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Motivated by a recent paper of Pennisi (2021) in the relativistic framework,we revisit the previous approach of two hierarchies of moments critically andpropose a new natural physical hierarchy of moments to describe classicalrarefied non-polytropic polyatomic gas in the framework of Molecular RationalExtended Thermodynamics. The differential system of the previous approach isproved to be a \emph{principal subsystem} of the present one. The main idea isthat at the molecular level, the total energy is the kinetic energy plus theenergy of internal mode due to the rotation and vibration, and the increasingmoments contain this total energy as power in which the power index increaseswith the number of tensorial indexes. In particular, we consider the case of$15$ moments, and we close the system using the variational method of theMaximum Entropy Principle. We prove the convexity of entropy and thepossibility to put the system in symmetric form. This more rich kineticframework may be interesting also as possible applications to biomathematics orother fields in which kinetic models were applied recently with success.

Which moments are appropriate to describe gases with internal structure in Rational Extended Thermodynamics?