We study the kinetic theory of a multi-component plasma. We introduce a scaling for a coupled electron-ion-neutral Boltzmann system, specifically adapted for modeling multi-species plasmas such as those found in Hall effect thrusters. The physics of the thruster is fundamentally characterized by significant variations in certain key physical quantities of interest, such as species densities or the electric field. This leads us to consider different zones corresponding to different physical regimes. As a result, we characterize the following regimes: a regime that corresponds to the neutral gas injection zone, a second regime for the gas ionization zone, and finally, a regime for the ion acceleration zone and plasma plume. We take into account the multi-component nature of the plasmas in our scaling, resulting in the appearance of small parameters, such as the mass ratio between heavy species and electrons, or the density ratio between charged species and the neutral species. This allows us for proceeding with a Hilbert expansion of the distribution functions in these small parameters. The system is then studied at successive orders of approximation, each corresponding to a characteristic physical time. Acknowledgement : This work is funded by CIEDS through OpenNumDef project.