  An alternative to the well-known complete form of the Mie-Gr\"uneisen
equation of state (EOS) for water is suggested. A closed analytical description
of the self-consistent EOS for an arbitrary medium based only on the first law
of thermodynamics and on a new form of virial theorem is obtained. This form of
the virial theorem (allowing a variable power-law exponent of the particles
interaction potential) is a result of the generalization of the known method of
similarity (Feynman et al., 1949). In the new EOS, the description of the
internal potential energy as a solution of a nonlinear Riemann-Hopf type
equation is proposed.
