The method of moments is used to obtain solutions to the Boltzmann equation describing flow and thermal relaxation of binary atomic mixtures undergoing adiabatic free jet expansion. Differential moment equations are solved within the framework of an anisotropic Maxwellian distribution function. Numerical integration of these equations yields the temporal dependence of the four moments of the distribution function for each flow component; density, hydrodynamic velocity, parallel temperature, and perpendicular temperature. Results are presented for mixtures of the rare gases which yield new insight into the driving forces behind the phenomena of velocity and temperature slip. The model can easily be extended to ternary and higher atomic mixtures.