For a quantum system ofn identical spins of magnitudej, we introduce an integrated density of states of definite total spin angular momentum. The underlying sequence $$\{ \mathbb{K}_n^j :n = 1,2,...\} $$ of probability measures satisfies Varadhan's large deviation principle, and converges to a degenerate distribution. We use the Berezin-Lieb Inequalities to obtain upper and lower bounds for the limiting specific free-energy of the spins interacting with a second quantum system under specified conditions on the Hamiltonian. The method is illustrated by applications to the BCS model and to the Dicke maser model.