58 PROCEEDINGS OF THE AMERICAN ACADEMY. 



changing the pressure on the solvent and without preventing the sub- 

 stance from passing freely into or out of the solvent. The osmotic • 

 pressure of the saturated solution depends upon the pressure on the 

 substance. If the latter is represented by P and the former by 11, then 

 for P + d P the osmotic pressure will be 11 + </ 11. We may moreover 

 represent the molecular volume of the substance by v at pressure P, 

 by V — dv aX pressui'e P -\- d P ; the molecular volume in the solution 

 by v' at osmotic pressure H, hy v' — d v' Sit H ■\- d 11. If a gram-mole- 

 cule of the substance at pressure Pis (1) dissolved against the osmotic 

 pressure 11, (2) its solution concentrated to 11 -f c? TI, (3) removed from 

 solution against the pressure P -{■ d P and (4) allowed to expand from 

 P + d P to P, a,u isothermal cycle is formed, and if each step is made 

 reversible the total work of the cycle is zero. The work obtained in the 

 several steps may be represented by Wi, JV^, etc. 



Wi = Uv' - Pv, 



W^ = — Ild v>, 



W^ ^(P-\- dP) (v- dv) - (n-\- dU) (v' — dv'), 



W, = Pdv. 



Writing the sum equal to zero, 



vdP—v'dU = 0, 



or expressing in the equation the constancy of T, 



(l^),-^- <'^> 



This is an exact general equation connecting the osmotic pressure of a 

 saturated solution and the pressure upon the pure solute. It is entirely 

 analogous to equation (6). Since we may choose a solvent in which the 

 solute is as slightly soluble as desired we will choose one in which the 

 solution may be regarded as infinitely dilute. Then, 



PT 



n = 



v' 



from equation (2). Combining this equation with (12) we obtain 

 From equation (8), if/ = pU. Therefore In i/^ = In IT + In p, and 



\9P'j,~\ 9P ); 



