In the present study, a development of the paper [Can. J. of Phy., 2012, 90(2): 137-149] is introduced. The non-stationary BGK (Bhatnager- Gross- Krook) model of the Boltzmann nonlinear partial differential equations for a rarefied gas mixture affected by nonlinear thermal radiation field, for the first time, are solved instead of the stationary equations. The travelling wave solution method is used to get the exact solution of the nonlinear partial differential equations. These equations were produced from applying the moment method to the unsteady Boltzmann equation. Now, nonlinear partial differential equations should be solved in place of nonlinear ordinary differential equations, which represent an arduous task. The unsteady solution gives the problem a great generality and more applications. The new problem is investigated to follow the behavior of the macroscopic properties of the gas mixture such as the temperature and concentration. They are substituted into the corresponding two stream Maxiwallian distribution functions permitting us to investigate the non-equilibrium thermodynamic properties of the system (gas mixture + the heated plate). The entropy, entropy flux, entropy production, thermodynamic forces, kinetic coefficients are obtained for the mixture. The verification of the Boltzmann H-theorem, Le Chatelier principle, the second law of thermodynamic and the celebrated Onsager’s reciprocity relation for the system, are investigated. The ratios between the different contributions of the internal energy changes based upon the total derivatives of the extensive parameters are estimated via the Gibbs formula. The results are applied to the Argon-Neon binary gas mixture, for various values of both of the molar fraction parameters and radiation field intensity. Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.