The flow field in the ionizing relaxation zone developed behind a normal shock wave in an electrically neutral, homogeneous, two temperature mixture of thermally ideal gases (molecules, atoms, ions, electrons) was numerically solved. The heat transfer between the electron gas and the other components was taken into account while all the other transport phenomena (molecular, turbulent and radiative) were neglected in the relaxation zone, since it is dominated by inelastic collisions. The threshold cross sections measured by L. T. Specht, for excitation of argon by electron collisions, were used. The calculated results show good agreement with the results of the shock tube experiments presented by I. I. Glass and W. S. Liu, especially in the electron avalanche region. A critical examination was made of the common assumptions regarding the average energy with which electrons are produced by atom-atom collisions and the relative effectiveness of atom-atom collisions (versus electron-atom collisions) in ionizing excited argon.