Carnahan-Starling type equations of state for stable hard disk and hard sphere fluids

Abstract

The well-known Carnahan-Starling (CS) equation of state (EoS) [N.F. Carnahan and K.E. Starling. J. Chem. Phys. 51 (2), 635–636 (1969). doi:10.1063/1.1672048] for hard sphere (HS) fluid was derived from a quadratic relationship between the integer portions of the virial coefficients, $B_{n,integer}$, and their orders, $n$. In this paper, the method is extended to cover the full virial coefficients, $B_n$, for the general D-dimensional case. We propose a (D-1)th order polynomial for the virial coefficients starting from the 4th order and EoS’s are derived from it. For the hard rod (D = 1) case, the exact solution is obtained. For the stable hard disk fluid (D = 2), the most recent virial coefficients, up to the 10th order, [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0] and accurate compressibility data [J. Kolafa and M. Rottner. Mol. Phys. 104 (22–24), 3435–3441 (2006). doi:10.1080/00268970600967963; J.J. Erpenbeck and M. Luban. Phys. Rev. A. 32 (5), 2920–2922 (1985). doi:10.1103/PhysRevA.32.2920] are employed to construct and to test the EoS. For the stable hard sphere (D = 3) fluid, a new CS-type EoS is obtained by using the most recent virial coefficients [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0; A.J. Schultz and D.A. Kofke. Physical Review E. 90 (2) (2014). doi:10.1103/PhysRevE.90.023301], up to the 11th order, along with highly-accurate compressibility data [S. Pieprzyk et al. Phys. Chem. Chem. Phys. 21 (13), 6886–6899 (2019). doi:10.1039/C9CP00903E; M.N. Bannerman et al. J. Chem. Phys. 132 (8), 084507 (2010). doi:10.1063/1.3328823; J. Kolafa, et al. Phys. Chem. Chem. Phys. 6 (9), 2335–2340 (2004). doi:10.1039/B402792B]. The simple new EoS’s prove to be as accurate as the Padé approximations based on all available virial coefficients, which significantly improve the accuracy of the CS-type EoS in the hard sphere case. This research also reveals that as long as the virial coefficients obey a polynomial function, any EoS derived from it will diverge at the non-physical packing fraction, $\eta =1$.

GRAPHICAL ABSTRACT

KEYWORDS:

• Equation of state
• hard sphere
• hard disc
• virial coefficients
• thermodynamics

Acknowledgements

The author is grateful to a reviewer of the manuscript for the valuable suggestions, corrections and new references, which helped immensely to improve the quality of the final version.

Disclosure statement

No potential conflict of interest was reported by the author(s).