Inequalities, Homogeneous Functions, and Scaling

With the failure of any analytic theory of the critical point, no unequivocal conclusions can be drawn about the precise values of the exponents a, jS, y, 8 for fluids and fluid mixtures, since no statistical-mechanical theory of fluids is 

However, some inequalities can be deduced from the general requirements of thermodynamic stability, of which the most important are 

No one has yet succeeded in proving that these inequalities must in fact be equalities, or that y+ = y , or even that the limiting exponent y for Cp.A.m oc I T — r must be the same as that for Kt,a / — p y. How- 

If one assumes that the appropriate thermodynamic function [e.g. Gm(T, p, x) for the binary mixture], or more accurately the singular part thereof, is a homogeneous function of the appropriate variables (e.g. T - 7 or p — and X — X ), one can show that inequalities (15) to (18) become equalities and that 

In the application to binary mixtures, one way of writing this scaled equation of state is 

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