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).     

Kinetic analysis of solid-state reactions: errors involved in the determination of the kinetic parameters calculated by one type of integral methods

The integral methods are extensively used for performing the kinetic analysis of solid-state reactions. As the Arrhenius integral function p(u) does not have an exact analytical solution, many approximations have been proposed. One popular type of approximations is called the exponent approximation which can be put in the form . In this study, a systematic analysis of the errors involved in the determination of the kinetic parameters calculated by the integral methods based on the exponent approximations for p(u) has been carried out. The results have shown that the precision of the kinetic parameters computed from the integral methods analyzed in this paper depends on u and the errors of the kinetic parameters determined from Doyle approach are the largest.     

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.     

Precision of integral methods for the determination of the kinetic parameters

In this paper, a systematic analysis of the errors involved in the determination of the kinetic parameters (including the activation energy and frequency factor) from five integral methods has been carried out. The integral methods analyzed here are Coats-Redfern, Gorbachev, Wanjun-Yuwen-Hen-Zhiyong-Cunxin, Junmeng-Fusheng-Weiming-Fang, Junmeng-Fang and Junmeng-Fang-Weiming-Fusheng method. The results have shown that the precision of the kinetic parameters calculated by the different integral methods is dependent on u (E/RT), that is, on the activation energy and the average temperature of the process.     

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.     

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.     

Kinetic analysis of solid-state reactions: Evaluation of approximations to temperature integral and their applications

The temperature integral, which has no exact analytical solution, is involved in the analysis of the experiment data obtained under nonisothermal conditions. Some approximations for the temperature integral have been proposed in the literature for the determination of the kinetic parameters, in particular the activation energy. Those approximations are classified into two categories, that is, exponential and rational approximations. The precision of them for estimating the temperature integral was evaluated within a certain continuous range rather than at several discrete points. Some applications of the approximations in the kinetic methods were presented. The relative errors of the activation energy and pre-exponential factor with four rational approximations by employing model-fitting method were calculated. The relative errors of the activation energy for a series of conversion rate with four rational and four exponential approximations by employing linear integral isoconversional methods were evaluated.     

Approximations for the temperature integral

The temperature integral cannot be analytically integrated and many simple closed-form expressions have been proposed to use in the integral methods. This paper first reviews two types of simple approximation expressions for temperature integral in literature, i.e. the rational approximations and exponential approximations. Then the relationship of the two types of approximations is revealed by the aid of a new equation concerning the 1^st derivative of the temperature integral. It is found that the exponential approximations are essentially one kind of rational approximations with the form of h(x)=[x/(Ax+k)]. That is, they share the same assumptions that the temperature integral h(x) can be approximated by x/Ax+k). It is also found that only two of the three parameters in the general formula of exponential approximations are needed to be determined and the other one is a constant in theory. Though both types of the approximations have close relationship, the integral methods derived from the exponential approximations are recommended in kinetic analysis.     

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.     

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。     

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.     

Kinetic studies on the thermal polymerization of N‐chloroacetyl‐11‐aminoundecanoate potassium salt

A potassium salt of N‐chloroacetyl‐11‐aminoundecanoate was thermally polymerized to obtain the corresponding poly(glycolic acid‐alt‐11‐aminoundecanoic acid). A kinetic study was then performed that was based on isothermal and nonisothermal polymerizations performed in a differential scanning calorimeter. The complete kinetic triplet was determined (the activation energy, pre‐exponential factor, and integral function of the degree of conversion). A kinetic analysis was performed with an integral isoconversional procedure (free model), and the kinetic model was determined both with the Coats–Redfern method (the obtained isoconversional value being accepted as the effective activation energy) and through the compensation effect. The polymerization followed a three‐dimensional growth‐of‐nuclei (Avrami) kinetic mechanism. Isothermal polymerization was simulated with nonisothermal data. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 1166–1176, 2005     

Kinetic and thermodynamic treatment of gasification process for some s-triazines

Solid-gas phase transition processes of some triazines were studied from kinetic and thermodynamic viewpoint. DSC measurements and Clausius-Clapeyron equation were used to determine enthalpy values related to these processes. Model-fitting methods (based on Arrhenius, Šatava equations and Šesták-Berggren equations) and model-free methods (based on Ozawa-Flynn-Wall and Kissinger equations) allow to hypothesis R2 mechanism. An attempt to determine the activation parameters (ΔH ^#, ΔG ^#, ΔS ^#) related to these processes was carried out. Accordance between the activation enthalpy values with those of activation energy obtained by means of kinetic methods and with the experimental (DSC) and calculated (Clausius-Clapeyron) enthalpy values was found. This revised version was published online in August 2006 with corrections to the Cover Date.     

New linear integral kinetic parameters assessment method based on an accurate approximate formula of temperature integral

This work concerns a proposition of a new assessment method to obtain kinetic parameters from nonisothermal solid-state kinetics, based on a new and accurate approximate formula of temperature integral. The new formula was derived numerically by a two-step linearly fitting process without using any further approximating series. The relative error involved in the activation energy has been estimated and found to be less than 0.001% in the practical range of 15 < x < 60. A comparison of the suggested approximations to published approximates has shown significant improvements in terms of accuracy at high and low x values. The validity of the new method has been confirmed by computing activation energy from experimental data. Moreover, two approaches have been proposed to determine the kinetic reaction model and preexponential factor based on the new approximate formula. The comparison of the obtained results arising from the application of the present method to others obtained by the most widely reported methods in the literature shows a remarkable preeminence of the new method.     

On the expedience for practice to search for a more accurate analytical solution of the integral in the Arrhenius equation in non-isothermal kinetics

The existing methods of approach to solve the integral in the Arrhenius equation (Coats-Redfern, Gorbachev, Zsakó, Balarin etc.), when the standard linearization method of the integral kinetic equation is applied in order to determine the value of the activation energyE, yield factually identical results. Hence attempts to find more accurate approaches have no practical sense.     

Kinetic analysis of solid-state reactions: A new integral method for nonisothermal kinetics with the dependence of the preexponential factor on the temperature (A = A0Tn)

In this study, we have proposed a new approximation for the general temperature integral, which frequently occurs in the nonisothermal kinetics with the dependence of the preexponential factor on the temperature and has no exact analytical solution. The validity of the new approximation has been tested by some numerical analyses. As the solution of the general temperature integral, the new approximation is more accurate than other approximations. Based on the newly proposed approximation, the corresponding integral method has been given. The precision of the integral methods for the determination of the activation energy has been calculated, and the results have shown that the relative error involved in the activation energy obtained from the new integral method is smaller than that from other integral methods. For applications, nonisothermal data obtained by theoretical simulation have been successfully processed using the new integral method.     

The Accuracy of Senum and Yang's Approximations to the Arrhenius Integral

The accuracy of the integral of the Arrhenius equation, as determined from the 1^st to the 4^th degree rational approximation proposed by Senum and Yang, has been calculated. The precision of the 5^th to 8^th rational approximations, here proposed for the first time, has also been analyzed. It has been concluded that the accuracy increases by increasing the order of the rational approximation. It has been shown that these approximations to the Arrhenius equation integral would allow an accuracy better than 10⁻⁸ % in the E/RT range generally observed for solid state reactions. Moreover, it has been demonstrated that errors closed to 10⁻² % can be obtained even for E/RT=1, provided that high enough degrees of rational approximation have been used. Thus, it would be reasonable to assume that high degree rational approximations for the Arrhenius integral could be used for the kinetic analysis of processes, like adsorption or desorption of gases on solid surfaces, which can take place at low temperatures with very low values of E/RT. This revised version was published online in August 2006 with corrections to the Cover Date.     

Identification of the effective distribution function for determination of the distributed activation energy models using the maximum likelihood method: Isothermal thermogravimetric data

The new procedure for identification of the effective distribution function for determination of the distributed activation energy models, which is based on use the maximum likelihood method (MLM), was established. The five different continuous probability functions (exponential, logistic, normal, gamma, and Weibull probability functions (the extended set of distributions)) were used for searching the best reactivity model for two heterogeneous processes: (a) the isothermal reduction process of nickel oxide under hydrogen atmosphere and (b) the isothermal degradation process of bisphenol‐A polycarbonate (Lexan) under nitrogen atmosphere. The MLM showed that for both processes, the most suitable reactivity model represents the Weibull distribution model. It was concluded that the values of Arrhenius parameters (ln A and E_a), evaluated from the Weibull distribution model, represent the effective kinetic values for both considered processes. This procedure enables identification the suitable distribution model for considered process only from the experimental data (based on the shapes of obtained integral kinetic curves), and this fact represents the advantage of established analysis. The established mathematical procedure, which is based on the MLM, can be applied as the preliminary analysis for evaluating the distribution of activation energies for complex heterogeneous processes. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 41: 27–44, 2009     

Pyrazine Complexes of Octacyanometallates of Mo(IV) and W(IV) With 8-hydroxyquinoline: Synthesis,characterisation and thermal studies

The complexes formed by photosubstitution of pyrazine (Pz) in octacyanomolybdate(IV) and -tungstate(IV) with 8-hydroxyquinoline have been assigned the formulae [Mo(CN)₂(OH)₂(Pz)₂(OX)] and [W(CN)₂(OH)₂(Pz)₂(OX)·1.5H₂O]. Coordination of Pz as an unidentate ligand by donating a lone pair of electron from nitrogen is shown by an absorption peak between 8–11 µ. Mechanism for the thermal decomposition of the complexes has been given. The formation of tungsten metal as residue in case of II has been confirmed by XRD analysis. The kinetic and thermodynamic parameters like activation energy (E _a), pre-exponential factor (A) and entropy of activation (S ^#) were calculated employing different integral methods of Doyle, Coats and Redfern and Arrhenius.H for each stage of decomposition was obtained from DSC.This revised version was published online in November 2005 with corrections to the Cover Date.     

Kinetics of the exothermic decomposition of 1-nitro-3-(β,β,β-trinitroethyl)-4,5-dinitroiminoimidazolidine-2-one

The kinetic parameters of the exothermic decomposition of the title compound in a temperatureprogrammed mode have been studied by means of DSC. The DSC data obtained are fitted to the integral, differential, and exothermic rate equations by the linear least-squares, iterative, combined dichotomous, and least-squares methods, respectively. After establishing the most probable general expression of differential and integral mechanism functions by the logical choice method, the corresponding values of the apparent activation energy (E _a), preexponential factor (A), and reaction order (n) are obtained by the exothermic rate equation. The results show that the empirical kinetic model function in differential form and the values of E _a and A of this reaction are (1 − α)^−4.08, 149.95 kJ mol⁻¹, and 10^14.06 s⁻¹, respectively. With the help of the heating rate and kinetic parameters obtained, the kinetic equation of the exothermic decomposition of the title compound is proposed. The critical temperature of thermal explosion of the compound is 155.71°C. The above-mentioned kinetic parameters are quite useful for analyzing and evaluating the stability and thermal explosion rule of the title compound. The text was submitted by the authors in English.