Relation between the Compressibility and the Coexistence Curve near the Critical Point

It is known that if a fluid at its critical density is cooled toward its critical temperature, then its compressibility becomes infinite proportionally to (T—Tc)—f, where T and Tc are the temperature and critical temperature, and where the exponent f depends only on the dimensionality of the system. It is also known that along the coexistence curve in the temperature—volume plane, Tc—T becomes proportional to | V—Vc |d near the critical point, where V and Vc are the volume and the critical volume, and where the exponent d, like f, depends only on the dimensionality of the system. If it is conjectured that two features of classical (i.e., infinite-dimensional) systems are in fact independent of dimensionality, it then follows that f=2(1–1/d). The pair f=1, d=2 and the pair f=7/4, d=8 are well known for classical and for two-dimensional systems, respectively. For three-dimensional systems it has often been conjectured that f=5/4, from which it would follow that d=8/3, while it has also often been conjectured that d=3, from which it would follow that f=4/3.