Some thermodynamic relations at the critical point in liquid-vapor systems

The relation of the symmetry of the thermodynamic functions with respect to the critical density rho(c) to the analyticity of the chemical potential above the critical temperature is discussed with the aid of thermo-dynamic considerations. Special consideration is given to the differential coefficients ([unk](2)mu/[unk]T(2))rho and ([unk](2)P/[unk]T(2))rho. The discussion is extended to the two-phase region, and some of the relations between critical exponents are analyzed. A proof is given that the vapor-pressure curve joins smoothly with the critical isochore. The behavior of the entropy along the coexistence curve is considered. Finally, an argument is presented that the Griffiths-Liberman inequality gamma' >/= beta(delta - 1) is an equality.