Exponential temperature change in thermal analysis at non-isothermal kinetics

The applying of exponential temperature programming: dT/dt=T orT=T ₀ e ^t in thermal analysis at non-isothermal kinetics is discussed. An approach for the integration of the temperature integral is presented.     

An approximation to the linear heating rate in non-isothermal kinetics

A near-linear representation of the linear heating rate is presented which converts the exponential integral into an integrable form and allows a simple determination of the activation energy to high accuracy.     

Study on thermal decomposition and oxidation kinetics of cation exchange resins using non-isothermal TG analysis

Kinetics of two successive thermal decomposition reaction steps of cationic ion exchange resins and oxidation of the first thermal decomposition residue were investigated using a non-isothermal thermogravimetric analysis. Reaction mechanisms and kinetic parameters for three different reaction steps, which were identified from a FTIR gas analysis, were established from an analysis of TG analysis data using an isoconversional method and a master-plot method. Primary thermal dissociation of SO₃H⁺ from divinylbenzene copolymer was well described by an Avrami–Erofeev type reaction (n = 2, g(α) = [?ln(1 ? α)]^1/2]), and its activation energy was determined to be 46.8 ± 2.8 kJ mol^?1. Thermal decomposition of remaining polymeric materials at temperatures above 400 °C was described by one-dimensional diffusion (g(α) = α ²), and its activation energy was determined to be 49.1 ± 3.1 kJ mol^?1. The oxidation of remaining polymeric materials after thermal dissociation of SO₃H⁺ was described by a phase boundary reaction (contracting volume, g(α) = 1?(1 ? α)^1/3). The activation energy and the order of oxygen power dependency were determined to be 101.3 ± 13.4 and 1.05 ± 0.17 kJ mol^?1, respectively.     

Integral dependent on parameter E in classical non-isothermal kinetics with linear heating rate

A simple procedure to obtain the derivative of the temperature integral with respect to the activation energy is presented.     

A solution of the exponential integral in the non-isothermal kinetics for linear heating

A simple and satisfactorily accurate solution of the exponential integral in the nonisothermal kinetic equation for linear heating is proposed: $$\mathop \smallint \limits_0^T e^{ - E/RT} dT = \frac{{RT^2 }}{{E + 2RT}}e^{ - E/RT} $$      

Thermal decomposition of polymeric materials: Characteristics of kinetics at non-isothermal heating

General conclusions are made based on the results of experimental and theoretical investigations of the decomposition kinetics of various polymeric materials in a wide range of heating rates. It was found that the decomposition process of the investigated materials proceeds within confined temperature intervals. Kinetic equations are suggested to describe the thermal decomposition processes of polymers and natural coals, based on the characteristic revealed.     

Algorism for the solution of the exponential integral in non-isothermal kinetics at linear heating

The solution of the exponential integral at linear heating for the general case that the activation energy linearly depends on temperature according toE(T)=E ₀+RBT is

Cu-Al尖晶石的合成及非等温生成动力学分析

以程序升温碳化法合成β-Mo₂C载体,采用原位沉淀法制备负载量不同的Au/β-Mo₂C催化剂,利用XRD、STEM和氮气吸附-脱附等手段对Au在载体表面的分散性、微观形貌及孔结构等进行表征,并在逆水煤气变换(RWGS)反应中对其高温热稳定性进行了研究。XRD表征结果表明,在34.44°、38.02°、39.44°、52.12°、61.53°、69.62°和74.65°处出现了β-Mo₂C对应的(100)、(002)、(101)、(102)、(110)、(103)和(200)晶面的X射线特征衍射峰;同时,未出现Au物种的特征吸收峰,说明Au负载量较低的0.1%和0.5%的催化剂上Au纳米粒子的分散性较好。STEM表征结果也显示,当负载量较低(0.5%、1.0%和2.0%)时,金纳米粒子以2 nm左右的原子簇形式均匀分散并锚定在β-Mo₂C载体上。氮气吸附-脱附表征结果表明,催化剂具有良好的介孔结构。反应评价结果表明,0.2%Au/β-Mo₂C催化剂在RWGS反应中具有较好的催化活...     

Philosophy of non-isothermal kinetics

The most debatable and discrepant viewpoints of non-isothermal kinetics are discussed in the form of twelve questions and answers. The reputation of non-isothermal kinetics when carried out by thermoanalysts; the consequences of simplified concepts transferred from the kinetics of homogeneous reactions; the physical meaning of basic kinetic parameters in solid-state processes; the kinetic compensative effect and interdependence of kinetic parameters using the Arrhenius rate constant; the mutual usefulness of differential and integral methods of kinetic data evaluation; their accuracy and correctness; the reliability of DTA measurements; non-isothermal versus isothermal investigations; equilibrium and kinetic data and their mutual effect; the extended discussion initiated by MacCallum and Tanner; non-isothermal data publication policy; and finally the use of computers.

---

Zusammenfassung In der Form von zwölf Fragen und Antworten werden die meist umstrittenen und auseinandergehenden Aspekte der nicht-isothermen Kinetik diskutiert. Insbesondere das Ansehen der nicht-isothermen Kinetik bei Durchführung durch Thermoanalytiker, die Konsequenzen der vereinfachten von der Kinetik homogener Reaktionen übertragenen Konzepte, die physikalische Bedeutung der kinetischen Grundparameter in Festphasenprozessen, der kinetische Kompensationseffekt und die gegenseitige Abhängigkeit der kinetischen Parameter bei Anwendung der Geschwindigkeitskonstante nach Arrhenius, die gegenseitige Nützlichkeit der Differential- und Integralmethoden der Auswertung kinetischer Daten, ihre Genauigkeit und Richtigkeit, die Zuverlässigkeit der DTA-Messungen, nichtisotherme gegenüber von isothermen Untersuchungen, Gleichgewichts- und kinetische Daten und ihre gegenseitige Wirkung, die Ausbreitung der von MacCallum und Tanner initierten Diskussion Politik der Veröffentlichung nicht-isothermer Angaben und, endlich, Einsatz und Hilfeleistung der Computer.

---

Résumé Les aspects les plus discutables de la cinétique non-isotherme sont traités sous forme de douze questions et réponses, en particulier: réputation de la cinétique non-isotherme appliquée par les thermoanalystes, conséquences des concepts simplifiés déduits de la cinétique des réactions en phase homogène, signification physique des paramètres cinétiques fondamentaux dans les processus en phase solide, effet de compensation cinétique et interdépendance des paramètres cinétiques lors de l'utilisation de la constante de vitesse d'Arrhenius, utilité mutuelle des méthodes différentielles et intégrales d'évaluation des données cinétiques, leur exactitude et leur degré de justesse, fiabilité des mesures par ATD, études non-isothermes vis-à-vis des méthodes isothermes, données à l'équilibre et en régime dynamique et leurs effets mutuels, extension de la discussion engagée par MacCallum et Tanner, politique de publication des données non-isothermes et finalement utilisation des ordinateurs et aide apportée par ceux-ci.

---

. , , , , , , , , , , , , , , , , .

     

Compensation effect in heterogeneous non-isothermal kinetics

A number of 1145 sets of kinetic parameters derived in our earlier papers from TG curves have been worked up. The apparent activation energy and pre-exponential factor values have been found to obey a linear compensation law (isokinetic relation) if the thermal decomposition begins in the same temperature interval, irrespective of the nature of the chemical reaction. The isokinetic temperatureT _i has been found to be very close to the mean value of the temperaturesT _0.1 at which the conversion becomes equal to 0.1 and atT _i the rate constant has been found to be approximately equal to 10^?3s^?1 in allT _0.1 intervals investigated. It is concluded that the kinetic compensation effect observed in heterogeneous non isothermal TG kinetics is not a true one.     

Nucleation parameters for polymer crystallization from non-isothermal thermal analysis

Differential scanning calorimetry (DSC) has been used to obtain kinetic and nucleation parameters for polymer crystallization under a non-isothermal mode of operation. The available isothermal nucleation growth-rate equation has been modified for non-isothermal kinetic analysis. The values of the nucleation constant (K _g ) and surface free energies (, _e ) have been obtained for i-polybutene-1, i-polypropylene, poly(L-lactic acid), and polyoxymethylene and are compared with those obtained from isothermal kinetic analysis; a good agreement in both is seen.     

Statistical evaluation of non-isothermal decomposition kinetics analysis methods for AIBN

Decomposition kinetics parameters were obtained from non-isothermal DSC data for the decomposition of AIBN by two data analysis methods. A statistical evaluation of the results showed that there was no difference between the results obtained by each data analysis method and that the heating rate did not significantly affect the values of the kinetics parameters. Both data analysis methods isolate the order of reaction,n, the activation energy,E, and the Arrhenius factor,A, from a single non-isothermal scan of the decomposition reaction.

---

Zusammenfassung Mittels zwei Datenauswertungsmethoden werden aus Ergebnissen von nichtisothermen DSC-Werten kinetische Parameter für die Zersetzung von AIBN ermittelt. Eine statistische Auswertung der Ergebnisse ergab, daß zwischen den durch die beiden Datenauswertungsmethoden erhaltenen Resultaten kein Unterschied besteht und daß die kinetischen Parameter von der Aufheizgeschwindigkeit nicht bedeutend beeinflußt werden. Beide Verfahren liefern auf der Basis eines einzigen, nichtisothermen Durchlaufes der Zersetzungsreaktion die Reaktionsordnungn, die AktivierungsenergieE und den ArrheniusfaktorA.

---

AIBN . , , . , , (), () ().

---

---

The authors acknowledge Dr. John J. Stansbrey for the analysis of covariance and the verification of the statistical results in this work and Mr. Leo Tischer for obtaining much of the experimental data.     

Dehydration kinetics by non-isothermal techniques

The kinetics of thermal degradation of some cementitious calcium aluminates, sulfoaluminates, sulfoferrites and carboaluminates were studied. Activation energies, obtained by Morris and Rogers and Kissinger techniques, were found to be in good agreement with each other.

---

Zusammenfassung Die thermische Zersetzungskinetik einiger im Zement befindlicher Kalziumaluminate, Sulfoaluminate, Sulfoferrite und Karboaluminate wurde untersucht. Die nach den Methoden von Morris und Rogers, sowie Kissinger erhaltenen Aktivierungsenergien stimmten gut überein.

---

Résumé On a étudié la cinétique de la décomposition thermique de plusieurs constituants du ciment tels les aluminates, sulfoaluminates, sulfoferrites et carboaluminates de calcium. Les énergies d'activation obtenues en appliquant la méthode de Morris et Rogers ainsi que celle de Kissinger sont concordantes.

---

-, -, . , , , , .

     

Linear and hyperbolic temperature programming in non-isothermal kinetics

The method suggested by several authors for determining the mechanism of solid-phase transformations by linearizing the function Ing() vs. 1/T is more correct for a hyperbolic temperature change than for a linear temperature change. In the latter case, the method yields reliable results only under the condition that the relationship Ing()/T ² vs. 1/T is linear. The well-known Horowitz-Metzger method is essentially suited for processing thermokinetic curves obtained under hyperbolic heating or cooling.

---

Zusammenfassung Die von mehreren Autoren vorgeschlagene Methode den Mechanismus von Umwandlungen in der Festphase durch Linearisierung der Funktion zwischen Ing() und 1/T zu bestimmen ist für hyperbolische TemperaturÄnderungen korrekter als für die lineare TemperaturÄnderung. In letzterem Falle ergibt die Methode nur unter der Bedingung zuverlÄssige Ergebnisse, da\ der Zusammenhang zwischen Ing()/T ² und 1/T linear ist. Die bekannte Methode nach Horowitz-Metzger eignet sich im Wesentlichen zur Bearbeitung thermokinetischer Kurven bei hyperbolischem Aufheizen oder Kühlen.

---

Résumé La méthode proposée par plusieurs auteurs pour déterminer le mécanisme des transformations en phase solide en linéarisant la fonction reliant In g() et 1/T est plus correcte lorsque les variations de température suivent un régime hyperbolique qu'elle ne l'est pour les régimes linéaires. Dans ce dernier cas, la méthode ne fournit des résultats fiables qu'à la condition que la relation entre Ing()/T ² et 1/T soit linéaire. La méthode bien connue d'Horowitz — Metzger s'applique essentiellement au traitement des courbes obtenues avec des lois d'échauffement ou de refroidissement hyperboliques.

---

ln g() vs 1/T , ln g()/T ² vs 1/T. — .

     

Model-free method for isothermal and non-isothermal decomposition kinetics analysis of PET sample

Pyrolysis, one possible alternative to recover valuable products from waste plastics, has recently been the subject of renewed interest. In the present study, the isoconversion methods, i.e., Vyazovkin model-free approach is applied to study non-isothermal decomposition kinetics of waste PET samples using various temperature integral approximations such as Coats and Redfern, Gorbachev, and Agrawal and Sivasubramanian approximation and direct integration (recursive adaptive Simpson quadrature scheme) to analyze the decomposition kinetics.The results show that activation energy (E_α) is a weak but increasing function of conversion (α) in case of non-isothermal decomposition and strong and decreasing function of conversion in case of isothermal decomposition. This indicates possible existence of nucleation, nuclei growth and gas diffusion mechanism during non-isothermal pyrolysis and nucleation and gas diffusion mechanism during isothermal pyrolysis. Optimum E_α dependencies on α obtained for non-isothermal data showed similar nature for all the types of temperature integral approximations.