Adiabatic lapse rate of nonideal gases: The role of molecular interactions and vibrations

Abstract

We report a formula for the dry adiabatic lapse rate that depends on the compressibility factor and the adiabatic curves. Then, to take into account the nonideal behavior of the gases, we consider molecules that can move, rotate, and vibrate and the information of molecular interactions through the virial coefficients. We deduce the compressibility factor in its virial expansion form and the adiabatic curves within the virial expansion up to any order. With this information and to illustrate the mentioned formula, we write the lapse rate for the ideal gas, and the virial expansion up to the second and third coefficient cases. To figure out the role of the virial coefficients and vibrations, under different atmospheric conditions, we calculate the lapse rate for Earth, Mars, Venus, Titan, and the exoplanet Gl 581d. Furthermore, for each one we consider three models in the virial expansion: van der Waals, square-well, and hard-sphere. Also, when possible, we compare our results to the experimental data. Finally, we remark that for Venus and Titan, which are under extreme conditions of pressure or temperature, our calculations are in good agreement with the observed values, in some instances.

Adiabatic lapse rate of real gases

“…In this section, we discuss the derivation of the DALR formula presented in Ref. [3], and its application to a family of cubic and the non-cubic Guevara EOS.…”

“…In this work, we attempt to improve previous theoretical approximations to the adiabatic lapse rate of dry atmospheres (DALR) by incorporating into its computation complete equations of state (EOS), rather than truncated virial expansions [2,3]. This is applied to the atmosphere of Titan, whose surface atmospheric conditions are close to the boiling point of N 2 (P ≈ 101 kPa, T ≈ 77 K).…”

“…To some extent, this incorporates information on interactions and molecular vibrations. However, from the conceptual point of view this is an incorrect procedure because the used lapse rate formula strongly depends on the adiabatic curves [2,3]. Another method to derive a DALR for real gases consists of considering the experimental values of the compressibility factor [6].…”

“…The shortcomings of resorting to experimental values, as described above, may be overcome by using a truncated virial expansion and modifying the adiabatic curves to incorporate both intermolecular interactions and molecular vibrations [2,3]. This method improves the value of Γ significantly in extreme atmospheric conditions and allows to identify the origin of the correction, namely, molecular vibrations (like in Venus), molecular interactions (like in Titan), or both [3].…”

“…In Sect. 2, we review the formula for the lapse rate in terms of Z, as obtained in [3], and apply it to the EOS of interest. The resulting expressions depend on the adiabatic curves of each model, which are computed in Sect.…”