a 



Fig. 



3. 





c. 



! 



• 1 





in the Analogy between Solutions and Gases. 85 



constant, the proportionality between osmotic pressure and 

 absolute temperature, the concentration being maintained 

 constant, is not so manifest. Yet proof can be furnished from 

 thermodynamical considerations ; and experimental data exist 

 winch are highly favourable to the results predicted on 

 thermodynamical grounds. 



Theoretical Proof. — It has been already mentioned that, by 

 means of a piston and a cylinder with semipermeable walls, 

 reversible processes can be conceived to occur. If such pro- 

 cesses are expressed in the way common as regards gases, 

 volume and pressures are indicated on the lines V and P 

 (fig. 3); but pressure in this case, 

 as before, must be taken as osmotic 

 pressure. The initial volume (V cub. 

 metres) is represented by OA ; the 

 initial pressure on the piston of area 

 1 square metre (P kilogr.) by Aa; 

 and the absolute temperature by T ; 

 the solution is then conceived to un- 

 dergo a minute increase of volume, ° ad bc v 

 dV cubic metres ( = AB), by moving the piston through dV 

 metres, while the temperature of the solution is maintained 

 constant by introduction of the requisite amount of heat. This 

 amount of heat can be at once determined, inasmuch as it is 

 equivalent to the external work performed, FdV, by the 

 motion of the piston. Internal work is absent, for the dilu- 

 tion is, by hypothesis, so great that the dissolved molecules 

 exercise no attraction on each other This isothermal change 

 ab is succeeded by the isentropic or adiabatic change bc, 

 during which heat is neither absorbed nor evolved ; the tem- 

 perature sinks by dT° ; and the original condition is then 

 Drought about by a second isothermal and a second adiabatic 

 change, cd and da, respectively. The second law of therrno- 



dynamics requires that the fraction, -7^ PrfV of the initially 



imparted heat charge TdY shall have been converted into 

 work ; this must be equivalent to the area abed ; and hence 



the equation follows: 7=- TdV = abcd=af. AB = afdV ; and 



dT 

 hence P ~r = af. But af represents the change of osmotic 



pressure, volume being kept constant, due to the change of 



temperature cTT ; ?*. e. ( -jm ) dT ; hence 



W/ T 



