This paper deals with further developments of a nonlinear theory for a nonviscous gas in the presence of heat flux, which has been proposed in previous papers, using extended thermodynamics. The fundamental fields used are the density, the velocity, the internal energy density, and the heat flux. Using the Liu procedure, the constitutive theory is built up without approximations and the consistence of the model is showed: it is shown that the model is determined by the choice of three scalar functions which must satisfy a system of partial differential equations, which always has solutions.
Different changes of field variables are carned out, using different Legendre transformations, passing from variables which are convenient from a mathematical pomt of view to intermediate nonequilibrium variables and finally to equilibrium variables. In particular, two possible extensions of the temperature far from equilibrium are considered: the "nonequilibrium temperature" of extended irreversible thermodynamics
[defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy) and the "thermodynamic temperature" by Miiller (defined as the mverse of the coefficient linking the nonequilibrium entropy flux to the heat flux). The link between these two quantities is established, general expressions for these quantities, as functions of the density, of the absolute temperature and of the heat flux are obtained.