272 PROCEEDINGS OP THE AMERICAN ACADEMY. 



a binary mixture may be found by mean- of the apparatus Bhown in 

 Figure i. A contains the mixture of X, and X^. 1) is a pi 

 which determines the pressure on A. Mis a membrane pern 

 toX x . B contains a solution of X x in its ideal solvent Pisa piston 

 permeable only to the latter. Above P is the p ire -'Kent. 



'I be pressure on the piston P is the osmotic pressure, n, of the i 

 solution in B. In general if the pressure, P, on 1) is changed, the 

 equilibrium will be disturbed and the Bubstance X! will pass through 

 B, unless at the same time the pressure on P is changed by a suitable 

 amount. Let us tin-l* tlie mathematical expression for the change in 1 1, 

 which just compensates a given change in P. 



Starting with the piston P at B and with a large (better, an infinite) 

 amount of the mixture in A, occupying the volume V, let us perform 

 isothermally the following cycle of reversible operations, 



(1) Keeping the pressure P constant on the piston D, and keeping 

 the pressure on Falso constant and equal to the corresponding osm 

 pressure, II, raise F until one mol of Xi passes into B, where it occu- 

 pies the volume >■'. The diminution in the volume of A we will denote 

 by the symbol ,-. The work done by the system by means of the pistons 

 P and L) is, therefore, 



,|, = 11,'- /v. 



(2) Now increase the pressure on the piston D to /' + dP, and at 

 the same time increase the pressure on P to II + dJl, ,/Il being the in- 

 crement in II which is necessary to prevent X, from passing in either 

 direction through E. The volume of A will change from ['— v 



' V — dV) — (v — d-v), and the volume of the solution will change 

 from /•' to ,•' — ilr'. The work done by the system by means of the 

 pistons F and 1) i>. 



A, = - ii,/,'- P(dV-d»). 



I 3 1 Keeping the pressures on the two pistons constant and equal to 

 ii ) d\ i and P + dP respectively, lower F to E, forcing the mol ol X, 



back into A. The work done by the two pistons is 



,i ; =-(n + ,/i i m,-' -,/,•') + </' + ,//>)<, •-,/;>. 



( li Change the pressure in A back to P. The piston F is station* 

 try, and the work done by the piston D is, 



.1, = PdV. 



