  This paper investigates the asymptotic behavior of the Helmholtz free energy
of mixtures at small compressibility. We start from a general representation
for the local free energy that is valid in stable subregions of the phase
diagram. On the basis of this representation we classify the admissible data to
construct a thermodynamically consistent constitutive model. We then analyze
the incompressible limit, where the molar volume becomes independent of
pressure. Here we are confronted with two problems:
  (i) Our study shows that the physical system at hand cannot remain
incompressible for arbitrary large deviations from a reference pressure unless
its volume is linear in the composition.
  (ii) As a consequence of the second law of thermodynamics, the incompressible
limit implies that the molar volume becomes independent of temperature as well.
Most applications, however, reveal the non-appropriateness of this property.
  According to our mathematical treatment, the free energy as a function of
temperature and partial masses tends to a limit in the sense of epi- or
Gamma-convergence. In the context of the first problem, we study the mixing of
two fluids to compare the linearity with experimental observations. The second
problem will be treated by considering the asymptotic behavior of both a
general inequality relating thermal expansion and compressibility and a
PDE-system relying on the equations of balance for partial masses, momentum and
the internal energy.
