332 Aggregation and Dissociation [CH. xvin 



402. The observed deviations from the laws obeyed by a perfect gas 

 must of course be attributed partly to aggregation, as has just been explained, 

 and partly to the causes which have already been discussed in Chapter VI. 

 The two sets of causes are not, however, independent; so that it is not 

 sufficient to consider the effects separately, and then add. The state of the 

 question is, perhaps, best regarded as follows. 



The effect of the forces of cohesion is too complex for an exact mathe- 

 matical treatment to be possible. We have therefore, in Chapter VI. and 

 the present chapter, examined their effect with the help of two separate 

 simplifying assumptions. In Chapter VI., following Van der Waals, we 

 regarded the gas as a single molecular cluster containing an infinite number 

 of molecules, and in replacing the whole system of the forces of cohesion by 

 a permanent average force, we virtually neglected the etfect of any formations 

 of small clusters inside the large cluster. In the present chapter, on the 

 other hand, we have been concerned solely with the formation of small 

 clusters, and have disregarded the large cluster altogether. As a conse- 

 quence of the omission of the former treatment to take account of the 

 formation of small clusters, this treatment led to the erroneous result 

 (equation (386)), that the internal pressure is exactly proportional to the 

 temperature, whereas as a consequence of the omission of the present treat- 

 ment to consider the clustering of the gas as a whole, we are led in the 

 present chapter to the erroneous conclusion that the internal pressure is 

 identical with the boundary pressure. The situation may then be summed 

 up by saying that the treatment of Chapter VI. considers only the tendency 

 to mass- clustering, while that of the present chapter considers only the 

 tendency to molecular- clustering. 



So long as the deviations from the behaviour of a perfect gas are small, 

 the two tendencies may be considered separately, and the total deviation 

 regarded as the sum of the deviations caused by the two tendencies separately. 

 On the other hand, as we approach the critical point the phenomena of mass- 

 clustering and molecular-clustering merge into one another and ultimately 

 become identical at the critical point. The two effects are no longer 

 additive, for each has become identical with the whole effect. 



By the side of these considerations must be placed the consideration that 

 we have only found an exact mathematical treatment of either effect to be 

 possible by making the assumption that the effect itself is small. In other 

 words, so far as our results apply, the effects are additive. It may be noticed 

 that the deviations from the laws of a perfect gas which were discussed in 



Chapter VI., fell off proportionally to and ~^ , whereas the deviations dis- 

 cussed in the present chapter fall off much more rapidly as the temperature 

 increases. 



