78 PHENOMENA, ATOMS, AND MOLECULES 



while the fraction c is a C-surface. Similarly h and d are the fractions of 

 the surface of the B molecule which are occupied by B- and D-surfaces 

 respectively, so that 



a-\- c = 1 and b -\- d = i ( lo) 



Let us now consider a binary mixture in which the mol-fraction of 

 A-molecules is A and the mol-fraction of B-molecules is B. The right- 

 hand section of Fig. 3 represents an A-molecule in such a binary mixture. 

 We assume as before that there is neither orientation nor segregation of 

 the molecules which surround a given molecule. Then the relative proba- 

 bilities that any point on one molecule shall be in contact with an A-mole- 

 cule or with a B-molecule is proportional to the total surface areas of the 

 two kinds of molecules in the mixture. It is thus useful to express the 

 concentrations of the two components A and B in terms of their surface 

 fractions which may be defined by the equations 



where Sa and Sb are the surface areas of the A- and B-molecules respec- 

 tively. The fractions a and § as well as A and B are related : ^ 



A^B —1 and a -f (3 = 1 (12) 



The surface energy of the A-molecule in the liquid is now found by 

 adding together the energies of each of the five interfaces : AB, AC, AD, 

 BC, and CD. Consider for example the interface AD. The area of the A 

 surface of the A-molecule is oSa- Of this surface, the fraction /5 is in con- 

 tact with B-molecules and of this area of contact the fraction c? is a contact 

 with the D-surface. The corresponding surface energy is thus SAa^dyad- 

 For all five terms the surface energy Ai is 



li =S[2aacyac + p(fl&Ya& + odja.i + bc^bc + cdycd)] (13) 



Now let us imagine that the A-molecule is removed to a vapor phase, 

 leaving a cavity in the liquid. The surface energy 1.2 of the molecule in the 

 vapor phase is 



h = S(aya + cyo) (14) 



and the surface energy of the cavity is 



h = S,.,[a(aya + eye) + ^{byb + dya)] 



When the cavity is allowed to collapse the surface energy A3 disappears 

 and new interfaces having the energy l^ are produced. The value of X4 is 

 found by the summation of 12 terms. The surface of the cavity, before 



