Analytical solution for the Kissinger equation
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An analytical solution for the Kissinger equation relating the activation energy, E, with the peak temperature of the reaction rate, Tm, has been found. It is accurate (relative error below 2%) for a large range of E/RTm values (from 15 to above 60) that cover most experimental situations. The possibilities opened by this solution are outlined by applying it to the analysis of some particular problems encountered in structural relaxation of amorphous materials and in kinetic analysis. I. INTRODUCTION
When the rate of a reaction is governed by a single limiting step, the evolution with time of the conversion degree, a, is described by a differential equation of the form1: da ¼ f ðaÞkðTÞ ; ð1Þ dt where f(a) depends on the type of rate-controlling process and k(T) is the temperature-dependent rate constant. Usually, k(T) follows an Arrhenius dependence: kðTÞ ¼ Ak eE=RT
;
ð2Þ
where Ak is constant, E is the activation energy, T the temperature, and R the gas constant. When the temperature varies with time, Eq. (1) still holds for spatially homogeneous reactions, whereas for heterogeneous reactions such as crystallization it is only approximate.2 The most common nonisothermal experiments involve constant heating conditions, i.e., T ¼ T0 þ bt
;
ð3Þ
where T0 is chosen low enough to have a negligible effect on the results. These kinds of experiments allow Ak and E to be determined provided that a particular kinetics [f(a)] is assumed. Despite the large number of kinetic models, there is a simple relationship between the kinetic parameters, E and Ak, and the temperature, Tm, at which the transformation rate is at its maximum. Namely, E A ¼ eE=RTm 2 RTm b
:
ð4Þ
A is equal to Ak for most kinetics or is proportional to it.3 This equation was first derived by Kissinger.4 It is
II. THE ANALYTICAL SOLUTION
The Kissinger Eq. (4) can be rewritten in terms of the reduced activation energy, x E=RT, and the reduced pre-exponential rate constant, ATm/b, as
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0366 J. Mater. Res., Vol. 24, No. 10, Oct 2009
http://journals.cambridge.org
widely used for the analysis of structural transformations as diverse as the dehydrogenation of carbon nanotubes,5 the crystallization of glasses,6,7 and the thermal analysis of lipids, proteins, and biological membranes,8 because the slope of Ln(b/Tm2) versus 1/RTm (Kissinger plot) is just the activation energy. It is exact only for first-order kinetics [f(a) = 1–a],9 and it is accurate for other kinetics provided that E/RTm is large enough (for most kinetics, the error in E is lower than 2% if E/RTm > 10).9 A literature review reveals that this does not represent a serious limitation to the applicability of Eq. (4): (i) E/RTm > 25 for most glass-crystal transformations10 and (ii) we have found values in the 8 to 35 range in reactions involving polymers11 and thermal decomposition of molecules5,12 (among them, values below 10 are scarce). Until now, the Kissinger Eq. (4) has mainly been used for
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