There are many mathematical methods to determine the activation energy from non-isothermal experiments. However, controversies arise over the different values obtained by these methods. We will show that the origin of these discrepancies is either inaccurate approximations of the so-called temperature integral or the occurrence of complex transformations. We will review and compare the most commonly used methods. For those methods that lack accuracy, we will introduce simple numerical modifications to make them exact. In addition, we will introduce a new method that allows easy and accurate determination of the activation energy.
1. Flynn, JH. The isoconversional method for determination of energy of activation at constant heating rates. J Therm Anal Calorim. 1983;27:95–102. .
2. Sbirrazzuoli, N, Girault, Y, Elégant, L. Simulations for evaluation of kinetic methods in differential scanning calorimetry. Part 3—peak maximum evolution methods and isoconversional methods. Thermochim Acta. 1997;293:25–37. .
3. Vyazovkin, S, Wight, CA. Isothermal and nonisothermal reaction kinetics in solids: in search of ways toward consensus. J Phys Chem A. 1997;101:8279–8284. .
4. Vyazovkin, S, Wight, CA. Kinetics in solids. Annu Rev Phys Chem. 1997;48:125–149. .
5. Brown, ME, Maciejewski, M, Vyazovkin, S, Nomen, R, Sempere, J, Burnham, A, Opfermann, J, Strey, R, Anderson, HL, Kemmler, A, Keuleers, R, Janssens, J, Desseyn, HO, Li, CR, Tang, TB, Roduit, B, Malek, J, Mitsuhashi, T. Computational aspects of kinetic analysis: part A: the ICTAC kinetics project-data, methods and results. Thermochim Acta. 2000;355:125–143. .
6. Starink, MJ. The determination of activation energy from linear heating rate experiments: a comparison of the accuracy of isoconversion method. Thermochim Acta. 2003;404:163–176. .
7. Roduit, B, Dermaut, W, Lunghi, A, Folly, P, Berger, B, Sarbach, A. Advanced kinetics-based simulation of time to maximum rate under adiabatic conditions. J Therm Anal Calorim. 2008;93:163–173. .
8. Brown, M, Dollimore, D, Galwey, A. Comprehensive chemical kinetics Bamford, C, Tipper, CFH, eds. Reactions in the solid state. 22 Amsterdam: Elsevier; 1980 41–113. .
9. Šesták, J. Thermophysical properties of solids, their measurements and theoretical analysis. vol 12D. Amsterdam: Elsevier; 1984.
10. Coats, AW, Redfern, JP. Kinetic parameters from thermogravimetric data. Nature. 1964;201:68–69. .
11. Farjas, J, Roura, P. Simple approximate analytical solution for nonisothermal single-step transformations: kinetic analysis. AIChE J. 2008;54:2145–2154. .
12. Farjas, J, Butchosa, N, Roura, P. A simple kinetic method for the determination of the reaction model from non-isothermal experiments. J Therm Anal Calorim. 2010;102:615–625. .
13. Ozawa, T. A new method of analyzing thermogravimetric data. Bull Chem Soc Jpn. 1965;38:1881–1886. .
14. Flynn, JH, Wall, LA. A quick direct method for determination of activation energy from thermogravimetric data. Polym Lett. 1966;4:323–328. .
15. Doyle, CD. Series approximations to equation of thermogravimetric data. Nature. 1965;207:290–291. .
16. Starink, MJ. A new method for the derivation of activation energies from experiments performed at constant heating rate. Thermochim Acta. 1996;288:97–104. .
17. Lyon, RE. An integral method of nonisothermal kinetic analysis. Thermochim Acta. 1997;297:117–124. .
18. Cai, J, Chen, S. A new iterative linear integral isoconversional method for the determination of the activation energy varying with the conversion degree. J Comput Chem. 2009;30:1986–1991. .
19. Walter G , Cahill WF. 5. Exponential integral and related functions. Handbook of mathematical functions. In: Abramowitz M, Stegun IA, editors. New York: Dover Publications Inc.; 1972 p. 228–231.
20. Kissinger, HE. Reaction kinetics in differential thermal analysis. Anal Chem. 1957;29:1702–1706. .
21. Akahira, T, Sunose, T. Trans. joint convention of four electrical institutes, Research report Chiba Institute of Technology. Sci Technol. 1971;16:22–31.
22. Murray, P. White kinetics of the thermal dehydration of clays, part IV: interpretation of the differential thermal analysis of clays. Trans Br Ceram Soc. 1955;54:204–238.
23. Vyazovkin, S. Evaluation of activation energy of thermally stimulated solid-state reactions under arbitrary variation of temperature. J Comput Chem. 1997;18:393–402. .
24. Vyazovkin, S, Dollimore, DJ. Linear and nonlinear procedures in isoconversional computations of the activation energy of nonisothermal reactions in solids. Chem Inf Comput Sci. 1996;36:42–45.
25. Vyazovkin, S. Advanced isoconversional method. J Therm Anal Calorim. 1997;49:1493–1499. .
26. Brent, RP. Algorithms for minimization without derivatives. Englewood Cliffs, NJ: Prentice-Hall; 1973.
27. Press, WH, Teukolsky, SA, Vetterling, WT, Flannery, B. Numerical recipes in C. Cambridge: Cambridge University Press; 1992.
28. Li, CR, Tang, TB. Dynamic thermal analysis of solid-state reactions. J Therm Anal Calorim. 1997;49:1243–1248. .
29. Li, CR, Tang, TB. A new method for analysing non-isothermal thermoanalytical data from solid-state reactions. Thermochim Acta. 1999;325:43–46. .
30. Boswell, PG. On the calculation of activation-energies using a modified Kissinger method. J Therm Anal Calorim. 1980;18:353–358. .
31. Friedman, HL. Kinetics of thermal degradation of char-forming plastics from thermogravimetry. Application to a phenolic plastic. J Polym Sci C. 1964;6:183–195.
32. Roduit, B, Folly, P, Berger, B, Mathieu, J, Sarbach, A, Andres, H, Ramin, M, Vogelsanger, B. Evaluating sadt by advanced kinetics-based simulation approach. J Therm Anal Calorim. 2008;93:153–161. .
33. Ortega, A. A simple and precise linear integral method for isoconversional data. Thermochim Acta. 2008;474:81–86. .
34. Farjas J , Roura P. Isoconversional analysis of solid state transformations: a critical review. II complex transformations. J Therm Anal Calorim. This issue. .
35. Vyazovkin, S. Modification of the integral isoconversional method to account for variation in the activation energy. J Comput Chem. 2001;22:178–183. .
36. Vyazovkin, S. Some confusion concerning integral isoconversional methods that may result from the paper by budrugeac and segal “some methodological problems concerning nonisothermal kinetic analysis of heterogeneous solid–gas reactions”. Int J Chem Kinet. 2002;34:418–420. .
37. Yinnon, H, Uhlmann, DR. Applications of thermoanalytical techniques to the study of crystallization kinetics in glass-forming liquids.1. Theory. J Non-Cryst Solids. 1983;54:253–275. .
38. Vyazovkin, S, Sbirrazzuoli, N. Isoconversional kinetic analysis of thermally stimulated processes in polymers. Macromol Rapid Commun. 2006;27:1515–1532. .
39. Lee, JW. Hydrogen storage and desorption properties of Ni-dispersed carbon nanotubes. Appl Phys Lett. 2006;88:143126-1–143126-30.
40. Padhi, SK. Solid-state kinetics of thermal release of pyridine and morphological study of [Ni(ampy)(2)(NO3)(2)]; ampy=2-picolylamine. Thermochim Acta. 2006;448:1–6. .
41. Roura, P, Farjas, J. Analytical solution for the Kissinger equation. J Mater Res. 2009;24:3095–3098. .
42. Cai, JM, Liu, RH. On evaluate of the integral methods for the determination of the activation energy. J Therm Anal Calorim. 2009;96:331–333. .
43. Órfão, JM. On the evaluation of the accuracy of activation energies calculated by integral methods: rebuttal of a putative correction. J Therm Anal Calorim. 2010;100:593–597. .
44. Órfão, JM. Review and evaluation of the approximations to the temperature integral. AIChE J. 2007;53:2905–2915. .
45. Senum, GI, Yang, RT. Rational approximations of integral of Arrhenius function. J Therm Anal Calorim. 1977;11:445–449. .
46. Heal, GR. Evaluation of the integral of the Arrhenius function by a series of Chebyshev polynomials: use in the analysis of non-isothermal kinetics. Thermochim Acta. 1999;340–341:69–76. .
47. Flynn, JH. The ‘temperature integral’—its use and abuse. Thermochim Acta. 1997;300:83–92. .