CORRESPONDENCE OF MOLECULAR STATES 223 



pressure was greater. The pressure P in the gas 

 equation is therefore too small by an amount which 

 depends upon the separations of the molecules, being 

 inversely as the square of the volume. For P we 

 write the expression (F+a/F 2 ). Making these sub- 

 stitutions, we have the equation of Van der Waals 



(P+a/V*)(V-b)=RT (3) 



or multiplying out 



The values of a and 6 may be determined by sub- 

 stituting in this equation the known value of R and 

 experimentally determined values of P, V, and T. 



Equation (4) evidently states the physical relation 

 by which we should expect to calculate the volume V 

 of a molecular mass upon which the pressure is P and 

 of which the absolute temperature is T. It is a gen- 

 eral theorem of mathematics that there are three roots 

 for an equation in which the highest power of the 

 unknown magnitude is a cube. Since V enters as a 

 cube we expect to obtain three numerical values for 

 it. But can a molecular mass under these conditions 

 fill three different volumes? If we consider V in the 

 perfect gas equation to be the unknown there is only 

 one possible value which it may have for any assigned 

 values of P and T. According to equation (4), under 

 the same conditions of pressure and temperature there 

 appear to be three different volumes, any one of which 

 the molecules might occupy equally well, so far as 

 concerns the physical conditions represented by the 

 equation. 



