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ArticleName Boltzmann distribution as a basic universal expression of activation energy for viscous flow, chemical reactions and mechanical failures
DOI 10.17580/tsm.2018.10.01
ArticleAuthor Malyshev V. P., Makasheva A. M.

Abishev Institute of Chemistry and Metallurgy, Karaganda, Kazakhstan:

V. P. Malyshev, Head of Entropy Analysis Laboratory, e-mail:
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.

keywords Boltzmann distribution, activation energy, heat barrier, Arrhenius equation, Frenkel equation, mechanical energy

1. Schick M., Brillo J., Ergy I., Hallstedt B. Viscosity of Al – Cu liquid alloys: measurement and thermodynamic description. Journal of Materials Science. 2012. Vol. 47, No 23. pp. 8145–8152.

2. Zimon A. D. Physical chemistry. Moscow : Krasand, 2015. 318 p.
3. Budanov V. V., Maksimov A. I. Chemical thermodynamics : Leaner’s guide. Saint Petersburg : Lan, 2016. 320 p.
4. Efremov Yu. S. Statistical physics and thermodynamics : Learner’s guide. 2nd revised edition. Moscow : Yurayt, 2017. 209 p.
5. Zarubin D. P. Physical chemistry : Learner’s guide. Moscow : Infra-M, 2017. 476 p.
6. Schmidt P., Schafer R. Methods in Physical Chemistry. Manchester : John Willey & Sons Limited, 2017. 370 p.
7. Atkins P. W., Julio de Paula. Elements of Physical Chemistry. Oxford : W. H. Freeman and Company, 2016. 656 p.
8. Korcemkina N. V., Pastukhov E. A., Selivanov E. N., Chentsov V. P. Copper melts structure and properties with aluminum, tin and lead. Yekaterinburg : UIPTs, 2014. 182 p.
9. Monk Paul M. S. Physical chemistry: understanding our chemical world. Manchester : John Wiley & Sons Ltd, Manchester Metropolitan University, UK, 2004. 619 p.
10. Rogers D. W. Concise Physical Chemistry. Brooklyn : John Wiley & Sons, Inc. (Canada), 2013. 405 p.
11. Boltzmann L. Selected writings. Kinetic theory of gases. Thermodynamics. Statistical mechanics. Radiation theory. General problems of physics. Moscow : Nauka, 1984. 590 p.
12. Malyshev V. P., Makasheva A. M., Zubrina Yu. S. On relationship and proportion between discrete and continuous dependencies. Proceedings of the National Academy of Sciences of the Republic of Kazakhstan. 2016. No. 1. pp. 49–56.
13. Malyshev V. P., Zubrina Ju. S., Makasheva A. M. Analytical determination of the statistic Sum and the Boltzmann distribution. Abstracts of the XXI International Conference on Chemical Thermodynamics in Russia, RCCT-2017. Novosibirsk : Akademgorodok, 2017. p. 74.
14. Malyshev V. P., Zubrina Yu. S., Makasheva A. M. Quantization of particle energy in the analysis of the Boltzmann distribution and entropy. Journal of Mathematics and System Science. 2017. No. 10. pp. 278–288.
15. Leontovich M. A. Introduction to thermodynamics. Statistical physics. Saint Petersburg : Lan, 2008. 432 p.
16. Frenkel Ya. I. Kinetic theory of liquids. Moscow – Izhevsk : Regulyarnaya i khaoticheskaya dinamika, 2004. 424 p.
17. Shpilrayn E. E., Fomin V. A., Skovorodko S. N., Sokol G. F. Looking at the viscosity of liquid metals. Moscow : Nauka, 1983. 243 p.
18. Khodakov G. S. Physics of grinding. Moscow : Nauka, 1972. 307 p.
19. Malyshev V. P., Yun A. B., Sinyanskaya O. M., Zubrina Yu. S. Adapting the probabilistic grinding model to the operation of ball-tube mills. Tsvetnye Metally. 2017. No. 10. pp. 17–24. DOI: 10.17580/tsm.2017.10.02
20. Malyshev V. P., Makasheva A. M., Zubrina Yu. S. Scaling factor effect upon grinding process rate in mills of different sizes. Obogashchenie Rud. 2016. No. 3. pp. 9–13. DOI: 10.17580/or.2016.03.02
21. Malyshev V. P., Makasheva A. M., Kaykenov D. A., Zubrina Yu. S. The randomness and a probabilistic model of the grinding process. Moscow : Nauchniy mir, 2017. 260 p.
22. Malyshev V. P., Bekturganov N. S., Turdukozhaeva A. M. Viscosity, fluidity and density of substances as a measure of their randomization. Moscow : Nauchniy mir, 2012. 288 p.

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