THE DISTRIBUTION OF MOLECULES 91 



1. = _ 00, liquation (40) is inapplicable, but by the method used 

 in the derivation of Equation (36) we see that all the molecules will be 

 oriented as in Case I and the surface energy is thus equal to li as given 

 in Equation (30). Practically speaking we have this case, if a is negative 

 and — a is large compared to unity, which occurs if — ^'^yo is large com- 

 pared to kT. 



2. a = + '^. This case is approached if yo is positive and Sbyo is 

 large compared to kT. The molecules are nearly all oriented as in Case II 

 so that the energy is A2 as given by (31) or 



X, = Xi + Xo (41) 



3. = 0. This corresponds to an entire absence of orienting force. 

 The molecules are thus arranged wholly at random. Equation (40) gives 



X, = ?., +5/'rYo (42) 



Substituting in this the value of A,i by (30) and the value of yo from (33) 

 we find 



X« = S{abYab + adyad + bcybc + cdycd) (43) 



Equation (13) was derived as an expression for the energy of a 

 molecule AC surrounded by molecules AC and BD without orientation 

 or segregation. For the case in hand, where AC is surrounded only by 

 molecules BD, we place a = o, /5 = i and find that Equation (13) then 

 reduces to Equation (43) as of course it should. 



4. a small compared to. ■unity. This is the case when Sbyo is small 

 compared to kT. Equation (40) then applies. Let us consider how the 

 energy X then dififers from 1r the energy corresponding to random orienta- 

 tion. From (40) and (42) we get 



l-l,- Sabc.y, = ?^^^ (44) 



This equation enables us to estimate what error we have made in the early 

 part of this paper by assuming a random distribution of molecules in 

 liquids, as for example, in the derivation of Equations (11) to (17). 



As a specific example, let us apply Equation (44) to calculate the effect 

 of orientation of ethanol molecules in various mixtures of ethanol with 

 hexane. We have already found that the inter facial energy y(R-OH) is 

 34 ergs per cm.^ The surface Sa of ethanol molecules may be taken to be 

 83.5A- while that of hexane molecules is 136A-. The surface SAa of the 

 hydroxyl group in the ethanol molecule is 30A-. With these data we 

 find from Equation (17) (p = 4.39 ergs per cm. for ethanol-hexane 

 mixtures and from this, by Equations (25) we can calculate the partial 



