Derivation of the second virial coefficient for the Lennard–Jones potential: analytical assessment and literature validation

Abstract The Virial equation of state provides a direct link between microscopic intermolecular potentials and macroscopic P-V-T fluid behavior, since Virial coefficients can be obtained empirically or derived from statistical thermodynamics. This work revisits and reconstructs the analytical derivations of all literature formulations for the second Virial coefficient B ( T ) based on the canonical Lennard-Jones (LJ) potential and proposes three new expressions. It also assesses truncation limits for summation-based expressions by benchmarking against reference reduced second Virial coefficient B * ( T * ) values and deriving an empirical truncation rule as a function of reduced temperature T * and acceptable remainder R n . Empirical truncation rules were fitted as functions of reduced temperature T * and acceptable remainder R n over 0.1≤ T * ≤200 and 0.0001≤| R n |≤2. Finally, combining literature B ( T ) expressions with commonly used Lennard-Jones parameters ( σ and ϵ ), predictive accuracy was assessed against a broad experimental database using RMSD and MAE. Agreement was further improved by re-estimating via a hybrid stochastic-deterministic optimization strategy. All analytical derivations were reconstructed, and three new expressions were proposd for evaluating B ( T ). Literature Lennard-Jones parameter sets showed substantial deviations for helium, hydrogen, propane, and especially ethane. A new parameter set was therefore obtained via hybrid optimization, reducing RMSD – particularly at low temperatures – while also underscoring the inherent limits of a two-parameter isotropic LJ model for complex molecules. Exploration of the B ( T ) for the LJ potential formulations available in literature enabled closing literature gaps of some expressions: deductive pathway, necessity of an appropriate truncation strategy and delimitation of prediction accuracy of the Lennard-Jones parameters widely used in literature.