  We study in more detail the properties of the generalized Beth Uhlenbeck
formula obtained in a preceding article. This formula leads to a simple
integral expression of the grand potential of the system, where the interaction
potential appears only through the matrix elements of the second order Ursell
operator $U_{2}$. Our results remain valid for significant degree of degeneracy
of the gas, but not when Bose Einstein (or BCS) condensation is reached, or
even too close from this transition point. We apply them to the study of the
thermodynamic properties of degenerate quantum gases: equation of state,
magnetic susceptibility, effects of exchange between bound states and free
particles, etc. We compare our predictions to those obtained within other
approaches, especially the ``pseudo potential'' approximation, where the real
potential is replaced by a potential with zero range (Dirac delta function).
This comparison is conveniently made in terms of a temperature dependent
quantity, the ``Ursell length'', which we define in the text. This length plays
a role which is analogous to the scattering length for pseudopotentials, but it
is temperature dependent and may include more physical effects than just binary
collision effects; for instance at very low temperatures it may change sign or
increase almost exponentially, an effect which is reminiscent of a precursor of
the BCS pairing transition. As an illustration, numerical results for quantum
hard spheres are given.
