The Rate at Which Ions Lose Energy in Elastic Collisions

The rate of energy loss of ions (including electrons) moving through a gas is rigorously calculated on the assumption that the ions and molecules are smooth elastic spheres with no attraction at a distance, having Maxwellian velocity distributions corresponding to the temperatures $T_i$ and $T_m$ respectively. The result is $f=\frac{8}{3}\frac{\mathrm{mM}}{{\mathrm{}(m+M)\mathrm{}}^2}\left(1-\frac{T_m}{T_i}\right)$ where $m$ and $M$ are the masses of ion and molecule respectively, $T_i$ and $T_m$ are their temperatures, and $f$ is the average energy loss per collision expressed as a fraction of the average ionic energy.