  For mixtures of Bose and Fermi alkali atoms, the Fermion degrees of freedom
can be integrated out in their {\bf finite temperature} partition function. The
final result can be expanded as a power series in the Boson density. Under
appropriate conditions, the pairwise interaction between the bosons can be
changed from positive to negative at a low enough temperature by the
Fermion-mediated term and the Boson cloud may then collapse. For attractive
Fermion-Boson interactions, there is also a collapse of the Fermion cloud. This
transition is first order (discontinuous) because the leading order nonlinear
interaction term between the Bosons is {\bf third order in Boson density with a
negative coefficient}. We discuss the finite temperature phase diagram of this
transition. Our result may provide for an explanation of recent experimental
observations by Modungo and coworkers.
