90 PHENOMENA, ATOMS, AND MOLECULES 



The probability that the molecule shall be oriented in Case II may be 

 found by the Boltzmann Equation (4) and is 



where (35) 



^~ kT ~~ kT 



Within the region corresponding to Case III the surface energy varies 

 linearly between two limits, so that we find the total probability that the 

 molecule shall come under Case III is 



r2b 



ax 



e ax 







Now since the molecule must be oriented in one of these 3 ways the sum 

 of these 3 probabilities is unity 



'2b 



l^P[^a-b)^{c-h)e--\- /Tb^^. (36) 







If we assume that ^0 is small enough, a will be small compared to unity so 

 that we may expand eo into a series taking only the terms involving the 

 (irst order in o. The foregoing equation thus becomes 



yvhich gives us the values of P. 



We may now determine the average surface energy obtained from all 

 possible orientations by weighting each orientation in proportion to its 

 probability. We thus find, taking only terms up to the first order in 



lr=zl,J^Pl[c-\-,{c-lh)] (38) 



Since b<,a<,c we may neglect Ysb in comparison with c. Substituting the 

 value of P from (37) we can reduce Equation (38) to 



X = X, + U(i + -'0 (39) 



and by (34) we obtain 



1 = 1, -{- She (\ -^ c>a)yo (40) 



This equation allows us to determine theoretically whether the mole- 

 cules in a given mixture are oriented to an appreciable extent, and if so, 

 we may then estimate the magnitude of the efifect produced. Let us con- 

 sider the following cases : — 



