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

Abstract

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 a calculation of the second virial coefficient, $B_2$, by an equation $B_2 = 2πN_0 \int_0^\infty \left(1- \exp \left(ω/ kT\right)\right) r^2 \text{d}r$, where $ω$ is the potential of the attraction between two molecules in the gas. Here we present an alternative approach that does not use this equation, and does not a-priori assume that the function is quadratic in concentration, $c$. Still, the quadratic dependence on concentration is also found. We analyze a gas consisting of argon at temperatures between 200 and 700 K. From the numerical calculations, we derive that the attraction parameter depends on temperature according to a -1/6 power scaling, and thus the attraction component to the second virial coefficient, $B_2$, scales with a -7/6 power to temperature. For the same conditions, the virial equation presented above results in a square root scaling of $B_2$ with temperature, which is less accurate.

Figures (1)

• Figure 1: The contribution of attraction to the second virial coefficient of argon, as function of temperature. Continuous line is based on evaluation of the London dispersion force by the method explained in this paper, and circles are literature data. Dashed line is a 7/6th scaling with temperature. These data are obtained from aggregate $B_2$-values from which we subtract $B_\text{2,hs}=34$ cm3/mol, which is the contribution of hard sphere, volumetric, interactions.