142, 143] Deviations from Boyle's Law 125 



boundary of the gas. The effect of this field of force upon the gas as 

 a whole will be to alter the density of gas at points near the boundary. If 

 d is the distance from the boundary of a point near the boundary, we may 

 assume a gas density of the form p =f(d). If we calculate the potential 

 of the supposed permanent field of force for this law of density we find 

 a potential %(d) at distance d from the boundary, and we now have, by 79, 

 a relation of the form 



f(d) = Ce-W .............................. (295) 



from which to determine f(d). If the intermolecular forces are small, we 

 can calculate %(rf) upon the supposition that p is constant throughout the 

 gas, and we then have for the value of %(d) an expression of the form p^r(d} 

 where ty is independent of p. In this case, we can write the density at the 

 boundary in the form 



and the mean density of the gas, which may be taken to be f(<x> ), will be 

 given by 



The elimination of C from these two equations gives 



po = per *P+ .............................. (296), 



where ^ is written for ^(0) -fy* (GO ), a quantity which is independent of p 

 and depends only on the law of force between the molecules. 



In the more general case in which it is not legitimate to assume as a first 

 approximation that p is constant throughout the gas, we shall find it con- 

 venient to suppose that the value of p is still given by equation (296), but 

 we must in this case regard ^ as a function of Jboth p and h. 



The significance of the distinction between p and p , expressed by equation 

 (296), is as follows. We are supposing a density p at the boundary, and have 

 calculated v b , and therefore p, on the supposition that p is the density 

 throughout the gas. Owing to cohesion, this density is p, so that the 

 equation 



p = Rv b T ................................. (297) 



expresses p in the terms of the boundary density p , instead of the true 

 density p. 



Hence the relation between p and p must be obtained by the elimination 

 of p between the equation 



p = v b (p )RT ........................... (298), 



in which v b (p ) denotes the effective density at the boundary for a homogeneous 

 gas of density p , and the equation 



Po = pe-^ .............................. (299). 



