The paper has the following structure. In Sect. 6.2, we give a geometrical description of the thermodynamic state. We consider the thermodynamic state as twodimensional Lagrangian manifold, which can be defined by two equations with compatibility condition. In Sect. 6.3, we study state equations and corresponding Lagrangian manifolds for van der Waals gases and get its applicable domains with a description of phase transitions. In Sect. 6.4,we look for solutions as asymptotic expansions and analyse the zeroth and the first-order approximations. In Sect. 6.5, we show space-time domains corresponding to different phases of the medium.