An analytical expression for the second virial coefficient based on an exponent-spline-Morse-spline-van der Waals (ESMSV) potential is presented here for use in defining the thermodynamic properties of rare gases. Our method is established based on a series expansion of the exponential function, Meijer function, gamma function, binomial function, and hypergeometric function. Numerical approaches have commonly been used for the evaluation of the second virial coefficient with the ESMSV potential in the literature. The general formula obtained here can be applied to estimate the thermal properties of rare gases. Our results for the second virial coefficient based on the ESMSV potential of He-He, He-Ne, He-Ar, and He-Xe rare gases are compared with numerical calculations and experimental data, and it is shown that our analytical expression can be successfully used for other gases.

In this study, we proposed an alternative analytical procedure for appraising the fugacity coefficient for gases using second virial coefficient with Lennard-Jones (12-6) potential. The present research is more efficient for determining of the fugacity coefficient of gases in wide ranges temperature and pressure ranges and related thermodynamic properties which is the novelty of this study. The fugacity coefficient for gases , , and has calculated in the temperature range from 30 K to 1000 K and pressure range from 0.1 atm to 200 atm by using the obtained analytical formula. To our knowledge, this work offers the first approximation to the evaluation of the fugacity coefficient for gases for and by second virial coefficients. The results obtained using the analytical formula of the fugacity coefficient is in good agreement with literature.