the Osmotic Theory of Solutions. 



615 



1st operation. — Keeping the pressure p constant, slightly 

 withdraw the solution piston until a mass dm o£ solvent has 

 passed from solvent to solution. Let this be done so slowly 

 that there is all the time osmotic equilibrium. 



2nd operation. — Increase pressure p top' and simultaneously 

 increase the pressure on the solvent so as to maintain the 

 osmotic equilibrium without any solvent passing across the 

 membrane. 



3rd operation. — Keeping the pressure p ! constant, push 

 the solvent piston in very slowly until a mass dm o£ solvent 

 has been driven into the solution. 



4:th operation. — Decrease the pressure p' to />, and simul- 

 taneously decrease the pressure on the solvent in such a 

 manner as to maintain the osmotic equilibrium. 



The initial state has now been restored, and the work done 

 by external forces on the system must vanish for the whole 

 cycle. 



In operation (I.) the work done on the solution is psdm. 

 The work done on the solvent is due to two causes : — 



(a) The concentration of the solution has increased by 



dc = ^dm, 

 and the osmotic pressure by 



which expresses the decrease of pressure in the mass M of 

 solvent, hence the work done is 



M dqo oc 



(b) The mass of solvent has increased by dm, so that 



we get , _ s 



-(/>-P)Wm. 



In the 2nd operation the work done on the solution is 



while if P' represent the osmotic pressure corresponding to a 

 concentration c of the solution, and a pressure p' upon it 



