Representation and evaluation of the Arrhenius and general temperature integrals by special functions

The non-isothermal analysis of reactions with a constant heating rate involves a temperature integral. This integral is popularized as the Arrhenius integral, but when the pre-exponential factor of the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. In this paper, it is shown that when the activation energy and exponent of power-law assumed to be constant, these integrals can be expressed and evaluated by several kinds of special functions and using commercial or free softwares. The special functions include exponential integral function, incomplete gamma function, confluent hypergeometric function, Whittaker function and generalized hypergeometric function. New rational approximations can be obtained by expansion of the special functions. MATLAB and GNU Octave were used to compute the special function models as well as numerical integration. The accuracy of these methods was examined by variable precision arithmetic. For both softwares, the special function models showed higher accuracies compared to the numerical integration methods and the approximation functions. In addition, new iterative isoconversional method for constant activation energy and exponent of power-law has been proposed that shows very high accuracy in calculation of the activation energy.

Automated summary

The non-isothermal analysis of reactions with constant heating rate involves temperature integral, which can be expressed and evaluated using special functions and software when assuming constant activation energy and power-law exponent. Special functions include exponential integral function, incomplete gamma function, etc. MATLAB and GNU Octave were used for computing special function models with high accuracy.