The classical description of the critical point leads to infinities in such properties as the compressibility and the heat capacity at constant pressure, Cp, as the critical point is approached along a path in the homogeneous fluid. However, this description does not require a singularity in the heat capacity at constant volume, Cv, and supposes that the Helmholtz free-energy, A, is everywhere an analytic function of volume, V, and temperature, T. It is now clear that this description is inadequate, that Cv is infinite at the critical point and that A is a non-analytic function of V and T. The classical description must, therefore, be abandoned and it is shown that we have, in its place, a set of inequalities between the degrees of the singularities in such thermodynamic functions as Cv, compressibility, coefficient of thermal expansion along the orthobaric curve, and the curvature of the vapour pressure line. These inequalities provide powerful tests of even the best modern measurements and enable us to give a fairly complete thermodynamic description of the singularities of the critical region. The transport properties are not so well understood. It is probable that the shear viscosity is non-singular whilst the thermal conductivity is known to be highly singular. This last singularity is probably related to the infinity in Cv. The dimensionless functions—Rayleigh number, Prandtl number, and Grashof number—all become infinite at the critical point.
A canonical form for an analytic function of several variables at a critical point
