Determining the expression for isobaric coefficient of volume expansion for some molecular crystals of nitro compounds

Yu. M. Kovalev

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 2, pp. 57-67

Résumé

A type of equations on the state of molecular crystals of nitro compounds is suggested based on dividing Helmholtz free energy into intramolecular and intermolecular components. It is suggested to approach the thermal part of the internal energy and pressure of a molecular crystal with Debye approximation for intermolecular component, and with Einstein approximation for intramolecular component. This division of Helmholtz energy allowed to obtain clear expressions for all thermodynamic quantities being part of the equations of state. The suggestion that isothermal sound speed of a molecular crystal at the temperature of 0 K is determined solely by elastic characteristics of the crystal made it possible to obtain the dependency of Gruneisen coefficient on volume. The dependencies of thermodynamic quantities on temperature and volume determined in this work were used to build an analog of Gruneisen equation for molecular crystals, and to determine the dependency of isobaric coefficient of crystal volume expansion on temperature. It turned out that to obtain calculated values of volumes of unit cells of triamyno trinitrobenzene (TATB) crystals while integrating Gruneisen equation as per temperature, a high-temperature approximation may be used for Debye specific thermal capacity function. The obtained theoretical dependency of isobaric coefficient of crystal volume expansion on temperature indicates at the automatic meeting the condition of its tending to zero while the temperature tends to zero. The comparative analysis of the calculated and experimental values of volumes of unit cells of molecular crystals of 1,3,5-2,4,6-trinitrobenzene (TATB) depending on the temperature showed that they concur satisfactorily to the accuracy of no more than 3 %.

Détail

Keywords: equation of state, molecular crystal, Helmholtz energy, Boltzmann's constant, Einstein approximation.
Mots-clés : Planck's constant, Debye approximation

Bibliographie

[1] Fortov V. E., Equations of state of matter: from an ideal gas to a quark-gluon plasma, Fizmatlit Publ., M., 2013, 492 pp. (in Russ.)

[2] Khishchenko K. V., Fortov V. E., “Investigation of equations of state of materials at high concentration of energy”, Proceeding of the Kabardino-Balkarian State University, IV:1 (2014), 6–16 (in Russ.)

[3] Kuropatenko V. F., Models of Continuum Mechanics, Izd-vo ChelGU Publ., Chelyabinsk, 2007, 302 pp. (in Russ.)

[4] Kovalev Yu.M., Kovaleva E. A., “A Mathematical Study of the Conservation Equation for Two-Phase Mixtures”, Bulletin of the South Ural State University, Series “Mathematical Modelling, Programming Computer Software”, 7:2 (2014), 29–37 | DOI

[5] Bushman A. V., Fortov V. E., “Model equations of state”, Sov. Phys. Usp., 26 (1983), 465–496 | DOI | DOI

[6] Kitaygorodskiy A. I., Molecular Crystals, Nauka Publ., M., 1971, 424 pp. (in Russ.)

[7] Zharkov V. N., Kalinin V. A., Equations of state of solids at high pressures and temperatures, Nauka Publ., M., 1968, 311 pp. (in Russ.)

[8] Zhirifal'ko L., Statistical Solid State Physics, Mir Publ., M., 1975, 382 pp. (in Russ.)

[9] Bazarov I. P., Thermodynamics, Vysshaya shkola Publ., M., 1991, 375 pp. (in Russ.)

[10] Zel'dovich Ya.B., Rayzer Yu.P., Physics of shock waves and high-temperature hydrodynamic phenomena, Fizmatlit Publ., M., 2008, 652 pp. (in Russ.)

[11] Kovalev Yu.M., Fizika goreniya i vzryva, 20:2 (1984), 102–107 (in Russ.)

[12] Voskoboynikov I. M., Afanasenkov A. N., Bogomolov V. M., Fizika goreniya i vzryva, 3:4 (1967), 585–593 (in Russ.)

[13] P.J. Miller, S. Block, G.J. Piermarini, “Effect of Pressure on the Vibration Spectra of Liquid Nitromethane”, J. of Physical Chemistry, 93 (1989), 462–466 | DOI

[14] Khishchenko K. V., Lomonosov I. V., Fortov V. E., Shlenskiy O. F., Doklady Akademii nauk, 349:3 (1996), 322–325 (in Russ.)

[15] Landau L. D., Lifshits E. M., Statistical physics, v. I, Nauka Publ., M., 1976, 584 pp. (in Russ.)

[16] Kovalev Yu.M., “Determination of Form of the Gruneisen Coefficient for Molecular Crystals”, Doklady Akademii nauk, 403:4 (2005), 475–477 (in Russ.) | MR | Zbl

[17] Kovalev Yu.M., Voprosy atomnoy nauki i tekhniki. Seriya: Matematicheskoe modelirovanie fizicheskikh protsessov, 2005, no. 2, 55–59 (in Russ.)

[18] Molodets A. M., “Gruneisen function and the zero-temperature isotherm of three metals up to pressures of 10 TPa”, JETP, 80:3, 467–471

[19] Molodets A. M., Doklady Akademii nauk, 341:6 (1995), 753–754 (in Russ.)

[20] Shchetinin V. G., Khimicheskaya fizika, 18:5 (1999), 90–95 (in Russ.) | Zbl

[21] Kovalev Yu.M., Kuropatenko V. F., Inzhenerno-fizicheskiy zhurnal, 91:2 (2018), 297–306 (in Russ.)

[22] B.M. Dobrats, P.C. Crawford, LLNL Explosives Handbook. Properties of Chemical Explosives and Explosive Simulants, University of California, Livermore, California, 1985

[23] Stankevich A. V., Smirnov E. B., Kostitsyn O. V., Ten K. A., Shmakov A. N., Tolochko B. P., “Ispol'zovanie sinkhrotronnogo i teragertsovogo izlucheniya dlya issledovaniya vysokoenergeticheskikh materialov”, Proc. Using synchrotron and terahertz radiation to study high-energy materials, IYaF SO AN Publ., Novosibirsk, 2015, 46–49 (in Russ.)

[24] Olindzher B., Keydi G., “Shock compressibility of ten, TATB, CO$_2$ and H$_2$O at pressures up to 10 GPa, calculated on the basis of experimental data on hydrostatic compression”, Detonatsiya i vzryvchatye veshchestva, Mir Publ., M., 1981, 203–219 (in Russ.)

[25] Kovalev Yu.M., “Mathematical Modelling of the Thermal Component of the Equation of State of Molecular Crystals”, Bulletin of the South Ural State University, Series “Mathematical Modelling, Programming Computer Software”, 6:1 (2013), 34–42 (in Russ.)

[26] Klark T., Computer chemistry, Mir Publ., M., 1990, 381 pp. (in Russ.)

[27] Stepanov N. F., Novakovskaya Yu.V., Rossiyskiy Khimicheskiy Zhurnal, LI:5 (2007), 5–17 (in Russ.)

[28] T.R. Gibbs, A. Popolato, Last explosive property data. Los Alamos series on dynamic material properties, University of California Press, Berkeley–Los Angeles–London, 1980