States, and the Thermodynamic State-Equation. 151 



in the systems with respect to the relation of the given 

 quantities. 



This may be expressed by saying that the systems 

 "correspond proportionally" with respect to the given quan- 

 tities, corresponding states being defined by equal values of 

 certain fixed, specific ratios of the quantities. As was seen, 

 the mathematical condition for this proportional corre- 

 spondence is that the number of specific constants in the 

 equation connecting the quantities be not greater than the 

 number of independent dimensions. The geometrical ex- 

 pression is the superimposability of the logarithmic curves 

 and surfaces, as shown by Raveau for the (p, v, ^-state- 

 equation. 



It may at first sight be objected to this generalized idea 

 of corresponding states that it is purely theoretical, and 

 cannot be put to the test of experiment or to empirical uses. 

 This is, however, not true. The logarithmic test of Raveau, 

 or a test equivalent to it, can in all cases be applied with the 

 greatest simplicity, and with any desired accuracy. And 

 the manifold theorems which have been set up about different 

 functions at corresponding temperatures or pressures, in the 

 old theory, indicate the great advantage of being able to 

 define corresponding sets of values, in different systems, of 

 other physical quantities. Most of the applications are cases 

 of the general fact (pointed out by K. Onnes * for the 

 (j9, v, T)- state-equation) tliat any function of the given 

 quantities which is of zero dimension has the same value in 

 all the systems at states corresponding with respect to the 

 given quantities. 



It may safely be postulated that, for all phenomena con- 

 sisting in the interaction of energies in several material 

 systems, there exist and can be found factors and functions 

 of the energies involved which are connected by a common 

 or non-specific reduced equation — l. e., with respect to which 

 the systems correspond proportionally. Thus, considering 

 the relation of heat- and volume-energies in the systems 

 defined as ideal gases, there is no proportional correspond- 

 ence with respect to the quantities Energy, Entropy, and 

 Volume f ; but if the variables Pressure, Temperature, and 

 Volume be chosen, there is correspondence of states with 

 respect to these, as expressed in the reduced equation : 

 7T(f> = 6. It is a fundamental problem of physical chemistry 

 to find, for each set of phenomena, the quantities with respect 

 to which correspondence of states exists. So, in the case of 



* Loc. cit. 



t See below, p. 153. 



