  In this paper, we consider the kinetic model of continuous type describing a
polyatomic gas in two different settings corresponding to a different choice of
the functional space used to define macroscopic quantities. Such a model
introduces a single continuous variable supposed to capture all the phenomena
related to the more complex structure of a molecule having more than one atom,
such as internal degrees of freedom in a collision. In particular, we provide a
direct comparison of these two settings, and show their equivalence after the
distribution function is rescaled and the cross section is reformulated. We
then focus on the kinetic model for which the rigorous existence and uniqueness
result in the space homogeneous case is recently proven. Using the cross
section proposed in that analysis together with the maximum entropy principle,
we establish macroscopic models of six and fourteen fields. In the case of six
moments, we calculate the exact, nonlinear, production term and prove its total
agreement with extended thermodynamics, as it satisfies the entropy residual
inequality on the whole range of model validity. Moreover, for the fourteen
moments model, we provide new expressions for relaxation times and transport
coefficients in a linearized setting, that yield both matching with the
experimental data for dependence of the shear viscosity upon temperature and a
satisfactory agreement with the theoretical value of the Prandtl number, on the
room temperature range when only translational and rotational modes of
molecules are taken into account, as much as on higher temperatures when
vibrational modes appear as well.
