Adiabatic lapse rate estimation using a van der Waals-type equation of state

We revisit a family of temperature-dependent van der Waals-type equations of state (EOS) to improve the estimation of the adiabatic lapse rate in planetary atmospheres. These EOS generalize the classical van der Waals and Berthelot models by introducing a single parameter that modulates the temperature dependence of intermolecular interactions. We analyze their thermodynamic properties, including critical behavior, spinodal and coexistence curves, and entropy. The adiabatic curves are computed by incorporating explicitly the contribution of molecular vibrational and rotational degrees of freedom. Using a generalized expression for the adiabatic lapse rate, we estimate the adiabatic lapse rate in the troposphere of Titan and Venus. Our results show that the van der Waals-type EOS reproduce observed lapse rates more accurately than the van der Waals EOS.