Low Temperature Second Virial Coefficients for a Lennard-Jones Gas

For low temperatures, the second virial coefficient $$\mathrm{B(\theta )=}\frac{2}{3}{\mathrm{\pi Nr}}_0^3\mathrm{G(\theta )}$$ of a gas which follows a Lennard‐Jones 6—s potential, $$U_s{\text{(r)={ε/(s−6)}[6(r}}_0{\mathrm{/r)}}^s{\mathrm{-s(r}}_0{\mathrm{/r)}}^6]$$ where θ=ε/kT, can be represented by $$G_s{\mathrm{(\theta )\sim -e}}^{\mathrm{+\theta }}{\text{(3π/sθ)}}^{\frac{1}{2}}{\displaystyle \sum _{\text{n=0}}}{\beta }_n{\mathrm{(s)/\theta }}^n.$$ In two previous studies, formulas for β_n(9) and β_n(12) have been presented. In this paper, the relation for G_s(θ) is developed for the general case, using the method of steepest descents, and the first three coefficients β₀(s), β₁(s), and β₂(s) are computed. The use of a Hooke's law potential and the quasi‐ideal gas hypothesis yields the correct leading term in G_s(θ), but results in errors in β₁(s) and β₂(s).