THE DISTRIBUTION OF MOLECULES 89 



These two orientations correspond to the extreme vakies of the surface 

 energy ; when A and B are partly in contact with each other the energy X 

 has intermediate values. The difference 1.2 — ^i which we shall refer to 

 as Ao is thus the greatest surface energy which can be effective in causing 

 orientation. 



Xo = 6"&(Ya<I + Y6c — Yab — Ycd) (32) 



For convenience let us place 



Yo = yai + Y6C — Ya6 — Yc'J (33) 



so that 



Xo = Shyo (34) 



The unique occurrence of h in these expressions is due to the fact that h 

 is the smallest of the surface fractions a, h, c and d because of the con- 

 vention of nomenclature that we have adopted. 



We may represent any given orientation of a molecule AC in Fig. 4 

 by a given position of the vector OA with respect to the vector OB. If 

 AC passes through all possible orientations we may imagine OA as passing 

 over all points on a spherical surface. We may represent all orientations 

 within any given solid angle by the ratio of the spherical surface described 

 by the vector OA, to the total spherical surface, this ratio being called a 

 surface fraction. 



The difficulties in the complete solution of the 3-dimensional problem 

 of orientation are considerable and we may therefore content ourselves 

 with treating it as if it were a 2-dimensional problem. We shall assume 

 therefore that the molecule AC in Fig. 4 has C3'lindrical symmetry and can 

 rotate only about its axis. 



There are three cases to consider. 



I. The molecule is so oriented that the whole of the B-surface is in 

 contact with A. Surface fraction {a — h). 



II. The whole of the B-surface is in contact with C. Surface fraction 

 {c-h). 



III. Part of B is in contact with A while the remainder is in contact 

 with B. Surface fraction 2&. 



We may let P be the probability per unit surface fraction that the 

 molecule may be oriented as in Case I. The total probability that a molecule 

 shall be so oriented is thus P(a — b). 



