Error evaluation of integral methods by consideration on the approximation of temperature integral

Summary In this paper, the integral methods in general use are divided into two types in terms of their different ways to in order to deal with the temperature integral p(x): for Type A the function h(x)=p(x)x²e^x is regarded as constant vs. x, while for Type B h(x) varies vs. x and ln[p(x)] is assumed to have the approximation form of ln[p(x)]=alnx+bx+c (the coefficients a, b, and c are constant). The errors of kinetic parameters calculated by these two types of methods are derived as functions of x and analyzed theoretically. It is found that Type A methods have the common errors of activation energy, while the Coats-Redfern method can lead to more accurate value of frequency factor than others. The accuracy of frequency factor can be further enhanced by adjusting the expression of the Coats-Redfern approximation. Although using quite simple approximation of the temperature integral, the Coats-Redfern method has the best performance among Type A methods, implying that usage of a sophisticated approximation may be unnecessary in kinetic analysis. For Type B, the revised MKN method has a lower error in activation energy and an acceptable error in frequency factor, and thus it can be reliably used. Comparatively, the Doyle method has higher error of activation energy and great error of the frequency factor, and thus it is not recommended to be adopted in kinetic analysis.     

Dependence of the preexponential factor on temperature

Summary The dependence of the preexponential factor on the temperature has been examined and the errors involved in the activation energy calculated from isothermal and non-isothermal methods without considering such dependence have been estimated. It has been shown that the error in the determination of the activation energy calculated ignoring the dependence of Aon Tcan be rather large and it is dependent on x=E/RT, but independent of the experimental method used. It has been also shown that the error introduced by omitting the dependence of the preexponential factor on the temperature is considerably larger than the error due to the Arrhenius integral approach used for carrying out the kinetic analysis of TG data.     

Study on kinetics of combustion of brick-shaped carbonaceous materials

Combustion of brick-shaped carbonaceous materials (carbon deposits from coke oven, coke and electrographite) was carried out in thermobalance in static air. Analysis of kinetics of the process was carried out using both classical (Arrhenius law) and newer (three-parametric equation) methods. In classical approach two types of kinetic equations were used in calculations: differential and integral. The results obtained show that, independently on kinetic variables (α – conversion degree or m – mass of sample) used in differential equations, kinetics of combustion of brick-shaped carbonaceous materials is characterized by only one pair of Arrhenius coefficients: activation energy (E) and pre-exponential constant (A). At the same time the integral equation demonstrates distinction in relation to methods based on differential equations, generating higher activation energies and separate isokinetic effect (IE). Parallel IE shows that kinetic analysis has to encompass activation energy in connection to second coefficient, pre-exponential constant A, depending on assumptions made for kinetic equations. On the other hand three-parametric equation allows describing kinetic of combustion in alternative way using only one experimental value – initial temperature in form of point of initial oxidation (PIO) – and also offers new methods of interpretation of the process.     

鬼臼毒素及其衍生物热分解反应的动力学研究

The thermal behavior and thermal decomposition kinetic parameters of podophyllotoxin （1） and 4 derivatives, picropodophyllin （2）, deoxypodophyllotoxin （3）, fl-apopicropodophyllin （4）, podophyllotoxone （5） in a temperature-programmed mode have been investigated by means of DSC and TG-DTG. The kinetic model functions in differential and integral forms of the thermal decomposition reactions mentioned above for first stage were established. The kinetic parameters of the apparent activation energy Ea and per-exponential factor A were obtained from analy- sis of the TG-DTG curves by integral and differential methods. The most probable kinetic model function of the decomposition reaction in differential form was （1- a）^2 for compounds 1-3,2/3·a^-1/2 for compound 4 and 1/2（1-a）·[-In（1-a）]^-1 for compound 5. The values of Ea indicated that the reactivity of compounds 1-5was increased in the order： 5〈4〈2〈1〈3. The values of the entropy of activation △S^≠, enthalpy of activation △H^≠ and free energy of activation △G^≠ of the reactions were estimated. The values of △G^≠ indicated that the thermal stability of compounds 1-3 with the samef（a） was increased in the order： 2〈3〈1.     

Kinetic analysis of solid‐state reactions: Precision of the activation energy calculated by integral methods

The integral methods are extensively used for the kinetic analysis of solid‐state reactions. As the Arrhenius integral function [p(x)] does not have an exact analytical solution, different approximated equations have been proposed in the literature for performing the kinetic analysis of experimental integral data. Since the first approximation of Van Krevelen, a large number of equations have been proposed with the objective of increasing the precision in the determination of the Arrhenius integral, as checked from the standard deviation of the approximated function with regard to the real exact value of the integral. However, the main application of these equations is the determination of the kinetic parameters, in particular activation energies, and not the computation of the Arrhenius integral. A systematic analysis of the errors involved in the determination of the activation energy from these integral methods is still missing. A comparative study of the precision of the activation energy as a function of x and T computed from the different integral methods has been carried out. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 658–666, 2005     

Errors involved in the activation energy calculated by integral methods when the frequency factor depends on the temperature ( A = A 0 T m)

The dependence of the frequency factor on the temperature (A=A ₀ T ^m) has been examined and the errors involved in the activation energy calculated from some integral methods without considering such dependence have been estimated. Investigated integral methods are the Coats-Redfern method, the Gorbachev-Lee-Beck method, the Wanjun-Yuwen method and the Junmeng-Fusheng method. The results have shown that the error in the determination of the activation energy calculated ignoring the dependence of the frequency factor on the temperature can be rather large and it is dependent on x=E/RT and the exponent m.     

Kinetics study of norbixin’s first stage thermal decomposition,using dynamic method

Cis-norbixin isomer obtained by hydrolysis of cis-bixin and isolated by solvent extraction from annatto seeds. The thermal decomposition data of the cis-norbixin samples were analyzed by thermogravimetric analysis at different heating rates in the 25–900°C temperature range. DSC curves showed that thermal decomposition reactions for cis-norbixin occurred in the solid phase. The kinetic parameters, such as activation energy and pre-exponential factor were determined using integral and approximate methods: Coats–Redfern, Madhusudanan, Horowitz–Metzger and Van Krevelen. F1 mechanism describes well the first stage of the thermal decomposition.     

Non-isothermal kinetics in solids

The integral methods, which are obtained from the various approximations for the temperature integral, have been extensively used in the non-isothermal kinetic analysis. In order to obtain the precision of the integral methods for the determination of the activation energy, several authors have calculated the relative errors of the activation energy obtained from the integral methods. However, in their calculations, the temperature integral at the starting temperature was neglected. In this work, we have performed a systematic analysis of the precision of the activation energy calculated by the integral methods without doing any simplifications. The results have shown that the relative error involved in the activation energy determined from the integral methods depends on two dimensionless quantities: the normalized temperature θ=T/T ₀, and the dimensionless activation energy x ₀=E/RT ₀ (where E is the activation energy, T is the temperature, T ₀ is the starting temperature, R is the gas constant).     

Kinetics of the Exothermic Decomposition Reaction of N-Methyl-N-nitro-2,2,2- trinitroethanamine

The thermal behavior and kinetic parameters of the exothermic decomposition reaction of N-methyl-N-nitro-2,2,2-trinitroethanamine in a temperature-programmed mode have been investigated by means of differential scanning calorimetry (DSC).The kinetic equation of the exothermic decomposition process of the compound is proposed. The values of the apparent activation energy (Ea), pre-exponential factor (A), entropy of activation (ΔS^≠ ), enthalpy of activation (ΔH^≠ ), and free energy of activation (ΔG^≠ ) of this reaction and the critical temperature of thermal explosion of the compound are reported. Information is obtained on the mechanism of the initial stage of the thermal decomposition of the compound.     

Comparative kinetic study of the non-isothermal thermal curing of <Emphasis Type="Italic">bis</Emphasis>-GMA/TEGDMA systems

The thermal polymerization kinetics of dimethacrylate monomers was studied by differential calorimetry using non-isothermal experiments. The kinetic analysis compared the following procedures: isoconversional method (model-free method), reduced master curves, the isokinetic relationship (IKR), the invariant kinetic parameters (IKP) method, the Coats-Redfern method and composite integral method I. Although the study focused on the integral methods, we compared them to differential methods. We saw that even relatively complex processes (in which the variations in the kinetic parameters were only slight) can be described reasonably well using a single kinetic model, so long as the mean value of the activation energy is known (E). It is also shown the usefulness of isoconversional kinetic methods, which provide with reliable kinetic information suitable for adequately choosing the kinetic model which best describes the curing process. For the system studied, we obtained the following kinetic triplet: f(α)=α^0.6(1−α)^2.4, E=120.9 kJ mol⁻¹ and lnA=38.28 min⁻¹.     

Dependence of the frequency factor on the temperature: a new integral method of nonisothermal kinetic analysis

A new integral method of nonisothermal kinetic analysis has been developed with the dependence of the frequency factor on the temperature (A = A ₀ T ^m ). The new integral method is obtained from the newly proposed approximation for the general temperature integral, which is more accurate than the other existed approximations. For applications, nonisothermal thermoanalytical data obtained by theoretical simulation have been processed. The results have shown that the newly proposed integral method is an ideal solution for the evaluation of kinetic parameters from nonisothermal thermoanalytical data with the frequency factor dependent the temperature.     

Thermal properties,induction period,interfacial energy and nucleation parameters of solution grown benzophenone

The kinetic parameters (reaction order, n, activation energy, E, pre-exponential factor, A, constant rate, k) for the dehydration step due to elimination of osmotic water and hydrogen-bounded water with the carboxylic groups, and for the anhydrifying step owing to the dehydration of two neighboring (-COOH) groups, were determined under non-isothermal conditions for some carboxylic resins with acrylic-divinylbenzene (DVB) matrix. The kinetic parameters were evaluated by means of isoconversional methods from (TG/DTG) thermal analysis data. The results show a dependence of the apparent kinetic parameters on the cross-linking degree, granulation, gel/macroporous matrix nature, exchange capacity and heating rate.     

Improved version of Doyle integral method for nonisothermal kinetics of solid-state reactions

An improved version of Doyle integral method for the determination of the kinetic parameters from nonisothermal thermoanalytical data has been presented. The relative errors involved in the activation energy and frequency factor determined from Doyle integral method and its improved integral method have been estimated. The results have shown that the precision of the improved version of Doyle integral method for the determination the kinetic parameters (including the activation energy and frequency factor) is much higher than that of Doyle integral method.     

Comparative method to evaluate reliable kinetic triplets of thermal decomposition reactions

Reliable kinetic information for thermal analysis kinetic triplets can be determined by the comparative method: (1) An iterative procedure or the KAS method had been established to obtain the reliable value of activation energy E _a of a reaction. (2) A combined method including Coats-Redfern integral equation and Achar differential equation was put forward to confirm the most probable mechanism of the reaction and calculate the pre-exponential factor A. By applying the comparative method above, the thermal analysis kinetic triplets of the dehydration of CaC₂O₄·H₂O were determined, which apparent activation energy: 81±3 kJ mol^-1, pre-exponential factor: 4.51·106-1.78·10⁸ s^-1, the most probable mechanism function: f(α)=1 or g(α)=α, which the kinetic equation of dehydration is dα/dt=Ae^-E _a ^/RT. This revised version was published online in August 2006 with corrections to the Cover Date.     

Thermal decomposition study of bismuth (III) trichloride complex with 1,10-phenanthroline as the ligand

The complex BiCl₃·L (L = 1,10-phenanthroline) was synthesized and characterized by elemental analysis and infrared spectroscopy (IR). Infrared spectroscopy data suggested that the nitrogen atom of the aromatic ring is bonded to the bismuth atom. The kinetic study of thermal degradation was determined by non-isothermal thermogravimetry. Two methods based on integral equation of Coats-Redfern, were necessary for determining the kinetic trip: the fitting method, known as the checking model and an iso-conversional method. The latter gives the activation energy for each degree of conversion and the first, the kinetic model gives activation energy and the pre-exponential factor for thermal decomposition processes that occur through a single simple mechanism. The kinetic parameters, E _a and log A for the heating rates of 5, 10, and 15 min K⁻¹, were determined considering the decomposition model denoted by F0/R1 in the range of degree of conversion between 0.065 and 0.71.     

Kinetic Analysis of Thermal Decomposition of Ca(OH)2 Formed During Hydration of Commercial Portland Cement by DSC

In this paper, evaluation of kinetic parameters (the activation energy – E,the pre-exponential factor – A and the reaction order – n) with simultaneous determination of the possible reaction mechanism of thermal decomposition of calcium hydroxide (portlandite), Ca(OH)₂ formed during hydration of commercial Portland-slag cement, by means of differential scanning calorimetry (DSC) in non-isothermal conditions with a single heating–rate plot has been studied and discussed. The kinetic parameters and a mechanism function were calculated by fitting the experimental data to the integral, differential and rate equation methods. To determine the most probable mechanism, 30 forms of the solid-state mechanism functions, f(α_c) have been tried. Having used the procedure developed and the appropriate program support, it has been established that the non-isothermal thermal decomposition of calcium hydroxide in the acceleratory period (0.004<α_c<0.554) can be described by the rate equation: d α_c/dT=A/βexp(−E/RT)f(α_c), which is based on the concept of the mechanism reaction:f(α_c)=2(α_c)^1/2. The mechanism functions as well as the values of the kinetic parameters are in good agreement with those given in literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

Non-isothermal kinetic study on the decomposition of Zn acetate-based sol-gel precursor

The paper presents a non-isothermal kinetic study of the decomposition of Zn acetate-based gel precursors for ZnO thin films, based on the thermogravimetric (TG) data. The evaluation of the dependence of the activation energy (E) on the mass loss (Δm) using the isoconversional methods (Friedman (FR), Flynn-Wall-Ozawa (FWO) and Kissinger-Akahira-Sunose (KAS)) has been presented in a previous paper. It was obtained that the sample dried at 125°C for 8 h exhibits the activation energy independent on the heating rate for the second decomposition step. In this paper the invariant kinetic parameter (IKP) method is used for evaluating the invariant activation parameters, which were used for numerically evaluation of the function of conversion. The value of the invariant activation energy is in a good agreement with those determined by isoconversional methods. In order to determine the kinetic model, IKP method was associated with the criterion of coincidence of the kinetic parameters for all heating rates. Finally, the following kinetic triplet was obtained: E=91.7 (±0.1) kJ mol⁻¹, lnA(s⁻¹)=16.174 (±0.020) and F1 kinetic model.     

The Kissinger law and the IKP method for evaluating the non-isothermal kinetic parameters

The following problems concerning the apparent compensation effect (CE) (lnA=a+bE, where A is the pre-exponential factor, E is the activation energy, a and b are CE parameters) due to the change of the conversion function and on which the invariant kinetic parameters method (IKP method) is based, are discussed: (1) the explanation of this kind of CE; (2) the choice of the set of conversion functions that checks CE relationship; (3) the dependencies of CE parameters on the heating rate and the temperature corresponding to the maximum reaction rate. Using the condition of maximum of the reaction rate suggested by Kissinger (Kissinger law), it is pointed out that, for a certain heating rate, the CE relationship is checked only for reaction order (Fn) and Avrami-Erofeev (An) kinetic models, and not for diffusion kinetic models (Dn). Consequently, IKP method, which is based on the supercorrelation relationship between CE parameters, can be applied only for the set Fn+ An of kinetic models. The dependencies of a and b parameters on the heating rate and T _m (temperature corresponding to maximum reaction rate) are derived. The theoretical results are discussed and checked for (a) TG simulated data for a single first order reaction; (b) TG data for PVC degradation; (b) the dehydration of CaC₂O₄·H₂O.     

Kinetic parameters determination in non-isothermal conditions for the crystallisation of a silica-soda-lead glass

Two integral isoconversional methods (Flynn–Wall–Ozawa and Kissinger–Akahira–Sunose) and the invariant kinetic parameters method (IKP) were used in order to examine the kinetics of the non-isothermal crystallisation of a silica-soda-lead glass. The objective of the paper is to show the usefulness of the IKP method to determine both the activation parameters and the kinetic model of the investigated process. Thismethod associated with the criterion of coincidence of kinetic parameters for all heating rates and some procedures of the evaluation of the parameter from Johnson–Mehl–Avrami–Erofeev–Kolmogorov (JMAEK) equation led us to the following kinetic triplet: activation energy, E=170.5±2.5 kJ mol^–1 , pre-exponential factor, A=1.178±0.350·10 10 min^–1 and JMAEK model (A _m) m=1.5.     

非等温反应过程中新的动力学方程

对于非等温过程中的动力学方程，正确的Arrhenius方程的温度积分应该是从T₂到T₁，但是许多动力学方程中的温度积分是从T到0 K，例如Ozawa等方程。我们的研究指出对于某些反应，这些方程中的活化能存在较大的误差，因此我们提出了一个新的动力学方程。凭借等转化率法，应用新的方程可以精确求解线性或非线性加热过程中化学反应的活化能。用新方程对2个经典反应（聚酰胺的热裂解和一水草酸钙的热分解）的研究表明：Ozawa方程的活化能有时是精确的，有时偏差太大。