Numerical investigation of the uncertainty of Arrhenius parameters

The temperature dependence of rate coefficient k is usually described by the Arrhenius expression ln k = ln A − (E/R)T ⁻¹. Chemical kinetics databases contain the recommended values of Arrhenius parameters A and E, the uncertainty parameter f (T) of the rate coefficient and temperature range of validity of this information. Taking ln k as a random variable with known normal distribution at two temperatures, the corresponding uncertainty of ln k at other temperatures was calculated. An algorithm is provided for the generation of the histogram of the transformed Arrhenius parameters ln A and E/R, which is in accordance with their 2D normal probability density function (pdf). The upper and the lower edges of the 1D normal distribution of ln k correspond to the two opposite edge regions of the 2D pdf of the transformed Arrhenius parameters. Changing the temperature, these edge regions move around the 2D cone. The rate parameters and uncertainty data belonging to reactions H + H₂O₂ = HO₂ + H₂ and O + HO₂ = OH + O₂ were used as examples.     

On the errors of Arrhenius parameters and estimated rate constant values

A detailed statistical study is presented, based on simulated experimental data, on the estimation of activation parameters using the Arrhenius equation: k = A exp(B/T). The close correlation of the two parameters is shown, which requires the computation of the covariance matrix for the representation of uncertainties. This matrix facilitates the correct estimation of the confidence interval for interpolated (or extrapolated) values of rate coefficients. It is proposed that the full correlation matrix should be published in any article dealing with the determination of Arrhenius parameters. The importance of correct weighting is emphasized. Nonlinear fitting to the Arrhenius equation can be carried out without weighting only in case the (absolute) error of rate coefficient is independent of the temperature. Simulated experiments show that noncorrect weighting shifts the average values of fitted parameters and increases the variance of the parameters as well. With respect to the modified Arrhenius equation: k = A · Tⁿ exp(B/T), statistical analysis shows that the physically meaningful estimation of all three parameters is impossible. Nonlinear fitting of three parameters is suggested for interpolation (and extrapolation) of rate coefficients, whereas in case of activation parameter estimation, the fixing of “n” on the basis of theoretical considerations is advised followed by the estimation of the remaining two parameters.     

Kinetics and Mechanism of the Reactions Between Triphenylphosphine,Dialkyl Acetylenedicarboxilates and a NH-Acid,Pyrazole, by UV Spectrophotometry

Kinetic studies were made of the reactions between triphenylphosphine and dialkyl acetylenedicarboxylates in the presence of a NH-acid such as pyrazole. To determine the kinetic parameters of the reactions, the reaction progress was monitored by UV spectrophotometry. The second-order fits were automatically drawn and the values of the second-order rate constant (k ₂) were automatically calculated using standard equations. In the temperature range studied, the dependence of ln k ₂ on the reciprocal temperature was consistent with the Arrhenius equation. Furthermore, useful information was obtained from studies of the effect of solvent, structure of the reactants (different alkyl groups within the dialkyl acetylenedicarboxylates), and also the concentration of reactants on the rate of reaction. The mechanism was confirmed to involve a steady-state condition with the first step of the reaction being the rate-determining step.     

Single-step kinetics approximation employing non-Arrhenius temperature functions

Summary Solid-state reactions ordinarily demonstrate a tangled interplay of various chemical and physical processes. The single-step kinetics approximation resides in substituting a generally complex set of kinetic equations by the sole single-step kinetics equation. It enables to describe the kinetic hypersurface in a simple way irrespective of the complexity of the overall process. The kinetic hypersurface is the dependence of conversion on temperature and time. The functions describing the temperature and conversion components of the hypersurface should be separable. For a complex process, the adjustable parameters in the temperature function have no mechanistic significance so that there is no reason to be confined to the Arrhenius relationship. Two groups of isoconversional methods based on non-Arrhenius temperature functions are presented and the corresponding formulas for isothermal, integral, differential and incremental isoconversional methods are derived. As an example of the method using the explicit expression of the conversion function, the first-order kinetics is treated. Comparing with the methods based on the Arrhenius relationship, the greatest advantage of the methods presented here is that the problems with calculating the temperature integral are eliminated since the corresponding integrals can be expressed in a closed form.     

Kinetic Investigation of the Reaction between Triphenylphosphine,Dialkyl Acetylenedicarboxylate,and Carbazole by the UV Spectrophotometry Technique

The major objectives of the work undertaken were to carry out kinetic studies of the reaction between triphenylphosphine and dialkyl acetylenedicarboxylate in the presence of strong NH-acids, such as carbazole. To determine the kinetic parameters of the reaction, it was monitored by the UV spectrophotometer technique. The values of the second order rate constant (k ₂ ) were automatically calculated using standard equations within the program when the second order fits of the mentioned reactions were automatically drawn by the software associated with a Cary UV spectrophotometer model Bio-300 at an appropriate wavelength. At the temperature range studied, the dependence of the second order rate constant (Ln k) on reciprocal temperature was in agreement with the Arrhenius equation. This provided the relevant plots to calculate the activation energy of all reactions. Furthermore, useful information was obtained from studies of the effect of solvent and different alkyl groups within the dialkyl acetylenedicarboxylates on the rate of reactions.     

Temperature-dependent reaction kinetics of NH (a1Δ)

The gas-phase reactions of NH(a¹Δ) with H₂ and selected saturated and unsaturated hydrocarbons have been studied over the 250-600 K temperature range. Olefin reactions proceed at near the gas kinetic collision rate and show no temperature dependence. H₂ and saturated hydrocarbons show temperature-dependent reactions rates, with activation energies of = 0.8-2 kcai/mole. No evidence of electronic quenching of NH(a¹Δ) to the ground state was observed with any of the hydrocarbons studied. First-order reactions rates, Arrhenius A factors and activation energies for the reactions are reported. We discuss a mechanistic interpretation of the kinetics in view of earlier kinetic and reaction-product studies and ab initio SCF Cl calculations.     

Chemical kinetic simulations behind reflected shock waves

Chemical kinetic simulations that more accurately consider reaction conditions behind reflected shock waves in a high pressure shock tube have been conducted by accounting for (1) time‐dependent temperature and pressure variations in contrast to assuming constant temperature and pressure, (2) the inclusion of reactions during quenching by cooling in contrast to the assumption of zero kinetic contributions, and (3) real gas behaviors in contrast to assuming ideal gas conditions. The primary objective of the current work is to assess the degree of uncertainty associated with assuming constant temperature and pressure and that no reactions occur during the finite time of quenching and prefect gas behavior. The assessment of the subsequent effect of the uncertainty on chemical kinetic modeling is evaluated by conducting extensive comparative studies. In order to achieve this purpose, available CHEMKIN II and CHEMKIN Real Gas codes were utilized and modified to adopt the proposed approaches. From our computational experiments, it is found: (1) For shock tube experiment with less than a 15% endwall pressure increase, the conventional assumptions lead to reasonable accuracy in predicting stable species; (2) during reaction quenching, the consumption of radical species occurs efficiently and is nearly complete once the pressure drops to 50% of its highest value, but concentrations of stable species are insignificantly perturbed by reactions occurring during quenching; and (3) at elevated pressures, the real gas effects, which are a combination of nonideal P–V–T (state variables), thermodynamic, and kinetic behaviors, affect kinetics by speeding the reaction progress up slightly and do not significantly influence the development or validation of a detailed kinetic model from shock tube data that are obtained in a wide temperature range. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 38: 75–97, 2006     

UV spectrophotometric study of the kinetics and mechanism of the reactions between triphenylphosphine, dialkyl acetylenedicarboxylates and NH-acid

To determine the kinetic parameters of the reactions between triphenylphosphine and dialkyl acetylenedicarboxylates in the presence of an NH-acid, such as 2,3-di-hydroxybenzaldehyde, the reactions were monitored by UV spectrophotometry. The second order fits were automatically drawn and the values of the second order rate constants (k₂) were calculated using standard equations as part of the program. The dependence of the second order rate constant (lnk₂) on the reciprocal temperature was in agreement with the Arrhenius equation, in the temperature range studied, providing the relevant plots to calculate the activation energy of all reactions. Furthermore, we evaluated the effects of solvent, structure of different alkyl groups within the dialkyl acetylenedicarboxylates, and their concentration on the rates of reactions. The proposed mechanism was confirmed by experimental results and steady-state approximation. The first step (k₂) of the reaction was recognized as the rate determining step on the basis of experimental data.     

Kinetic parameter estimation of solvent‐free reactions monitored by 13C NMR spectroscopy,a case study: Mono‐ and di‐(hydroxy)ethylation of aniline with ethylene carbonate

The kinetics of solvent‐free reactions can be followed in situ by ¹³C nuclear magnetic resonance (NMR) spectroscopy, provided that the reaction mixture can be maintained liquid at the monitoring temperature. The pros and cons of the technique and the correct translation of the signal intensities into concentrations are discussed. A good model for this investigation is the reaction of ethylene carbonate ( 1 ) with aniline ( 2 ) at 140°C, two alkylation products of N‐mono‐ and N, N‐bis‐(2‐hydroxy)ethylation of aniline form (compounds 3 and 4 , respectively). The overall reaction occurs with heavy volume shrinking, so that the physical as well as the chemical features evolve during the course of the process. The chemical evolution is described by the kinetic constants k₁ and k₂ of the two N‐alkylation steps, the physical evolution by the time‐dependent activity coefficients α(t). Two complementary procedures are utilized for the determination of these parameters. © 2011 Wiley Periodicals, Inc. Int J Chem Kinet 43: 154–160, 2011     

Liquid-phase autoxidation of organic compounds at elevated temperatures. Absolute rate constant for intermolecular hydrogen abstraction in hexadecane autoxidation at 120–190°C

The absolute rate constants for the intermolecular hydrogen abstraction reactions of secondary hydrogens by secondary alkylperoxy radicals in hexadecane autoxidation, k₃, have been determined in the temperature range of 120–190°C using the stirred flow reactor technique. Absolute rate constants determined in this study for hexadecane are in good agreement with those determined for other hydrocarbons in liquid phase, on a per hydrogen basis, at lower temperatures. Arrhenius parameters for k₃/H derived from this study are A = 10^8.6 M^?1 s^?1 and E_a = 16.0 kcal/mol. The values of these parameters provide experimental confirmation for previous estimates made from both lower temperature reactions in the liquid phase and higher temperature reactions in the gas phase. © 1994 John Wiley & Sons, Inc.     

Bayesian uncertainty quantification of recent shock tube determinations of the rate coefficient of reaction H + O2→ OH + O

We analyze the ignition delay in hydrogen–oxygen combustion and the important chain ‐branching reaction H + O₂→ OH + O that occurs behind the shock waves in shock tube experiments. We apply a stochastic Bayesian approach to quantify uncertainties in the theoretical model and experimental data. The approach involves a statistical inverse problem, which has four “components” as input information: (a) model, (b) prior joint probability density function (PDF) of the uncertain parameters, (c) experimental data, and (d) uncertainties in the scenario parameters. The solution of this statistical inverse problem is a posterior joint PDF of the uncertain parameters from which we can easily extract statistical information. We first perform a parametric study to investigate how the level of the total uncertainty (which we define as the sum of model uncertainty and experimental uncertainty) affects the uncertainty in the rate coefficient k₁ of the reaction H + O₂→ OH + O, which is “most likely” expressed by k₁=1.73×10²³T^?2.5exp(?11550/T) cm³ mol^?1 s^?1 over the experimental temperature range 1100–1472 K. We also introduce the idea of “irreducible” uncertainty when considering other parameters in the system. After statistically calibrating the parameters modeling the rate coefficient k₁, we predict its 95% confidence interval (CI) for different temperature regimes and compare the CI against the values of k₁ obtained deterministically. Our results show that a small uncertainty in gas temperature (±5 K) introduces appreciable uncertainty in k₁. © 2012 Wiley Periodicals, Inc. Int J Chem Kinet 44: 586–597, 2012     

Free-radical chain dechlorination of chloroethanes in liquid triethylsilane

The kinetics and mechanism of the free-radical chain dechlorination of C₂Cl₆, C₂Cl₅H, and sym-C₂Cl₄H₂ in Et₃SiH wereinvestigated over a wide temperature range. The propagation step of the dechlorination of chloroethanes (C₂Cl_x H_6?x) proceeds by the following reactions: Analysis of the temperature dependence of product formation gave the Arrhenius expressions for k₄/k₃ which in turn were utilized for the estimation of the absolute Arrhenius parameters for hydrogen abstraction from Et₃SiH. Our results show that the values of E_abs are lower in Et₃SiH than in c-C₆H₁₂ by about 2–3 kcal/mol, while the A factors are almost equal. In competitive studies k₂ was determined versus Br abstraction from n-C₅H₁₁Br. The relative Arrhenius parameters determined by this method show that variations in both A factors and activationenergies are responsible for the reactivity trends observed in the Cl transfer reactions of Et₃Si radicals.     

Temperature‐dependent kinetics study of the gas‐phase reactions of atomic chlorine with acetone, 2‐butanone,and 3‐pentanone

A laser flash photolysis–resonance fluorescence technique has been employed to study the kinetics of the reactions of atomic chlorine with acetone (CH₃C(O)CH₃; k₁), 2‐butanone (C₂H₅C(O)CH₃; k₂), and 3‐pentanone (C₂H₅C(O)C₂H₅; k₃) as a function of temperature (210–440 K) and pressure (30–300 Torr N₂). No significant pressure dependence is observed for any of the reactions studied. Arrhenius expressions (units are 10^?11 cm³ molecule^?1 s^?1) obtained from the data are k₁(T) = (1.53 ± 0.19) exp[(?594 ± 33)/T], k₂(T) = (2.77 ± 0.33) exp[(+76 ± 33)/T], and k₃(T) = (5.66 ± 0.41) exp[(+87 ± 22)/T], where uncertainties are 2σ and represent precision only. The accuracy of reported rate coefficients is estimated to be ±15% over the entire range of pressure and temperature investigated. The room temperature rate coefficients reported in this study are in good agreement with a majority of literature values. However, the activation energies reported in this study are in poor agreement with the literature values, particularly for 2‐butanone and 3‐pentanone. Possible explanations for discrepancies in published kinetic parameters are proposed, and the potential role of Cl + ketone reactions in atmospheric chemistry is discussed. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 40: 259–267, 2008     

Pressure and temperature dependence of reactions proceeding via a bound complex. An approach for combustion and atmospheric chemistry modelers. Application to HO + CO → [HOCO] → H + CO2

Elementary bimolecular processes that involve formation of a chemically activated intermediate species are common. We address the general problem of modeling these processes and describe the necessary and sufficient information that must be specified to assure that a kinetics model will extrapolate the rate constants for those reactions over wide ranges of temperature and pressure. The approach is illustrated for the system centered around the HOCO intermediate. Here, specification of the temperature and pressure dependence of three rate constants, k(T,P) and the temperature dependence of two equilibrium constants, K(T), is necessary and sufficient, viz: Rate constants are cast in the form of an analytical expression, suggested by Troe, and appropriate parameters are tabulated.     

Kinetic study of the gas‐phase reaction of the nitrate radical with methyl‐substituted thiophenes

The gas‐phase reactions of the NO₃ radical with 2‐methylthiophene, 3‐methylthiophene, and 2,5‐dimethylthiophene have been studied, using relative and absolute methods at 298 K. Determination of relative rate was performed using Teflon collapsible bag as the reaction chamber and gas chromatography as the analytical tool. For the absolute method, experiments were carried out using fast‐flow‐discharge technique with detection of NO₃ by laser‐induced fluorescence. The temperature dependence was studied by the absolute technique for the reactions of NO₃ with 2‐methylthiophene and 3‐methylthiophene in the range 263–335 K. The proposed Arrhenius expressions for the reaction of the nitrate radical with 2‐methylthiophene and 3‐methylthiophene are k = (4 ± 2) × 10^?16 exp[?(2200 ± 100)/T]] cm³ molecule^?1 s^?1 and k = (3 ± 2) × 10^?15 exp[?(1700 ± 200)/T]] cm³ molecule^?1 s^?1, respectively. © 2003 Wiley Periodicals, Inc. Int J Chem Kinet 35: 286–293, 2003     

Kinetic isotope effects and rate constants for the gas‐phase reactions of three deuterated toluenes with OH from 298 to 353 K

The rate constants for the gas‐phase reactions of three deuterated toluenes with hydroxyl radicals were measured using the relative rate technique over the temperature range 298–353 K at about 1 atm total pressure. The OH radicals were generated by photolysis of H₂O₂, and helium was used as the diluent gas. The disappearance of reactants was followed by online mass spectrometry, which resulted in high time resolution, allowing for a large amount of data to be collected and used in the determination of the Arrhenius parameters. The following Arrhenius expressions have been determined for these reactions (in units of cm³ molecule^?1 s^?1): k=(6.42_?0.99^+1.17)×10^?13exp [(661±54)/T] for toluene‐d₃, k=(2.11_?0.69^+1.03)×10^?12exp [(287±128)/T]for toluene‐d₅, and k=(1.40^+0.44_?0.33)×10^?12exp [(404±88)/T]for toluene‐d₈. The kinetic isotope effects (KIEs, k_H/k_D) of these reactions were 1.003 ± 0.042 for all three compounds at 298 K. The KIE for toluene‐d₃ was temperature dependent; at 350 K, its KIE was 1.122^+0.048_?0.046. The KIE of toluene‐d₅ and toluene‐d₈ did not vary significantly with temperature. These KIE results suggest that methyl H‐atom abstraction is more important than aromatic OH addition at higher temperatures. © 2012 Wiley Periodicals, Inc. Int J Chem Kinet 44: 821–827, 2012     

Thermoset Cure Kinetics by Isoconversional Methods

The curing kinetics of thermosets based on unsaturated polyester resin crosslinked with styrene was studied by differential scanning calorimetry (DSC). The isoconversional kinetic analysis was applied to non-isothermal data. The results obtained show a dependence of the activation energy (E_α) on conversion (α) that proves the existence of a multistep process and a complex kinetic scheme that must be interpreted in terms of chemical and physical mechanisms. The interrelationship of the Arrhenius parameters obtained from the isoconversional kinetic data has been used as a tool to investigate the production of free radicals by the action of a promoter (cobalt octoate) and the temperature on the initiator (methyl ethyl ketone peroxide). An optimum promoter/initiator ratio has been found. This revised version was published online in August 2006 with corrections to the Cover Date.     

Development of a Joint Hydrogen and Syngas Combustion Mechanism Based on an Optimization Approach

A comprehensive and hierarchical optimization of a joint hydrogen and syngas combustion mechanism has been carried out. The Kéromnès et al. (Combust Flame, 2013, 160, 995–1011) mechanism for syngas combustion was updated with our recently optimized hydrogen combustion mechanism (Varga et al., Proc Combust Inst, 2015, 35, 589–596) and optimized using a comprehensive set of direct and indirect experimental data relevant to hydrogen and syngas combustion. The collection of experimental data consisted of ignition measurements in shock tubes and rapid compression machines, burning velocity measurements, and species profiles measured using shock tubes, flow reactors, and jet‐stirred reactors. The experimental conditions covered wide ranges of temperatures (800–2500 K), pressures (0.5–50 bar), equivalence ratios (? = 0.3–5.0), and C/H ratios (0–3). In total, 48 Arrhenius parameters and 5 third‐body collision efficiency parameters of 18 elementary reactions were optimized using these experimental data. A large number of directly measured rate coefficient values belonging to 15 of the reaction steps were also utilized. The optimization has resulted in a H₂/CO combustion mechanism, which is applicable to a wide range of conditions. Moreover, new recommended rate parameters with their covariance matrix and temperature‐dependent uncertainty ranges of the optimized rate coefficients are provided. The optimized mechanism was compared to 19 recent hydrogen and syngas combustion mechanisms and is shown to provide the best reproduction of the experimental data.