The second virial coefficient and critical point behavior of the Mie Potential

Aspects of the second virial coefficient, b2, of the Mie m : n potential are investigated. The Boyle temperature, T0, is shown to decay monotonically with increasing m and n, while the maximum temperature, Tmax, exhibits a minimum at a value of m which increases as n increases. For the 2n : n special case T0 tends to zero and Tmax approaches the value of 7.81 in the n → ∞ limit which is in quantitative agreement with the expressions derived in Rickayzen and Heyes [J. Chem. Phys. 126, 114504 (2007)] in which it was shown that the 2n : n potential in the n → ∞ limit approaches Baxter's sticky-sphere model. The same approach is used to estimate the n - dependent critical temperature of the 2n : n potential in the large n limit. The ratio of T0 to the critical temperature tends to unity in the infinite n limit for the 2n : n potential. The rate of convergence of expansions of b2 about the high temperature limit is investigated, and they are shown to converge rapidly even at quite low temperatures (e.g., 0.05). In contrast, a low temperature expansion of the Lennard-Jones 12 : 6 potential is shown to be an asymptotic series. Two formulas that resolve b2 into its repulsive and attractive terms are derived. The convergence at high temperature of the Lennard-Jones b2 to the m = 12 inverse power value is slow (e.g., requiring T ≃ 10(4) just to attain two significant figure accuracy). The behavior of b2 of the ∞ : n and the Sutherland potential special case, n = 6, is explored. By fitting to the exact b2 values, a semiempirical formula is derived for the temperature dependence of b2 of the Lennard-Jones potential which has the correct high and low temperature limits.