  A detailed comparison is carried out between two popular equations of state
(EOS), the Mihalas-Hummer-Dappen (MHD) and the OPAL equations of state, which
have found widespread use in solar and stellar modeling during the past two
decades. They are parts of two independent efforts to recalculate stellar
opacities; the international Opacity Project (OP) and the Livermore-based OPAL
project. We examine the difference between the two equations of state in a
broad sense, over the whole applicable rho-T range, and for three different
chemical mixtures. Such a global comparison highlights both their differences
and their similarities.
  We find that omitting a questionable hard-sphere correction, tau, to the
Coulomb interaction in the MHD formulation, greatly improves the agreement
between the MHD and OPAL EOS. We also find signs of differences that could stem
from quantum effects not yet included in the MHD EOS, and differences in the
ionization zones that are probably caused by differences in the mechanisms for
pressure ionization. Our analysis do not only give a clearer perception of the
limitations of each equation of state for astrophysical applications, but also
serve as guidance for future work on the physical issues behind the
differences. The outcome should be an improvement of both equations of state.
