330 Aggregation and Dissociation [CH. xvm 



398. Regarded as a series in terms of A, the series (772) is a power 

 series, and so has a single radius of .convergence. Thus for a given value 

 of h, say h , there is a single value of A, say A , such that the series is 

 convergent for all values of A less than A and is divergent for all values 

 of A greater than A . In other words, corresponding to a given temperature, 

 there is a definite density at which the substance liquefies. This of course 

 is the vapour-density corresponding to this temperature. Clearly as h 

 increases, A decreases, and conversely, so that an increase of pressure is 

 accompanied by a rise in the boiling point of the substance. 



399. For very small values of h, as we have seen, v and i/j become almost 



/ h?7Y$ 



identical, so that we can replace A by v A/ r , and the series (772) becomes 

 very approximately, 



the exponential factor, which is very nearly equal to unity, being omitted. 



The integrals are now all independent of h, and we can suppose the 

 integration with respect to V in the first integral and similar variables in 

 the other integrals to be performed, so that we are left with a series of 

 integrals each extending over a sphere of molecular action integrals, then, 

 which depend solely on the structure of the molecules and which may 

 therefore be treated as constants. The series is now a power-series in 

 ascending powers of vh$, and therefore becomes divergent when vh$ exceeds 

 a certain value. 



We are considering only the limit in which h vanishes, so that for the 

 series to become divergent, v must, in the limit, be infinite. This is, however, 

 inconsistent with the supposed finite extension of the spheres of molecular 

 action. Thus when h is very small there is no possibility of the series which 

 we have been discussing becoming divergent. In other words, at very high 

 temperatures, a substance will not liquefy no matter how great the pressure. 



The Critical Point. 



400. We have already found that corresponding to every value of h 

 there is a definite value of A at which a substance liquefies. This leads 

 to a functional relation between v and T, say 



f(v,T) = .............................. (773), 



expressing the relation between v and T, at the boiling point of a liquid. 

 We have now seen that corresponding to T = oo , there is only a single root 

 of equation (773), namely v=cc. This value of v is, however, prohibited 

 by the molecular structure of the gas, from exceeding a certain critical 



