The problem of redistribution of translational and rotational energy has been solved for diatomic gases within the framework of the Chapman-Enskog method and the Parker model in the general case of the arbitrary energy exchange ratio. The nonequilibrium gasdynamic equations, transport coefficients and relaxation time have been found for rotational-translational processes in a diatomic gas. The calculations of relaxation time, viscosity, thermal conductivity, and diffusion coefficients are carried out in the temperature range from 200 K to 10,000 K for nitrogen. The calculated parameters and coefficients are compared with the values obtained by the Mason-Monchick approximate method as well as data from experiments in ultrasonic, shock-wave, and vacuum devices. The correlation of the theoretical and experimental data is satisfactory. The applicability of one-and two-temperature relaxation models is discussed. The numerical solutions of the obtained system of the Navier-Stokes equations are analyzed for the cases of nitrogen flow in underexpanded jets and supersonic rarefied gas flow near a sphere.