22 PROCEEDINGS OF THE AMERICAN ACADEMY. 



stant volume the same in the liquid and its vapor, the equation assumes 

 the simpler form 



RT\n^^ =PV- U- HT, (34) 



where Vy and v<, represent the molecular volumes in the liquid and gase- 

 ous states respectively. From the previous section we know that the 

 term H will enter simply as a volume correction. The exact manner in 

 which it so enters may be best shown by finding the free energy of the 

 process of liquefaction from the work that might be done if the vapor 

 were compressed isothermally and continuously until it reached the liquid 

 condition. This work would be equal to J^j) dv^ and ■p can be found in 

 terms of v from the equation of condition which holds good throughout 

 the process. From equation (27) 



RT rfm 



■f{v) dv 



Therefore 



= / pdv= I — dv- I dm 



R T 



The integration of -rj-z dv is only possible when the form of f{v\ 



v —f{v) ^ ^ '' ^ ' 



is known ; but since y(y) does not change greatly, and since it is only an 



important part of the expression when v is small, it may be regarded as 



a constant, and equal to the value oi f(v) in the liquid state, or b^. This 



value may be found from equation (33). Then the above equation 



becomes 



■Vi — Oi 

 and for equilibrium from equation (11), A = P V. 



R Tin "^ ~ f^ = PV- U. (35) 



^1 — bi 



Since bi is but small compared with v^, we may replace v„ — bi by Vo. 

 Since U represents the change of internal energy in vaporization, and 

 P Fthe external work, — U + P V will be equal to the ordinary heat of 

 vaporization per gram-molecule, including the external work. This whole 

 quantity may be designated by L. Then 



RT\u—^=L. (36) 



Vi — Ol 



