Calculating liquid-phase entropy using a real gas model within the QM/PCM framework

In conventional quantum mechanical/polarizable continuum model (QM/PCM) method-based thermodynamic calculations, the entropy of liquid-phase molecules is typically evaluated using the ideal gas model, leading to systematic overestimation. The study reported herein demonstrates a real gas model (RGM) that uses free volume values to generate more physically reasonable liquid-phase entropies. This RGM produces a more accurate translational entropy by replacing the volume term in the translational partition function with the free volume, defined as the region accessible to the molecular center of mass. In the case of pure liquids, the free volume was obtained from the relationship between the liquid molar volume and the molecular volume under an idealized cubic packing assumption. For solutes in solution, the effective excluded volume of the solvent molecules was also considered. Although this model requires experimental density data for liquids, the associated computational cost is comparable to that of the ideal gas model. The pressure equation derived from the RGM partition function was shown to be formally identical to the van der Waals equation of state without the attractive term, implying that the liquid phase is equivalent to a real gas under high pressure in this model. Entropy calculations were performed for 20 pure liquids and 37 solutes in solution using QM/PCM computations at the ωB97X-D/6-311++G(d,p)/IEF-PCM level, followed by RGM-based corrections. The calculated entropies reproduced experimental values with root-mean-square deviations within the range of chemical accuracy. To assess the impact of entropy accuracy on Gibbs free energy values and the temperature dependence of these values, the Gibbs free energies of gaseous and liquid methanol and ethanol were evaluated as functions of temperature at the G4//ωB97X-D/6-311++G(d,p) level, and the intersection points of the resulting curves were used to estimate boiling points. The predicted boiling points were in reasonable agreement with experimentally derived data, demonstrating that the RGM considered here substantially improves the predictive accuracy compared with conventional models.