Thermodynamics deals with the macroscopic properties of matter. Thermodynamics is not dependent on the fundamental nature of matter. That is, thermodynamics is valid whether matter is made up of atoms and molecules or whether matter is made up of some kind of continuum fabric. Quantum mechanics deals with the behavior of matter on the microscopic scale. In quantum mechanics we calculate the energy and behavior of a few atoms or small molecules.
Statistical mechanics is a bridge between quantum mechanics and thermodynamics.
We seek to use statistical mechanics to compute macroscopic properties from the quantum mechanical information about the atoms and molecules of interest. Usually EOS are cast in some analytic functional form. Guess some functional form, . Measure as much data as you possibly can for a specific fluid. Regress the parameters in the EOS in some least squares fashion.