330 HEAT. 



partition permeable to the gas only, and a solid piston against which 

 any desired pressure can be exerted. Let us begin with A at the 

 bottom and C close against B, and let the pressure of the gas 

 necessary to keep the solution at its actual strength be P ; let the 

 osmotic pressure be p. Then A and C are in equilibrium if the pressures 

 on them are p and P. Let V be the volume of the liquid, and let the 

 dissolved gas have volume nV at pressure P. Now move the piston A 

 upward through the solution, pure liquid being left behind it, and at 

 the same time allow the gas to pass through B, and C to move forward 

 so that the solution is always of the same strength between A and B. 

 Then C must move n times as far as A, and ultimately when A has 

 reached B the volume of gas between B and C is nV. The work done 

 is ^;Y PnV. Now, allow the gas between B and C to 

 expand isothermally to some new volume riV + V and 

 pressure P'. Let the osmotic pressure, of the gas in 

 6 solution under pressure P' =p' . Now let the pressure on 

 A be made equal to p, and let be moved down under 

 pressure P' while A is moved down under pressure p. C 

 moves through riV while A moves back through V. 

 Then we have nV volumes of gas dissolved under pres- 

 sure P' and V volumes still outside. If we now force 

 the piston C down till all the gas is dissolved, at every 

 pressure n volumes at that pressure will be in the liquid, 

 so that the gas compresses into the liquid just as if we 



vapour 



solvent 

 Pi 



solution 



started with wV + V volumes at P, and ended with V 

 FIG. 188. at P. Hence the work done on the gas in this last com- 



pression is just that done by the gas in the expansion from 

 nV to riV + V. We have now gone through a one-temperature reversible 

 cycle, and have total work zero, or 



pV - PwV - p'V - P'riV + equal and opposite 

 works in change from nV to nV + V = 0. 



or p nP = constant. 



But evidently if P = 0, p = 0, so that the constant = 

 and p = riP. 



No method has yet been found of extending this proof to dilute 

 solutions of solids and liquids, and in the absence of anything correspond- 

 ing to Henry's law we cannot expect such extension. But we might 

 perhaps expect the result to hold, inasmuch as in dilute solutions of 

 solids and liquids the molecular distances of the solutes are of the order 

 of gaseous molecular distances, even if we did not know it from the 

 direct investigation of osmotic pressures by Pfeffer. 



It now only remains to show that Raoult's value of the lowering of 

 the vapour pressure in solutions follows from Van t'Hoff's law of the gas 

 value of osmotic pressure. 



Let us suppose that we have a solvent of density />, vapour pressure 

 a), and density of vapour <r. Let it form a solution in which the osmotic 

 pressure is P while the vapour pressure is a/ and its density cr'. Let the 



