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

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, Bn, integer, and their orders, n. In this paper, the method is extended to cover the full virial coefficients, Bn, 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 Pade 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,. = 1. [GRAPHICS]

Automated summary

This study extends the Carnahan-Starling (CS) equation of state method to propose a new equation for describing hard sphere and hard disk fluids, using the latest virial coefficients and compressibility data for verification and comparison. The research reveals that the simple new equations are as accurate as the Pade approximations, significantly improving the accuracy of the CS-type equation of state in the hard sphere case.