The problem is characterized by so-called critical exponents. For example, the vapour or liquid density of a pure fluid along the vapour pressure curve can be described by |ρ - ρc| = B |T - Tc|^β in the vicinity of the critical point. It can be shown that any analytical equation of state yields β = 0.5 . The experimental value, however is about 0.33 . The departure is explained by long-range density fluctuations.

There are basically two ways to deal with this problem in EOS: (a) using an analytical EOS with so many terms that non-classical critical exponents are obtained in the experimentally accessible range (and classical ones where there are no experimental data); (b) introduce some nonanalytical terms.

Most of the latter attempts require that the location of the critical point is known exactly – a condition not fulfilled for real fluids. Moreover, most of these attempts cannot be generalized to mixtures, where there are critical curves (for binary mixtures), and sometimes more than one.

Thank you Ulrich Deiters . The need to introduce critical exponents (were they first introduced by Van Der Waals?) could be also also stated: "because the critical point non analytical behavior, the Helmholtz free energy could not be expanded in Taylor series of the density and temperature around the critical point". This not analytical behavior due to long-range density fluctuations corresponds for instance to the experimental evidence of critical opalescence (Einstein, 1910) or to the statistical mechanics 2D ising model (Onsanger, 1944) for continuous, 2nd order phase transition.