796 Mr. R. H. Fowler on tie 



point in the gas which is still (practically) at an infinite 

 distance from the boundary z = 0. In the same way %(0) 

 means the work done by the attractions when the molecule 

 is brought to the boundary itself. Whatever the infinity 

 conditions, the work W done against the attractions when a 

 molecule is brought to the boundary must be given by 



W=x(~)- X (0)= x (0), 

 and we have from (5), 



v % Jo 



From (19) we can calculate a without much trouble for 

 any reasonable law of force. 



§ (9). The correct value of van der Waals' b. — The correct 

 value of the second virial coefficient is of course given by 

 van der Waals' or Dieterici's equations of state in the form 



NkTb-a, 



provided correct values of a and b are used. The correct 

 value of a is given by (19). It is usually assumed that 

 & = §7rNo- 3 , but when intermolecular attractions exist, this is 

 no longer correct, but we must have 



5 = l7rNo- 3 / in{<r) , ..... (20) 



a result which is often overlooked. Consider, for example, 

 the proof of the usual formula for b given by Jeans *. He 

 first obtains expressions for the excluded regions of the usual 

 generalized space, and allows for them on the usual assump- 

 tion that the a priori probability for the presence of a 

 molecule in any element of generalized space is the same. 

 To allow, for example, for the excluded region, 



on this assumption of the equal probability of all positions of 

 the 6-molecule, we must exclude a fraction J7rcr 3 /fl of the 

 total generalized space I2 N . But when there are central 

 forces acting between the molecules we are no longer at 

 liberty to assume that this a priori equal probability for 

 elements of generalized space implies equal probabilities for 

 equal volume elements in ordinary space. The correcting 

 term must be multiplied by the probability of another 

 molecule being at a distance <r from the selected molecule — 



* Loc. cits J). 157. 



