  We conjecture the inversion relations for thermalized solvable interaction
round the face (IRF) two dimensional lattice models. We base ourselves on an
ansatz for the Baxterization described by the author in the 90's. We solve
these inversion relations in the four main regimes of the models, to give the
free energy of the models, in these regimes. We use the method of Baxter in the
calculation of the free energy of the hard hexagon model. We believe these
results to be quite general, shared by most of the known IRF models. Our
results apply equally well to solvable vertex models. Using the expression for
the free energy we calculate the critical exponent $\alpha$, and from it the
dimension of the perturbing (thermal) operator in the fixed point conformal
field theory (CFT). We show that it matches either the coset ${\cal O}/{\cal
G}$ or ${\cal G}/{\cal O}$, where $\cal O$ is the original CFT used to define
the model and $\cal G$ is some unknown CFT, depending on the regime. This
agrees with known examples of such models by Huse and Jimbo et al.
