  We calculate the ratio of the viscosity to the entropy density for both Bose
and Fermi gases in the unitary limit using a new approach to the quantum
statistical mechanics of gases based on the S-matrix. In the unitary limit the
scattering length diverges and the S-matrix equals -1. For the fermion case we
obtain eta/s > 4.7 times the proposed lower bound of \hbar/4 pi k_B which came
from the AdS/CFT for gauge theories, consistent with the most recent
experiments. For the bosonic case we present evidence that the gas undergoes a
phase transition to a strongly interacting Bose-Einstein condensate, and is a
more perfect fluid, with eta/s > 1.3 times the bound.
