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 II, then 

 for P -f d P the osmotic pressure will be II + d II. We may moreover 

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

 by v — d v at pressure P + dP; the molecular volume in the solution 

 by v' at osmotic pressure II, by v' — d v' at II + d II. If a gram-mole- 

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

 pressure II, (2) its solution concentrated to II -f d II, (3) removed from 

 solution against the pressure P + d P and (4) allowed to expand from 

 P + d P to P, an 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 W x , W 2 , etc. 



W x = 1TV - Pv, 

 Wz = -Ildv>, 



w 3 = (P + d P) (v - dv) - (n + d n) 0' - dv<), 



W i = Pdv. 



Writing the sum equal to zero, 



vdP— v'dU = 0, 

 or expressing in the equation the constancy of T, 



(3n\ v_ 

 \dPJ T ~ v<' 



(12) 



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, 



n 



v' 



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



From equation (8), t/r = p II. Therefore In \p = In II + In p, and 



\JP~)*~~ \9P Jt 



