the Osmotic Theory of Solutions. 611 



pistons a and a\ and containing solutions of the same con- 

 centration and pressure as at the corresponding levels in the 

 column) be fixed behind q and q f as shown in the figure. 



In the thermodynamic cycJe now to be considered, the final 

 pressures and concentrations are the same as the initial 

 ones, and all operations are performed isothermally; hence, 

 although the changes of pressure cause changes in the 

 volume of the contents of the cylinders and thus give rise to 

 terms in the expression for the work done, these, in so far as 

 they relate to definite masses of solution of definite concen- 

 tration, must eliminate. Accordingly these terms will not be 

 taken into account when writing down the work done in the 

 cycle. 



(I.) Let the piston a, under a pressure which never differs 

 more than infinitesimally from p, be pushed in until a mass 

 dm of solvent has passed through q into the column, and 

 simultaneously let the piston a' be withdrawn, so that at each 

 instant the mass of solvent which has left the column by q f is 

 precisely equal to that which lias entered by q. Moreover, 

 suppose the operation to be performed so slowly that the 

 column of solution is everywhere sensibly in a state of 

 equilibrium ; the solvent thus streams slowly from q to q' 

 without affecting the concentration at any level, and without 

 producing, on the whole, any perceptible displacement of the 

 solute. 



Adding up the work done by the external forces, we have 

 for the two pistons 



ps x dm-(p + '•'- Sh) [ «, + ( I 1 ' f/ + | 5j •" ->/» ] dm 

 V '" / I \O e l "" OP ioJ J 



or 

 01- 



( ton dc, B-'i .'A g~\ S7 , 



= \- p [ T 2 dh + ^w)- s >wr hdm ' 



and the work done by our ideal field of force is — Fi-8/u/m, 



since — Fjq/ifl is the force acting per unit mass of solvent. 



(II.) Close the membranes q and q f by shutters, and 

 change the pressure on the solution in V so that it would be 

 in osmotic equilibrium with that in b, the pressure in V has 



fiT* rf « 



now become p + — - 1 ~ ^ l -> but the work thus done may be 



OC 2 (III 



disregarded as above explained. 



(III.) Withdraw the cylinders (carrying the membranes 

 with them) * from the column and place the solution in V in 



* The use of shutters, where necessary, is to be understood. 

 2 T 2 



