Analysis of the Rankine attraction term in an equation of state based on the London dispersion force

The attraction term in an equation of state for gases, $-a c^2$, proposed by Rankine in 1854, is generally related to the London dispersion force via the equation for the second virial coefficient, $B_2$, given by $B_2 = 2πN_0 \int_0^\infty \left(1- \exp \left(-ω/ kT\right)\right) r^2 \text{d}r$, where $ω$ is the attraction energy between two molecules in the gas. This equation works very well to describe $B_2$ and thus $a=B_2 RT$ as function of temperature, but the derivation is complicated. Here we present two other methods to derive $B_2$ and thus $a$ from the London equation, which have a more intuitive background. (The more simple of the two we expect must be available in literature.) We analyze these three models for the gas argon at temperatures between 150 and 900 K. All three methods fit the data quite closely while an analytical approximation fits data even better. The temperature dependence of $a$ is well described by a proportionality of $a$ with a term $1+T^*/T$ where $T^*$ is a constant that depends on the type of gas.