Boltzmann equation, summation invariant

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

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