Equations of state suitable for modeling compression of solids under the hydrodynamic assumption are presented. Specifically derived are reductions of the Lagrangian, Eulerian, and logarithmic theories developed in the prior three chapters to cases wherein deviatoric stress can be ignored. In such cases, scalar equations of state are obtained that relate pressure, volume, and temperature or entropy. Model predictions are compared with planar shock data for finite compression of ductile metals, demonstrating suitability of the hydrodynamic approximation as well as superiority of the Eulerian equation of state, which is equivalent to that of Birch and Murnaghan. The logarithmic equation of state is found suitable for modeling hydrostatic compression of several less ductile polycrystalline minerals. This chapter concludes with an overall assessment of the three thermoelastic formulations, where the Eulerian model is deemed preferable for ductile solids with a relatively low ratio of shear to bulk modulus and the logarithmic model for brittle solids with a higher ratio of shear to bulk modulus.