  We consider a non reactive two component gas mixture. In a macroscopic
description of a gas mixture we expect four physical coefficients
characterizing the physical behaviour of the gases to appear. These are the
diffusion coefficient, the viscosity coefficient, the heat conductivity and the
thermal diffusion parameter in the Navier-Stokes equations. We present a
Chapman-Enskog expansion of a kinetic model for gas mixtures by Klingenberg,
Pirner and Puppo, 2017 that has three free parameters in order to capture three
of these four physical coefficients. In addition, we propose several possible
extensions to an ellipsoidal statistical model for gas mixtures in order to
capture the fourth coefficient.
