  An equation of state for the domain extending from hot gases to cool-dense
fluids is formulated for a hydrogen-helium mixture. The physical processes take
account of temperature ionization and dissociation, electron degeneracy,
Coulomb coupling and pressure ionization. Pressure ionization and Coulomb
coupling are studied with simple and comprehensive modeling. A single and
complete algorithm is achieved with explicit expressions available for the
whole domain from hot gases to cool dense fluids ($T>10^2% K$). Pressure
ionization and Coulomb coupling have been examined for their contributions to
the pressure and internal energy. The result reveals that their contributions
smooth the variation of the pressure and internal energy in the region of
pressure ionization even at very low temperatures.

Astrophysics