392-394] Degree of Aggregation 327 



where the integrations extend over all configurations in which the inter- 

 molecular forces are appreciable. Now if, when the configuration of two 

 molecules is selected at random from all possible configurations, W aa is as 

 likely to be positive as negative, then the whole of the first integral can be 

 expressed as a sum of terms of the form 



W p-2hW aa _ W 



' * GLO.^ ' " 



this term being obtained by combining two configurations in which the 

 values of W aa are equal in magnitude but opposite in sign. This expression 

 is, however, negative for all values of W aa . The second integral can be 

 similarly treated, so that we arrive at the final result that <J> is negative. 



If now r} 1} r) z . .. are any of the 's which determine orientation (or any 

 other coordinates which enter into the intermolecular potential, but not into 

 the internal energy of a single molecule), then the first integral in expression 

 (769) can be written in the form 



and this again can be written as 



A* 



where M^ is an average intermolecular potential for pairs of molecules of 

 which all the coordinates except 77!, tj 2 ... are specified, and is negative. 



Degree of Aggregation. 



394. Consistently with what has been said, we may now simplify the 

 problem by regarding molecules as point-centres of force, acting on one 

 another with a force depending only on their distance apart. The chance 

 of finding a free molecule of class A inside an element of volume dxdydz is 

 now, by equation (768) 



Ae- hmc *dudvdwdxdydz ........................ (770), 



while the chance of finding two molecules of classes A and B in adjacent 

 elements dxdydz and dx'dy'd^ is 



A*e~ hm ^+ c '^ -^dudvdwdxdydzdu'dv'dw'dx'dy'dz'. 



If we take the element dx' dy ' dz to be a spherical shell of radii r and 

 r + dr surrounding the centre of the first molecule, this last expression 

 becomes 



A 2 e - hm (c+ c' 2 ) - zh* fin dv dw du' dv'dw' 4?r r 2 dr dx dydz, 



^ being a function of r. If, as in 30, we use the transformations 

 u = ^ (u + u') etc., a = u' - u, etc., 



