative mass accuracy of better than 10 ppm for masses up to 140 u. The effects of extrapolation beyond the calibration range are discussed and a recommended procedure is given to ensure that accurate mass is achieved within a selectable uncertainty for large molecules. Additionally, we can alternatively operate our instrument in a regime with good energy discrimination (i.e., poor energy compensation) to study the fragmented energies of molecules. This leads to data that support previous concepts developed in G-SIMS.     

Estimation of uncertainty in routine pH measurement

A procedure for estimation of measurement uncertainty of routine pH measurement (pH meter with two-point calibration, with or without automatic temperature compensation, combination glass electrode) based on the ISO method is presented. It is based on a mathematical model of pH measurement that involves nine input parameters. Altogether 14 components of uncertainty are identified and quantified. No single uncertainty estimate can be ascribed to a pH measurement procedure: the uncertainty of pH strongly depends on changes in experimental details and on the pH value itself. The uncertainty is the lowest near the isopotential point and in the center of the calibration line and can increase by a factor of 2 (depending on the details of the measurement procedure) when moving from around pH 7 to around pH 2 or 11. Therefore it is necessary to estimate the uncertainty separately for each measurement. For routine pH measurement the uncertainty cannot be significantly reduced by using more accurate standard solutions than ±0.02 pH units – the uncertainty improvement is small. A major problem in estimating the uncertainty of pH is the residual junction potential, which is almost impossible to take rigorously into account in the framework of a routine pH measurement.¹ Received: 11 August 2001 Accepted: 22 February 2002     

Spectrophotometric determination of organic dye mixtures by using multivariate calibration

The simultaneous determination of organic dye mixtures by using spectrophotometric methods is a difficult problem in analytical chemistry, due to spectral interferences. By using multivariate calibration methods such as partial least-squares regression (PLSR), it is possible to obtain a model adjusted to the concentration values of the mixtures used in the calibration stage. In this study, the calibration model is based on absorption spectra in the 350-650-nm range for a set of 16 different mixtures of reactive red 195, reactive yellow 145 and reactive orange 122 dyes, and made the determination of the dye concentrations possible in a validation set with significantly greater accuracy than the conventional univariate calibration method. By using the developed model it was possible to monitor the decolorization kinetic of one dye (reactive orange 122), when the mixture of the three dyes was previously submitted to an ozonation process.     

Uncertainty analysis of correlated parameters in automated reaction mechanism generation

Uncertainty analysis is a useful tool for inspecting and improving detailed kinetic mechanisms because it can identify the greatest sources of model output error. Owing to the very nonlinear relationship between kinetic and thermodynamic parameters and computed concentrations, model predictions can be extremely sensitive to uncertainties in some parameters while uncertainties in other parameters can be irrelevant. Error propagation becomes even more convoluted in automatically generated kinetic models, where input uncertainties are correlated through kinetic rate rules and thermodynamic group values. Local and global uncertainty analyses were implemented and used to analyze error propagation in Reaction Mechanism Generator (RMG), an open-source software for generating kinetic models. A framework for automatically assigning parameter uncertainties to estimated thermodynamics and kinetics was created, enabling tracking of correlated uncertainties. Local first-order uncertainty propagation was implemented using sensitivities computed natively within RMG. Global uncertainty analysis was implemented using adaptive Smolyak pseudospectral approximations as implemented in the MIT Uncertainty Quantification Library to efficiently compute and construct polynomial chaos expansions to approximate the dependence of outputs on a subset of uncertain inputs. Cantera was used as a backend for simulating the reactor system in the global analysis. Analyses were performed for a phenyldodecane pyrolysis model. Local and global methods demonstrated similar trends; however, many uncertainties were significantly overestimated by the local analysis. Both local and global analyses show that correlated uncertainties based on kinetic rate rules and thermochemical groups drastically reduce a model's degrees of freedom and have a large impact on the determination of the most influential input parameters. These results highlight the necessity of incorporating uncertainty analysis in the mechanism generation workflow.     

Universal calibration in GPC

The M[η]-elution volume calibration curve for gel-permeation chromatography (GPC) is based on the implicit assumption that the hydrodynamic volume of a solvated polymer species in the GPC columns is that which pertains at infinite dilution. This is not true of highly solvated high molecular weight fractions and results in apparent failure of this calibration in some instances. A model is presented to estimate hydrodynamic volumes of polymers at finite concentrations. The parameters required are polymer concentration, molecular weight, amorphous density, and the Mark-Houwink constants for the particular polymer–solvent combination. The calculated log (hydrodynamic volume)–elution volume relation provides a universal GPC calibration. The model accounts for the occasional shortcomings of the infinite dilution calibration and is essentially equivalent to it in noncritical cases. The use of the proposed calibration method is illustrated.     

On the Statistical Calibration of Physical Models

We introduce a novel statistical calibration framework for physical models, relying on probabilistic embedding of model discrepancy error within the model. For clarity of illustration, we take the measurement errors out of consideration, calibrating a chemical model of interest with respect to a more detailed model, considered as “truth” for the present purpose. We employ Bayesian statistical methods for such model‐to‐model calibration and demonstrate their capabilities on simple synthetic models, leading to a well‐defined parameter estimation problem that employs approximate Bayesian computation. The method is then demonstrated on two case studies for calibration of kinetic rate parameters for methane air chemistry, where ignition time information from a detailed elementary‐step kinetic model is used to estimate rate coefficients of a simple chemical mechanism. We show that the calibrated model predictions fit the data and that uncertainty in these predictions is consistent in a mean‐square sense with the discrepancy from the detailed model data.     

Worst case uncertainty estimates for routine instrumental analysis

A methodology for the worst case measurement uncertainty estimation for analytical methods which include an instrumental quantification step, adequate for routine determinations, is presented. Although the methodology presented should be based on a careful evaluation of the analytical method, the resulting daily calculations are very simple. The methodology is based on the estimation of the maximum value for the different sources of uncertainty and requires the definition of limiting values for certain analytical parameters. The simplification of the instrumental quantification uncertainty estimation involves the use of the standard deviation obtained from control charts relating to the concentrations estimated from the calibration curves for control standards at the highest calibration level. Three levels of simplification are suggested, as alternatives to the detailed approach, which can be selected according to the proximity of the sample results to decision limits. These approaches were applied to the determination of pesticide residues in apples (CEN, EN 12393), for which the most simplified approach showed a relative expanded uncertainty of 37.2% for a confidence level of approximately 95%.     

Petroleum resins adsorption onto quartz sand: near infrared (NIR) spectroscopy study

In this paper we have tried to evaluate adsorption parameters of petroleum resins. Near infrared (NIR) spectroscopy is applied for resins bulk concentration evaluation during adsorption process. NIR experimental scheme and parameters are provided. NIR spectra range of 9000-13,000 cm(-1) is chosen. Quartz sand (0.2-0.8 mm fraction) is used as adsorbent; benzene is used as solvent. Different approaches of "NIR spectra-resins concentration" calibration model building are discussed. Partial least squares (PLS) regression method is used. Langmuir model is chosen for experimental data fitting. Combined usage of kinetic and isothermic data gives us ability to evaluate the maximal adsorbed mass density, the equilibrium constant of adsorption, and the rate constants of adsorption (and desorption). The rate constants of resins adsorption and desorption are found to be concentration independent.     

Scale-up of batch kinetic models

The scale-up of batch kinetic models was studied by examining the kinetic fitting results of batch esterification reactions completed in 75 mL and 5 L reactors. Different temperatures, amounts of catalysts, and amounts of initial starting reagents were used to completely characterize the reaction. A custom written Matlab toolbox called GUIPRO was used to fit first-principles kinetic models directly to in-line NIR and Raman spectroscopic data. Second-order kinetic models provided calibration-free estimates of kinetic and thermodynamic reaction parameters, time dependent concentration profiles, and pure component spectra of reagents and product. The estimated kinetic and thermodynamic parameters showed good agreement between small-scale and large-scale reactions. The accuracy of pure component spectra estimates was validated by comparison to collected NIR and Raman pure component spectra. The model estimated product concentrations were also validated by comparison to concentrations measured by off-line GC analysis. Based on the good agreement between kinetic and thermodynamic parameters and comparison between actual and estimated concentration and spectral profiles, it was concluded that the scale-up of batch kinetic models was successful.     

Estimation of the standard uncertainty of a calibration curve: application to sulfur mass concentration determination in fuels

The construction of a calibration curve using least square linear regression is common in many analytical measurements, and it comprises an important uncertainty component of the whole analytical procedure uncertainty. In the present work, various methodologies are applied concerning the estimation of the standard uncertainty of a calibration curve used for the determination of sulfur mass concentration in fuels. The methodologies applied include the GUM uncertainty framework, the Kragten numerical method, the Monte Carlo method (MCM) as well as the approximate equation calculating the standard error of prediction. The standard uncertainty results obtained by all methodologies agree well (0.172?C0.175?ng???L^?1). Aspects of inappropriate use of the approximate equation of the standard error of prediction, which leads to overestimation or underestimation of calculated uncertainty, are discussed. Moreover, the importance of the correlation between calibration curve parameters (slope and intercept) within GUM, MCM and Kragten approaches is examined.     

Modeling of branching ratio uncertainty in chemical networks by Dirichlet distributions

Validation of complex chemical models relies increasingly on uncertainty propagation and sensitivity analysis with Monte Carlo sampling methods. The utility and accuracy of this approach depend on the proper definition of probability density functions for the uncertain parameters of the model. Taking into account the existing correlations between input parameters is essential to a reliable uncertainty budget for the model outputs. We address here the problem of branching ratios between product channels of a reaction, which are correlated by the unit value of their sum. We compare the uncertainties on predicted time-dependent and equilibrium species concentrations due to input samples, either uncorrelated or explicitly correlated by a Dirichlet distribution. The method is applied to the case of Titan ionospheric chemistry, with the aim of estimating the effect of branching ratio correlations on the uncertainty balance of equilibrium densities in a complex model.     

Using free energy perturbation to predict effects of changing force field parameters on computed conformational equilibriums of peptides

We propose a method that may allow data about the conformational equilibriums of peptides to enter the parameter calibration phase in force field developments. The method combines free energy perturbation with techniques for extensive sampling in the conformational space. It predicts shifts in computed conformational equilibriums in response to separate or combined perturbations of force field parameters. As an example we considered a force field associated with an implicit solvent model. We considered two different approaches to define conformational states of four peptides. One is based on reaction coordinates and two-dimensional free energy surfaces. The other is based on the clustering analysis of sampled conformations. Effects of perturbing various model parameters on the equilibriums between nativelike states with other conformational states were considered. For one type of perturbation predicted to have consistent effects on different peptides, the predictions have been verified by actual simulations using a perturbed model.     

New approaches to studying the kinetics of the radical reaction of N-phenyl-1,4-benzoquinone monoimine with 2-mercaptobenzothiazole

The kinetics of the reaction of N-phenyl-1,4-benzoquinone monoimine with 2-mercaptobenzothiazole in chlorobenzene at 298 and 343 K under argon has been studied by kinetic spectrophotometry. The reaction orders have been determined. The influence of tetraphenylhydrazine and azobisisobutyronitrile initiators has been investigated. An accelerating effect of one of the reaction products, 4-hydroxydiphenylamine, has been discovered. The accelerating effect strenthens with an increasing 2-mercaptobenzothiazole concentration. Two methods are proposed for determination of the kinetic parameters of the reaction. The first method uses data on the consumption rate of quinoneimine in the noninitiated reaction in the presence of 4-hydroxydiphenylamine. In the second method, the accumulation of quinone monoimine is studied during initiator decomposition in the presence of mixtures of 4-hydroxydiphenylamine and 2-mercaptobenzothiazole and parameters are estimated from the dependences of the limiting concentration of accumulated quinone monoimine on the initiation rate and on the concentrations of the reactants Using the proposed approaches, the numerical values of a number of kinetic parameters of the radical reactions of the quinone compounds with thiols have been determined for the first time for the reaction studied.     

Estimation of uncertainty in electrochemical amperometric measurement of dissolved oxygen concentration

A procedure for the estimation of measurement uncertainty of dissolved oxygen (DO) concentration measurement based on the ISO approach is presented. It is based on a mathematical model that involves 14 input parameters. The uncertainty of DO concentration strongly depends on changes in experimental details (temperature difference between calibration and measurement, the time interval between calibration and measurement, etc.). The relative measurement uncertainty is, however, practically independent of the DO concentration itself. The uncertainty is the lowest if the calibration and the measurement are done at the same temperature and on the same day. A calculation tool is provided (in the form of a GUM Workbench file) for practitioners that can be used for uncertainty calculation of DO concentrations at very different experimental conditions.Electronic Supplementary Material The uncertainty calculation example is available as a GUM Workbench calculation file C_O2_meas.smu (GUM Workbench ver. 1.3.3, Metrodata GmbH) together with its data file Input_values.xls (MS Excel 97). For those users who do not have GUM Workbench, the full report of the GUM Workbench calculation is available as a PDF file C_O2_meas.pdf. This material is available via the Internet at .     

Problems and risks occurred during uncertainty evaluation of a quantity calculated from correlated parameters: a case study of pH measurement

Parameters of a model describing a measurement process obtained during a calibration experiment allow one to calculate a measurement result, but a simple estimation of measurement uncertainties of the parameters is not sufficient to assess the uncertainty of the result. In this paper, an example of a pH measurement conducted using an ion-selective electrode is presented, in which the uncertainty is evaluated taking into consideration the existing correlation between the parameters of the electrode. The calculations apply either covariances or correlation coefficients that have to be computed additionally. The example presented in this paper illustrates that there are some problems with rounding of variables which, because of the existing very strong correlations, significantly changes the sought uncertainty. This approach is compared with other approaches, that is, usage of uncorrelated variables and Monte Carlo simulations that are described in an earlier work. It is concluded that the approach of uncertainty evaluation, in which covariances or correlation coefficients are explicitly calculated, is work-consuming and may cause significant discrepancies between correct and obtained assessments if some roundings or approximations are done, or if the correlation coefficient is obtained experimentally based on data including random errors.     

Uncertainty due to the quantification step in analytical methods

The quantification step is an important source of uncertainty in analytical methods, but it is frequently misunderstood and disregarded. In this paper, it is shown how this uncertainty is closely related to the linear response range of a method, and to the Pearson correlation coefficient of the calibration line. So, if there is a need for a pre-fixed quantification uncertainty, the linear response range will be affected. Some practical cases are given showing the quantification uncertainty significance. The theoretical equation giving the value of the quantification uncertainty is deduced from which new conclusions can be taken out. Because of that, the quantification uncertainty can easily be calculated and the parameters that really affect its value are shown along the paper. Some final considerations about detection limits and two-point calibration lines are also given. The paper can also be considered a reflection on uncertainty owed to calibration and on their consequences on the analytical methodology.     

Evaluation of combined measurement uncertainty of the determination of Mn content in sediment samples

Combined uncertainties of an analysis of elemental content of sediment samples were evaluated. A monitoring system has been designed and implemented for the characterization of the environmental conditions of Lake Balaton in Hungary. Sediments samples were collected and an acidic digestion method was used to determine the concentration of elements. For the calculation of the result of each measurement three different approaches were considered, namely a.) the calculation of the result using a calibration curve and estimating the confidence limit by the Student t-distribution, b.) calculation of the combined uncertainty and c.) estimation of the sampling errors using the transport and field blanks. The latter approach gave the most reliable result since it included all the parameters which had to be considered regarding sampling and sample handling, and measurement. Determination of acid soluble Mn content in sediment samples has been chosen as an example, and the combined uncertainty is calculated using blanks for sampling. Received: 17 March 2000 Accepted: 4 October 2000     

The use of FT-NIR for API content assay in organic solvent media: a single calibration for multiple downstream processing streams

The use of near infrared spectroscopy (NIRS) in downstream solvent based processing steps of an active pharmaceutical ingredient (API) is reported. A single quantitative method was developed for API content assessment in the organic phase of a liquid–liquid extraction process and in multiple process streams of subsequent concentration and depuration steps. A new methodology based in spectra combinations and variable selection by genetic algorithm was used with an effective improvement in calibration model prediction ability. Root mean standard error of prediction (RMSEP) of 0.05 in the range of 0.20–3.00% (w/w) was achieved. With this method, it is possible to balance the calibration data set with spectra of desired concentrations, whenever acquisition of new spectra is no longer possible or improvements in model's accuracy for a specific selected range are necessary. The inclusion of artificial spectra prior to genetic algorithms use improved RMSEP by 10%. This method gave a relative RMSEP improvement of 46% compared with a standard PLS of full spectral length.