The question of agreement of the kinetic moment theory with the first principle continuum thermodynamics is considered. The characteristics of distribution function describing local non-equilibrium state of monatomic gases are analyzed. In the first approach on non-equilibrium variables the equations making more exact the results of Chapman and Enskog are received. The paradox of heat conduction is eliminated and the coincidence of theoretical and experimental data on dispersion of sound speed in inert gases is improved.