268 PHENOMENA, ATOMS, AND MOLECULES 



forces a molecule of palmitic acid vapor will assume a nearly spherical form 

 in which the carboxyl group is buried as completely as possible within the 

 sphere. Thermal agitation and in.ternal stresses in the molecule will tend to 

 prevent the molecule from being truly spherical, but with a molecule as 

 large as palmitic acid the effect of the surface energy should predominate 

 and make the molecule approximately spherical. Let us estimate the maxi- 

 mum and minimum surfaces which the molecule may possess. 



The minimum surface will correspond to the spherical form. The diam- 

 eter of a sphere having a volume * of 497 A^ is 9.83 A and its surface is 

 304 . A^. The maximum area occurs when the hydrocarbon chain is as nearly 

 straight as possible. The experiments 6n adsorbed films on water have 

 shown a minimum cross-section of 20 A^ for the hydrocarbon chain. If we 

 consider the molecule to be a cylinder having this cross-section and having 

 hemispherical ends, we find the diameter to be 5.04 A, the length of the 

 cylindrical part 21.5 A and the total surface 420. A^, which is 39 per cent 

 greater than the surface when the molecule is spherical. 



The difference in surface (ii6.A^) multipHed by the surface energy 

 of hydrocarbons (50 ergs per cm.^) gives the energy involved in the trans- 

 formation between the cylindrical and the spherical form. 



Let us then consider whether this energy of 58 X lO"^^ erg is sufficient 

 to cause the molecule to assume a nearly spherical form. 



The ratio P1/P2 of the probabilities of the occurrence of any particular 

 configuration of a molecule is given by the modified Boltzman equation 



Pl = te-if (5) 



P2 P2 



where A is the energy required to change the molecule from the state 2 to 

 state I, and px and p2 are the a priori probabilities of the two states, which 

 are largely determined by geometric factors and the definitions of the two 

 states. If we put 1 =-58 X 10"^*, T = 293, the exponent is — 14.4 and the 

 exponential factor is thus 6 X 10"''' so that nearly all the molecules must be 

 in a nearly spherical form unless pi/p2 has a very large value. If the surface 

 energy X were zero the ratio p\/p2 would measure the relative probabilities 

 of the two configurations. Of course the probability p2 would be zero if we 

 were to consider only absolutely spherical molecules. We should therefore 

 consider a range for each state, taking, for example, state 2 to be one in 

 which the surface area is not more than 5 per cent greater than that of a 

 perfect sphere, while state i is one in which the area is not less than 95 per 

 cent of that of the cylindrical molecule already considered. 



* For convenience we shall express molecular dimensions in terms of the Ang- 

 strom unit 10"® cm. ; thus 5 A^ = 5 X lO"^^ cm.^ and 4 A^ = 4 X 10"^'* cm.^ [A stands 

 for A.U. J. A] 



