Abishev Institute of Chemistry and Metallurgy, Karaganda, Kazakhstan:

V. P. Malyshev, Head of Entropy Analysis Laboratory, e-mail: eia_hmi@mail.ru
A. M. Makasheva, Principal Researcher at the Entropy Analysis Laboratory 

This paper illustrates the fundamental importance of the Boltzmann distribution for expressing any activation barriers in physical, chemical and mechanical processes and for calculating the corresponding properties, including those that can be determined with the help of the Arrhenius and Frenkel equations, as well as probabilistic models of the grinding process. For this purpose, the authors analysed discrete and continuous forms of the Boltzmann distribution and substantiated a single expression of the share of superbarrier particles for any chaotization border and, through this, the activation of a substance. This research saw the first time when a proportion coefficient was applied to the Boltzmann discrete and continuous distribution, which was proposed by the authors. With the help of the proportion coefficient, the authors were able to identify a mutual transition to each of the distributions and substantiate the single expression for thermal activation barriers. The paper proves that it would be more appropriate to use the Frenkel equation to express fluidity rather than viscosity, because in this case the resultant exponent would have a form that would be consistent with the exponent obtained under the Boltzmann probability distribution and similar to the Arrhenius equation. It would also result in a more accurate viscous flow activation energy calculated with the help of the Frenkel equation. The exponent itself, expressed more correctly with the temperature rising from zero to infinity, changes from zero to 1 and thus acquires a probability meaning of overcoming the fluidity activation barrier. In the conventional presentation of the Frenkel equation, under the same conditions the exponent changes from infinity to 1 and thus loses its probability meaning and its connection to the Boltzmann distribution. However, when the Frenkel equation is used to process the viscosity data, the formal activation energy appears to be the same as the activation energy that is calculated under the fluidity equation. This is due to an identity transformation between the Frenkel equation and the fluidity equation. It is shown that to avoid absurd results when deriving a probability grinding equation (such as the results that can be obtained when using the Frenkel equation to analyse the exponent), one should add the mechanical energy to the thermal energy of a substance rather than subtract it from the activation energy, thus broadening the meaning of the Boltzmann distribution.
This research was carried out as part of the Grant Project No. AR05130844/GF4 MON RK.

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