If the pressure, volume and temperature (absolute) of a fluid are expressed as fractions of the critical values, giving the so-called reduced pressure, π = p/pc, volume, φ = v/vc and temperature, θ = T/Tc, then fluids having the same reduced pressure, volume and temperature are said to be in corresponding states. If the equation of state of the fluid is based upon two characteristic properties of it (apart from the mass), then there is a relation of the form pr = f (vr, Tr) and it is sufficient to say that if any two independent reduced quantities are the same, the fluids are in corresponding states.
Other reductions are possible. In terms of intermolecular force parameters, p* = p(σ3/ε), v* = v/(Nσ3), T* = kT/ε, and if p* = f(v*,T*), then equality in v*, T* implies corresponding states [Hirschfelder, Curtiss & Bird (1964)]. For quantum fluids, p* = f(v*, T* , Λ*), where Λ* = h/σ while for polar fluids, ψ* = f(v*, T*, μ*), where ψ* is any property at μ* = μ/ = reduced dipole moment. In these equations N is Avogadro's Number, σ is the collision diameter, e is the well depth in the intermolecular potential function and μ is the dipole moment. Close to the critical point, "scaling" [see, e.g., Bejan (1988)] can be used. Extensive charts and tables for several thermodynamic properties in terms of pr Vr and Zc are available. For Compressibility Factor, see the separate section on that property.
Hirschfelder, J. O., Curtiss, C. F, and Bird, R. B. (1954) Molecular Theory of Gases and Liquids, Wiley, NY. DOI: 10.1016/0016-0032(55)91080-2
Bejan, A. (1988) Advanced Engineering Thermodynamics, Wiley Interscience, NY.