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 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 α, 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 $O/G$ or $G/O$, where $O$ is the original CFT used to define the model and $G$ is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.