New approximate formula for Arrhenius temperature integral

Abstract

In this paper a more precise approximate formula for Arrhenius temperature integral, i.e., $\text{−}\text{ln}\text{P(u)=0.37773896+1.89466100}\text{ln}\text{u+1.00145033u}$, is proposed, by using two-step linearly fitting process: (i) the linear dependence of $\text{d}\text{ln}\text{p(u)/}\text{d}\text{u}$ on 1/u and (ii) the linear dependence of ($\text{ln}\text{p(u)−c}\text{ln}\text{u}$) on u. Values of p(u) at different u were directly obtained from numerical integration of temperature integral without derivation from any approximating infinite series, and values of $\text{d}\text{ln}\text{p(u)/}\text{d}\text{u}$ were obtained by numerical differentiating. New equation for the evaluation of non-isothermal kinetic parameters has been obtained from the above dependence, which can be put in the form

$$\text{ln}\text{g(α)}\text{T}{}^{1.89466100}\text{=}\text{ln}\text{AE}\text{βR}\text{+3.63504095−1.89466100}\text{ln}\text{E}\text{−1.00145033}\text{E}\text{RT}$$

The validity of this formula was confirmed and its deviation from the values of numerical integrating was discussed. Compared with some previously published approximate formulae, our formula has the best result in the kinetics analysis of non-isothermal process.

Introduction

Last century had witnessed an accelerating use of thermal analysis techniques to investigate the kinetics of various isothermal and non-isothermal decompositions. Thermogravimetry (TG) may be one of the most popular thermal analysis techniques. And the reaction is carried out under a linear temperature program in the most of reported TG studies. An understanding of kinetic parameters, such as kinetic model, activation energy and the frequency factor, can be obtained by various approaches. The integral approach is generally believed to be more convenient, reliable, and accurate than the differential method [1]. It is usually assumed that the basic kinetic equation for solid-state decomposition process under non-isothermal conditions can be expressed as a function of the fractional conversion α (0<α<1) in the following form:

$$\text{d}\text{α}\text{d}\text{T}\text{=}\text{A}\text{β}\text{e}{}^{\mathrm{-E/RT}}\text{f(α)}$$

where A is the pre-exponential factor of the Arrhenius rate constant, E the apparent activation energy, β the heating rate, T the absolute temperature, R the gas constant, f(α) the reaction kinetics function.

Rearranging Eq. (1) and integrating both sides of the equation, the following expression is obtained:

$$\text{g(α)=}\text{∫}\text{0}\text{α}\text{d}\text{α}\text{f(α)}\text{=}\text{A}\text{β}\text{∫}\text{0}\text{T}\text{e}{}^{\mathrm{-E/RT}}\text{d}\text{T=}\text{AE}\text{βR}\text{p(u),}\text{p(u)=}\text{∫}\text{∞}\text{α}\text{−}\text{e}{}^{\mathrm{-u}}\text{u}{}^2\text{d}\text{u}$$

where p(u) is the Arrhenius temperature integral, u=E/RT. Unfortunately, the right side of Eq. (2) cannot be analytically integrated. Consequently, extensive efforts have been devoted to obtain better integral approximations. In a recent paper, Flynn [2] has reviewed various evaluations and approximations for the temperature integral and reappraised their accuracies and utilities to evaluate the temperature integral in thermal analysis kinetics.

In 1986, Madhusudanan et al. [3] proposed an approximate formula for temperature integrating calculation, which can be readily rearranged for using in the field of iso-conversional method. Because of its simplicity the formula is commonly used in integral methods of thermal analysis [4], [5], [6], [7], [8]. This formula is shown below:

$$\text{−}\text{ln}\text{P(u)=0.297580+1.921503}\text{ln}\text{u+1.000953u}$$

Two other equations [9] in the form of Eq. (3) were proposed for the evaluation of kinetic parameters from non-isothermal experiments later. These two equations reported are shown below:

$$\text{−}\text{ln}\text{P(u)=0.299963+1.920620}\text{ln}\text{u+1.000974u}$$

$$\text{−}\text{ln}\text{P(u)=0.389677+1.884318}\text{ln}\text{u+1.001928u}$$

Just as Heal [10] said, ‘Those early mathematical tables are themselves derived from some approximating infinite series, because there is, of course, no true value for the integral’. The question is not only “How precisely do the recent approximation fit the older standard tabulated data”, but also, “How reliable are the old data sets”. And he also pointed out that the tables used in the past, considering the date of publication, must have been produced without the aid of computers, and might contain errors. So he concluded that a new set of values, computer-calculated, were needed. When we reevaluated these literatures, we found that there was something to be improved.

In the literature [2], the authors declared that their approximate formula (Eq. (3)), which is called as MKN equation, was derived based on a two-term approximate formula, i.e. (e^−u/u²)[(u+1)/(u+3)], which is just but an approximation of Eq. (2). That is, their approximate formula (Eq. (3)) was not directly derived from the original numerical values of temperature integral. Using the same method, in another literature [8], , were derived based on the three-term approximate equation, i.e.

$$\text{e}{}^{\mathrm{-u}}\text{u}{}^2\text{1−}\text{2}\text{u+3}\text{−}\text{5}\text{(u+1)(u+2)(u+3)}$$

and the series approximate equation for Arrhenius temperature integral, i.e.

$$\text{e}{}^{\mathrm{-u}}\text{u}{}^2\text{1−}\text{2}\text{u+3}\text{−}\text{6}\text{(u+1)⋯(u+3)}\text{÷}\text{28}\text{(u÷1)⋯(u+4)}\text{−}\text{120}\text{(u+1)⋯(u+5)}\text{+}\text{496}\text{(u+1)⋯(u+6)}\text{−⋯}$$

respectively. Taking into account the generalized use of MKN equation for temperature integral, it seems to be of great interest to check the accuracy of these approximations.

As we know, the values of temperature integral at different temperature interval can be easily obtained by numerical integral on a PC computer nowadays. If an approximate formula is directly derived, its accuracy and reliability will be higher than that derived indirectly. Furthermore, in order to delete the ambiguity accompanied by the interpretation of kinetic data produced from thermal analysis measurements, it is important to use adequate computational methods and experimental conditions. With accurate measurements of temperature, the use of proper approximate formula for the temperature integral allow us to calculate kinetic parameters as precisely as possible [2]. So we think it is necessary to reevaluate these approximations and to derive a more precise approximate formula. The main objective of this study is to present a precise integral approximate formula for kinetic analysis of non-isothermal thermogravimetric analysis (TGA) data. Because of its simplicity in calculation, the form will be remained and its accuracy will be enhanced in the newly proposed approximate formula.