  We present a large $q$ expansion of the 2d $q$-states Potts model free
energies up to order 9 in $1/\sqrt{q}$. Its analysis leads us to an ansatz
which, in the first-order region, incorporates properties inferred from the
known critical regime at $q=4$, and predicts, for $q>4$, the $n^{\rm th}$
energy cumulant scales as the power $(3 n /2-2)$ of the correlation length. The
parameter-free energy distributions reproduce accurately, without reference to
any interface effect, the numerical data obtained in a simulation for $q=10$
with lattices of linear dimensions up to L=50. The pure phase specific heats
are predicted to be much larger, at $q\leq10$, than the values extracted from
current finite size scaling analysis of extrema. Implications for safe
numerical determinations of interface tensions are discussed.
