Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 14, 2021

Analytical treatment of the critical properties of a generalized van der Waals equation

  • Magdy E Amin EMAIL logo

Abstract

The two-parameter van der Waals (vdW) equation of state is generalized, by adding another two parameters to the attractive term. General relations between thermodynamic functions of the generalized vdW equation and the hard sphere gas are derived. The cubic equation of the generalized vdW is solved and the critical points (Pc, Vc, Tc) are obtained for general k. The critical properties of the vdW real gas such as the isothermal compressibility KT, the isobaric expansion coefficient α and the isobaric heat capacity CP are calculated exactly. The temperature dependence of KT, α and CP is investigated close to the critical point on the critical isobar path Pr = 1(P = Pc). Numerical calculations for KT and CP are presented above and below Pr.


Corresponding author: Magdy E Amin, Mathematics Department, Faculty of Science, Minia University, 61915El-Minia, Egypt, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] J. D. van der Waals, Ph.D. Thesis, Leiden Univ., 1873; English translation: J. D. van der Waals, On the Continuity of the Gaseous and Liquid States, Dover, Mineola, NY (1988).Search in Google Scholar

[2] L. D. Landau and E. M. Lifshitz, Statistical Physics, Oxford, Pergamon, 1975.Search in Google Scholar

[3] W. Greiner, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics, New York, Springer-Verlag, Inc., 1995.10.1007/978-1-4612-0827-3Search in Google Scholar

[4] D. V. Schroeder, An Introduction to Thermal Physics, San Francisco, Addison Wesley Lonngman, 2000.Search in Google Scholar

[5] O. Redlich and J. N. S. Kwong, “On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions,” Chem. Rev., vol. 44, p. 233, 1949. https://doi.org/10.1021/cr60137a013.Search in Google Scholar

[6] G. Soave, “Equilibrium constants from a modified Redlich-Kwong equation of state,” Chem. Eng. Sci., vol. 27, p. 1197, 1972. https://doi.org/10.1016/0009-2509(72)80096-4.Search in Google Scholar

[7] D.-Y. Peng and D. B. Robinson, “A new two-constant equation of state,” Ind. Eng. Chem. Fund., vol. 15, p. 59, 1976. https://doi.org/10.1021/i160057a011.Search in Google Scholar

[8] D. Jou and C. Pérez-García, “Generalized van der Waals equation for nonequilibrium fluids,” Phys. Rev. A, vol. 28, p. 2541, 1983. https://doi.org/10.1103/physreva.28.2541.Search in Google Scholar

[9] Y. Adachi, B. C.-Y. Lu, and H. Sugie, “A four-parameter equation of state,” Fluid Phase Equil., vol. 11, p. 29, 1983. https://doi.org/10.1016/0378-3812(83)85004-3.Search in Google Scholar

[10] M. M. Martynyuk and R. Balasubramanian, “Equation of state for fluid alkali metals: Binodal,” Int. J. Thermophys., vol. 16, no. 2, p. 533, 1995. https://doi.org/10.1007/bf01441919.Search in Google Scholar

[11] H. Hinojosa-Gómez, J. F. Barragán-Aroche, and E. R. Bazúa-Rueda, “A modification to the Peng-Robinson-fitted equation of state for pure substances,” Fluid Phase Equil., vol. 298, p. 12, 2010. https://doi.org/10.1016/j.fluid.2010.06.022.Search in Google Scholar

[12] A. A. Sobko, “Description of Evaporation Curve by the Generalized Van-der-Waals-Berthelot Equation. Part I, Journal of Physical Science and Application,” J. Phys. Sci. Appl., vol. 4, no. 8, p. 524, 2014.Search in Google Scholar

[13] J. S. Lopez-Echeverry, S. Reif-Acherman, and E. Araujo-Lopez, “Peng-Robinson equation of state: 40 years through cubics,” Fluid Phase Equil., vol. 447, p. 39, 2017. https://doi.org/10.1016/j.fluid.2017.05.007.Search in Google Scholar

[14] R. Balasubramanian and G. Theertharaman, “A new four-parameter generalized van der Waals equation of state: metastable state of group IV elements,” Int. J. Sci. Res., vol. 7, no. 4, p. 165, 2018.Search in Google Scholar

[15] C. N. Yang and T. D. Lee, “Statistical theory of equations of state and phase transitions. 1. Theory of condensation,” Phys. Rev., vol. 87, p. 410, 1952. https://doi.org/10.1103/physrev.87.404.Search in Google Scholar

[16] J.-H. Park and S.-W. Kim, “Existence of a critical point in the phase diagram of the ideal relativistic neutral Bose gas,” New J. Phys., vol. 13, p. 033003, 2011. https://doi.org/10.1088/1367-2630/13/3/033003.Search in Google Scholar

[17] K. Huang, Statistical Mechanics, New York,John Wiley & Sons. Inc., 1987.Search in Google Scholar

[18] M. E. Fisher, “The theory of equilibrium critical phenomena,” Rep. Prog. Phys., vol. 30, p. 615, 1967. https://doi.org/10.1088/0034-4885/30/2/306.Search in Google Scholar

[19] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, New York, NY, Oxford University Press, 1971.Search in Google Scholar

[20] L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, Singapore, World Scientific, 2000.10.1142/4016Search in Google Scholar

[21] K. Michaelian and I. Santamaría-Holek, “Critical analysis of negative heat capacity in nanoclusters,” Europhys. Lett., vol. 79, p. 43001, 2007. https://doi.org/10.1209/0295-5075/79/43001.Search in Google Scholar

[22](a) W. Thirring, “Systems with negative specific heat,” Z. Phys., vol. 235, p. 339, 1970. https://doi.org/10.1007/bf01403177.Search in Google Scholar

(b) D. Lynden-Bell, Proc. XXth IUPAP Int. Conf on Stat. Phys., Paris, July 20–24, 1998, condmat/9812172.Search in Google Scholar

[23] M. Ď. Agostino, F. Gulminelli, P. Chomaz, et al.., “Negative heat capacity in the critical region of nuclear fragmentation: an experimental evidence of the liquid-gas phase transition,” Phys. Lett. B, vol. 473, p. 219, 2000.10.1016/S0370-2693(99)01486-0Search in Google Scholar

[24](a) M. Schmidt, R. Kusche, W. Kronmller, B. v. Issendorff, and H. Haberland, “Experimental determination of the melting point and heat capacity for a free cluster of 139 sodium atoms,” Phys. Rev. Lett., vol. 79, p. 99, 1997.10.1103/PhysRevLett.79.99Search in Google Scholar

(b) M. Schmidt, R. Kusche, B. v. Issendorff, and H. Haberland, “Irregular variations in the melting point of size-selected atomic clusters,” Nature (London), vol. 393, p. 238, 1998.10.1038/30415Search in Google Scholar

(c) M. Schmidt, R. Kusche, T. Hippler, et al.., “Negative heat capacity for a cluster of 147 sodium atoms,” Phys. Rev. Lett., vol. 86, p. 1191, 2001.10.1103/PhysRevLett.86.1191Search in Google Scholar

[25] M. Ď. Agostino, R. Bougault, F. Gulminelli, et al.., “On the reliability of negative heat capacity measurements,” Nucl. Phys. A, vol. 699, p. 795, 2002.10.1016/S0375-9474(01)01287-8Search in Google Scholar

[26] D. H. E. Gross, “Microcanonical thermodynamics and statistical fragmentation of dissipative systems. The topological structure of the N-body phase space,” Phys. Rep., vol. 279, p. 119, 1997. https://doi.org/10.1016/s0370-1573(96)00024-5.Search in Google Scholar

[27] P. Chomaz, V. Duflot, and F. Gulminelli, “Caloric curves and energy fluctuations in the microcanonical liquid-gas phase transition,” Phys. Rev. Lett., vol. 85, p. 3587, 2000. https://doi.org/10.1103/physrevlett.85.3587.Search in Google Scholar

[28] L. G. Moretto, J. B. Elliott, L. Phair, and G. J. Wozniak, “Negative heat capacities and first order phase transitions in nuclei,” Phys. Rev. C, vol. 66, p. 041601, 2002. https://doi.org/10.1103/physrevc.66.041601.Search in Google Scholar

[29] C. B. Das, S. Das Gupta, and A. Z. Mekjian, “Negative specific heat in a thermodynamic model of multifragmentation,” Phys. Rev. C, vol. 68, p. 014607, 2003. https://doi.org/10.1103/physrevc.68.014607.Search in Google Scholar

[30] C. Das, S. Dasgupta, W. Lynch, A. Mekjian, and M. Tsang, “The thermodynamic model for nuclear multifragmentation,” Phys. Rep., vol. 406, p. 1, 2005. https://doi.org/10.1016/j.physrep.2004.10.002.Search in Google Scholar


Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/zna-2021-0002).


Received: 2021-01-03
Revised: 2021-03-29
Accepted: 2021-04-10
Published Online: 2021-05-14
Published in Print: 2021-07-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/zna-2021-0002/html
Scroll to top button