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

Aspects of the second virial coefficient, b 2, of the Mie m : n potential are investigated. The Boyle temperature, T 0, 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 T 0 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 T 0 to the critical temperature tends to unity in the infinite n limit for the 2n : n potential. The rate of convergence of expansions of b 2 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 b 2 into its repulsive and attractive terms are derived. The convergence at high temperature of the Lennard-Jones b 2 to the m = 12 inverse power value is slow (e.g., requiring T ≃ 10⁴ just to attain two significant figure accuracy). The behavior of b 2 of the ∞ : n and the Sutherland potential special case, n = 6, is explored. By fitting to the exact b 2 values, a semiempirical formula is derived for the temperature dependence of b 2 of the Lennard-Jones potential which has the correct high and low temperature limits.

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