394-397] Liquid and Gaseous States 329 



Similarly, if i> 2 is the molecular density of double molecules, we obtain by 

 integration of expression (771), 



= A * A/BIT-, 1 1 e-W^+w 167r 2 FFc2r, 

 V 8h*m*JJ 



in which the integration extends over all values of V and r for which 

 + 2^ is negative. 



The total number of constituent molecules per unit volume (v) is 



so that if we denote the fraction of the whole mass which is free by q, 

 we have 



v l 1 



L + viJj e 



Continuity of Liquid and Gaseous States. 

 397. At very high temperatures h is very small, so that A, or 



/h 3 m? 



"l A / 



V 7T 3 ' 



is very small. Also e -M4F2+2*] jg very ne arly-. equal to unity. Thus at 

 these temperatures the series (772) reduces to its first term, and q = 1. 

 There are therefore no molecules in permanent combination, as indeed 

 we should expect. 



At lower temperatures h is greater, so that not only is A greater, but the 

 exponential e - h ^ mV ' 2 +^\ in which it will be remembered that the index 

 is always positive, is also greater. The relative importance of the later 

 terms of the series (772) is therefore greater. Ultimately we reach low 

 values of the temperature for which h has so great a value that the series 

 (772) becomes divergent. At this point the molecules tend, according to our 

 analysis, to form into clusters, each containing an infinitely great number 

 of molecules. By the time this point is reached, however, the analysis 

 ceases to apply, as the assumption that the molecular clusters are small, 

 made in 83, is now invalidated. It is, however, easy to give a physical 

 interpretation of the point now reached : obviously it is the point at which 

 liquefaction begins, and the collection of molecular clusters is a " saturated 

 vapour." 



