A recently developed treatment of partition functions in the equation of state (Mihalas, Hummer, Daeppen, MH&D) has led to a substantial improvement in the agreement between observed and theoretically predicted solar p-mode oscillation frequencies. The MH&D equation of state is a realization of the free-energy-minimization method, based on the so-called 'chemical picture', in which ionization and dissociation reactions are assumed to be those that maximize entropy, or equivalently, minimize the free energy. An alternative equation of state has recently been developed at Livermore. It realizes a virial expansion of pressure, and is based on the 'physical picture', in which explicitly only fundamental species (i.e., electrons and nuclei) appear. Results of a first comparison between thermodynamic quantities of the MH&D and Livermore equations of state are presented. For simplicity, a mixture with only hydrogen and helium (90 percent H and 10 percent He by number) is chosen. The comparison is made for a low-density and a high-density case. In the first case, the conditions are those of the hydrogen and helium ionization zones of the sun, in the second case those of the solar center. In both cases, the MH&D and Livermore results agree strikingly, despite the very different formalisms they are based on.