  In this note, we propose the first mathematical derivation of a macroscopic
Baer-Nunziato type system for compressible two-phase flows allowing two
pressure state laws depending on the different phases. By doing so, we extend
the results obtained by the first author and M. Hillairet (Annales ENS 2019) to
cover this important physical situation. A relaxation term in the mass fraction
equation is obtained without closure assumptions contrarily to
theoretical-physics literature dedicated to mixture theory, see for instance
"Thermo-Fluid dynamics of Two Fluid Flows" by M. Ishii. The relaxation
parameter is linked to the viscosities of the different fluids (which may be
small for applications) and the relaxed quantity is linked to the laws chosen
at interfaces of the two-fluid system at a mesoscale.
