  The entropy and kinetic, potential, and interaction energies of an atomic
Fermi gas in a trap are studied under the assumption of thermal equilibrium for
finite temperature. A Feshbach resonance can cause the fermions to pair into
diatomic molecules. The entropy and energies of mixtures of such molecules with
unpaired atoms are calculated, in relation to recent experiments on molecular
Bose-Einstein condensates produced in this manner. It is shown that, starting
with a Fermi gas of temperature $T= 0.1 T_F^0$, where $T_F^0$ is the
non-interacting Fermi temperature, an extremely cold degenerate Fermi gas of
temperature $T \lesssim 0.01 T_F^0$ may be produced without further evaporative
cooling. This requires adiabatic passage of the resonance, subsequent sudden
removal of unpaired atoms, and adiabatic return. We also calculate the ratio of
the interaction energy to the kinetic energy, a straightforward experimental
signal which may be used to determine the temperature of the atoms and indicate
condensation of the molecules.
