The quantum statistical treatment of the Rutherford model, considering matter as a system of point charges (electrons and nuclei), is analyzed. First, in the historical context, the solutions of different fundamental problems, such as the divergence of the partition function, elaborated by Herzfeld, Planck, Brillouin and Rompe, are discussed. Beyond this, the modern state of art is presented and new results are given which explain why bound states according to a discrete part of the spectra occur only in a valley in the temperature-density plane. Based on the actual state of the quantum statistics of Coulomb systems, virial expansions within the canonical ensemble and the grand ensemble and combinations are derived. The following transitions along isotherms are studied: (i) the formation of bound states occurring by increasing the density from low to moderate values, (ii) the disappearance of bound state effects at higher densities due to medium effects. Within the physical picture isotherms of pressure for Hydrogen are calculated in a broad density region, and it is shown that in the region between 20 000 K and 100 000 K and particle densities below 1022 cm-3 the cross-over from full to partial ionization may be well described by the contributions of extended ring diagrams and ladder diagrams.