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.     

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 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.     

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.     

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.     

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.     

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.     

New approximations of the temperature integral for nonisothermal kinetics

The accuracy and scope of application of previously reported approximations of the temperature integral were evaluated. The exact solution was obtained independently by solving the temperature integral numerically be Simpson's rule, the trapezoidal rule and the Gaussian rule. Two new approximations have been proposed: $$\begin{gathered} P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 5.2/X^2 ) \hfill \\ P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 4.6/X^2 ) \hfill \\ \end{gathered} $$ whereX=E/RT. The first equation gives higher accuracy, with a deviation of less than 1% and 0.1% from the exact solution forX≥7 andX≥10, respectively. The second equation has a wider scope of application, with a deviation of less than 1% forX≥4 and of less than 0.1% forX≥35.     

A new iterative linear integral isoconversional method for the determination of the activation energy varying with the conversion degree

The conventional linear integral isoconversional methods may lead to important errors in the determination of the activation energy when the significant variation of the activation energy with the conversion degree occurs. Vyazovkin proposed an advanced nonlinear isoconversional method, which allows the activation energy to be accurately determined [Vyazovkin, J Comput Chem 2001, 22, 178]. However, the use of the Vyazovkin method raises the problem of the time‐consuming minimization without derivatives. A new iterative linear integral isoconversional method for the determination of the activation energy as a function of the conversion degree has been proposed, which is capable of providing valid values of the activation energy even if the latter strongly varies with the conversion degree. Also, the new method leads to the correct values of the activation energy in much less time than the Vyazovkin method. The application of the new method is illustrated by processing of theoretically simulated data of a strongly varying activation energy process. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

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 nonisothermal solid-state reactions: determination of the kinetic parameters by means of a nonlinear regression method

A nonlinear regression method has been proposed for a simultaneous calculation of the activation energy, frequency factor, and reaction order from a single TG curve. This method was based on the new temperature integral approximation proposed in this paper and the Levenberg–Marquardt method. The newly proposed nonlinear regression method was applied for determining the kinetic parameters from two simulated TG curves. The results of the calculations were compared with values obtained by the traditional method. It can be concluded from this comparison that the new nonlinear regression method is more accurate than the traditional method for the determination of the kinetic parameters of solid-state heterogeneous reactions.     

A new integral method for the kinetic analysis of thermogravimetric data

Methods for the calculation of activation energies, pre-exponential factors and reaction orders from thermogravimetric data are briefly reviewed. A new integral method is proposed for the determination of these kinetic parameters, using data from pairs of TG curves produced at different heating rates. Employing accurate values of the temperature integral of the Arrhenius equation, tabulated over a range ofE andT, and a simple graphical procedure, the method offers advantages of speed and accuracy over those previously reported. It is suggested that at least one of the kinetic parameters should be allowed to move freely in order to achieve the best possible fit between calculated and experimental traces.     

The correctional kinetic equation for the peak temperature in the differential thermal analysis

The peak temperature (T _p) and different temperature (ΔT) are the basic information in the differential thermal analysis (DTA). Considering the kinetic relation and the heat equilibrium in DTA, a correctional differential kinetic equation (containing T _p and ΔT parameter) is proposed. In the dehydration reaction of CaC₂O₄·H₂O, the activation energy calculated from the new equation showed some smaller than that from Kissinger equation, but some bigger than that from Piloyan equation.     

Chiral analysis by mass spectrometry using the kinetic method in flow systems

Chiral analysis is an important task of analytical chemistry. Besides separation techniques, mass spectrometry can be applied in this field. One mass spectrometric approach is based on Cooks' kinetic method. The method was successfully applied in a static system in which the concentration of the analyte as well as the chiral selector solution was constant during the experiment. The application of the kinetic method in dynamic systems (changing concentration of analyte) is presented. Such systems allow the speeding up of the analytical process (flow injection analysis (FIA)) or the use of the kinetic method for chiral detection after liquid chromatographic separation.The influence of the concentration of the components of the chiral selector solution as well as its flow rate on the recognition of enantiomers was evaluated. A new procedure for correction for the differences between ratio of enantiomers in the liquid phase and their observed ratio in the gas phase is also described. A significant improvement in accuracy using this procedure was achieved. Applicability of the method was demonstrated in the analysis of amino acids using FIA as well as HPLC/MS. After an achiral separation of leucine and isoleucine, chiral mass spectrometric detection was successfully used for enantiomeric recognition.     

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.     

Comprehensive method based on model free method and IKP method for evaluating kinetic parameters of solid state reactions

This article presents, firstly, a short review of methods for evaluating kinetic parameters of solid state reactions and a critical analysis of the isoconversional principle of model free methods. It shows theoretically that the activation energy for complex reactions is not only a function of the reaction degree but also of heating programs, and points out that any method that attempts to extract the dependences of activation energy on conversion degree without considering the dependences of heating programs is problematic. Then an analysis is given of the invariant kinetic parameters (IKP) method and recommends an incremental version of it. Based on the incremental IKP method and model free method, a comprehensive method is proposed that predicts the degree of the dependences of activation energy on heating programs, selects reliable values of activation energy and extracts the values of variable pre‐exponential factor. This comprehensive method is tested using both simulation data and experimental data, the results of which show it can not only give reliable values of kinetic parameters but also be helpful in explaining inconsistencies of kinetic results in solid state reactions. © 2012 Wiley Periodicals, Inc.     

Establishment and application of a new method for the determination of kinetic parameters by plug-plug kinetic capillary electrophoresis (ppKCE)

A novel plug-plug kinetic capillary electrophoresis method was established, which can be used to si-multaneously determine the kinetic parameters kon and koff in interaction systems. The method is comparatively simple and some restrictions in conventional ppKCE methods can be effectively avoided. The requirements for resolution and detection sensitivity in this method are much lower than those of conventional ppKCE. The successful determination of the kinetic parameters and the binding constant Kb between citalopram and BSA showed availability of this method. The results were confirmed by us-ing the time ratio method. The application field of kinetic capillary electrophoresis can be expanded with this new method.     

Dependence of the pure quadrupole resonance frequency on temperature for KBrO3

The pure quadrupole resonance frequency of KBrO₃ in the temperature range 4.17–300.16 K was determined experimentally and compared with the model developed by Stahl which contains three adjustable constants. Excellent agreement between the experimental data and the model was obtained when considering only the temperature range 4.17–52 K. As a result of the comparison, the following values are suggested for KBrO₃: the resonance frequency at 0 K is 179.50373 MHz, the Debye temperature, Θ_d, is 108 K, and the rotational energy, Iω_l², of each of the first two modes of oscillation of the BrO₃ group is 1.707 × 10^?18 kg m². Here I is the moment of inertia and ω_l is the lattice vibration frequency. Comparison of model and experimental data over the entire temperature range from 4.17 to 300.16 K shows that the suggested theory reflects very well the general trend of the experimental data but lacks in some detail. Copyright © 2003 John Wiley & Sons, Ltd.     

Kinetics of esterification of the levulinic acid with n‐hexanol,n‐octanol,and 2‐ethylhexanol in the presence of methanesulfonic acid as a catalyst under nonisothermal conditions

The levulinic acid was esterified with alcohol at an alcohol to acid molar ratio of 3:1, 5:1, and 10:1 in the presence of a 0.1 wt% methanesulfonic acid catalyst. During esterification, the temperature was changed linearly from 373 to 428 K and its average change was 4.5 K/min. The authors stated that reactions were of second order and that the activation energy (E) decreased from 61 to 46 kJ/mol in the following alcohol sequence: n‐hexanol > n‐octanol > 2‐ethylhexanol. The fitting errors varied between 3.8% and 6.4%. The time of experiment carried out under nonisothermal condition is five to 15 times shorter than that conducted under isothermal conditions. A smaller number of experimental series also determines a significantly lower cost of such research. The results of such study are the precise form of the kinetic equation, which is indispensable in design and optimization of industrial‐scale chemical reactors.