The form of the Boltzmann equation presently being used to describe phenomena dependent upon the internal angular momentum states of molecules in the gas phase is inconsistent in that angular momentum is not conserved. Thus, the internal angular momentum relaxation is correctly described by this equation but the resulting production of angular momentum in the translational degrees of freedom just does not appear at all. This work is aimed at extending the Boltzmann equation to give a consistent description of all conserved quantities. It is shown that this is not a trivial matter and that some truncation of an expansion in position gradients is required. The simplest choice is discussed. In the development, a central role is played by sum rules which arise from the assumed localized nature of the intermolecular potential.