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.     

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.     

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.     

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     

Study of the influence of a reactive diluent on the rheological properties of an epoxy-diamine system

The generalized temperature integral I(m, x) appears in non-isothermal kinetic analysis when the frequency factor depends on the temperature. A procedure based on Gaussian quadrature to obtain analytical approximations for the integral I(m, x) was proposed. The results showed good agreement between the obtained approximation values and those obtained by numerical integration. Unless other approximations found in literature, the methodology presented in this paper can be easily generalized in order to obtain approximations with the maximum of accurate.     

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.     

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.     

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.     

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.     

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.     

Supplement on applicability of the Kissinger equation in thermal analysis

The relative errors (e%) in the determination of the activation energy from the slope of the Kissinger straight line ln(β/βT _p²) vs. 1/T _p (β is the heating rate) are in-depth discussion. Our work shows that the relative errors is a function containing the factors of x _p and Δx _p, not only x _p (x _p = E/RT _p, E is the activation energy, T _p is the temperature corresponding to maximum process rate, R is the gas constant). The relative error between E _k and E _p will be smaller with the increase of the value of x and/or with the decrease of the value of Δx. For a set of different heating rates in thermal analysis experiments, the low and close heating rates are proposed from the kinetic theory.     

New algorithm for the numerical solution of the integro-differential equation with an integral boundary condition

In this paper, a sequence of approximate solution converging uniformly to the exact solution for a class of integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics and other field of physics and mathematical chemistry is obtained by using an iterative method. Its exact solution is represented in the form of series in the reproducing kernel space. The n-term approximation u _n (x) is proved to converge to the exact solution u(x). Moreover, the first derivative of u _n (x) is also convergent to the first derivative of u(x).     

Combined kinetic analysis of solid-state reactions: The integral method (ICKA)

In this work, we propose the first Integral method for the Combined Kinetic Analysis (ICKA) of solid-state reactions typically performed in a thermogravimetric analyzer. The ICKA method prevents the systematic inaccuracies inherent to all the differential methods, including the standard CKA method. Two main achievements have been made for implementing the method: (1) the most accurate approximation for the general temperature integral yet developed, and (2) a general integral form of the kinetic model of the type g (α) = (abcZ)⁻¹ [1 − (1 − α^a)^b]^c, where Z is a parameter evaluated together with the preexponential factor and a, b, and c are fitting parameters. This expression allows any known kinetic model to be exactly or very closely reproduced. Together, the two developments yield an equation for the conversion, α, that has been successfully fitted to simulated conversion values of single-step reaction processes following different kinetic models. The curve fitting resulted in the same values of the kinetic and model parameters as those from which the simulated conversion curves were originally built, proving the validity of the ICKA method.     

Preparation,Mechanical Properties and Thermal Degradation Kinetics of a Novel Poly(ester-ether-imide) Elastomer Based on N′,N-Bis(2-hydroxyethyl)-Pyromellitimide Unit

A novel poly(ester-ether-imide) (PEEI) based on N′,N-bis(2-hydroxyethyl)-pyromellitimide unit was synthesized via a conventional two-stage procedure with 1,4-butanediol (BD), dimethyl terephthalate (DMT) and poly(tetramethylene glycol) (PTMG1000). The structures of imide dihydric alcohol and PEEI were confirmed by FT-IR and ¹H-NMR spectra, respectively. The thermal properties and mechanical properties were investigated. The results show PEEI possesses good mechanical properties and excellent thermal stability with the 5% weight loss temperature of the PEEI at 367.3°C, and melting temperature of hard segments (T_mh) at 209.7°C. In addition, the kinetic parameters of thermal degradation of the PEEI were studied by thermogravimetric analysis (TGA) at different heating rates. The activation energy of the solid-state process was determined to be 174.83 and 175.83 kJ/mol using Kissinger and Flynn–Wall–Ozawa methods, respectively. The degradation mechanism model of PEEI was determined bythe Coats-Redfern method. Compared with the values obtained from the Kissinger and Flynn–Wall–Ozawa methods, the actual reaction mechanism of the novel PEEI is a F₁ type (Random nucleation with one nucleus on the individual particle nucleation) and growth model with integral g(a)=?ln(1?a)).     

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.     

Critical mathematical considerations involving the Abel integral equation for converting side-on experimental data in plasma spectroscopy—I: A study of basic analytical solutions

The particular form of the Abel integral equation used in plasma spectroscopy is investigated from the point of view of the computational error introduced in radial emission coefficient determinations from side-on experimental intensities. Since this emission coefficient is the basis of measurements of fundamental physical properties of the plasma, its values are to be obtained with minimal error.In this first part of this work, the mathematical properties of basic analytical solutions are investigated. Polynomial solutions resulting from positive powers of the variable in intensity profiles are described with reduced coordinates in terms of “rough parabolas” I(x) = 1 ?xⁿ (n = 2p, x = $\text{x}\text{?}$_exptl/$\text{x}\text{?}$_{max exptl}, I(x)=$\text{I}\text{?}$ ($\text{x}\text{?}$)_exptl/$\text{I}\text{?}$_{max exptl}) up to the 26th degree. For negative powers a new and original method is developed; it leads to ABEL inversion of intensity profites of the Lorcntzian kind. These profiles may be of interest when center-to-edgc intensity decay ends with a positive curvature.In following papers, all these results will then be applied to numerical approximation of theoretical standards and side-on experimental data in the spectroscopy of an inductivcly coupled plasma (ICP).     

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.     

Kinetic characterization of the reduction of silica supported cobalt catalysts

Abstact  The reduction process of silica supported cobalt catalyst was studied by thermal analysis technique. The reduction of the catalyst proceeds in two steps: