'266 The Earl of Berkeley : Notes 



Assume that they are not, and let the defect be in the 

 direction that more of substance A passes from (2) to (1) so 

 that (1) becomes more dilute ; but (1) and (3) were origi- 

 nally in equilibrium, and to restore this some A must pass 

 through into (3). On the other hand, the mixture in (2)> 

 has been concentrated. Hence some of A will be required 

 to pass from (3) to (2), and perpetual motion results. Now, 

 in ail these operations definite changes in volume are taking 

 place (this follows from our definition of $ a and crj, so that 

 the system could be harnessed to do external work, and con- 

 sequently the motion is impossible. A similar result follows 

 if we assume the defect in equilibrium to be in the opposite- 

 direction, and that a flow of substance A takes place from 

 (l)to(2). 



Applying the equivalence theorem to the system in fig. 1,, 

 it is easy to see that it is in equilibrium. For, taking com- 

 partments (1), (4), and (5), the solution is supposed to be in 

 osmotic equilibrium with the pure liquid in (4), and the 

 latter with its vapour in (1), so that (1) and (5) are in equi- 

 librium *. But the pressure it can be adjusted so that the 

 mixed vapours are in equilibrium with the solution ; hence 

 (1) and (2) are in equilibrium, similarly for (5), (6), and (3), 

 and, therefore, for (5), (2), and (3). 



In the foregoing notation the two components have been 

 regarded as miscible in all proportions. This seems to be 

 the most general case for osmotic pressures, but special cases 

 can be dealt with under the same notation. 



Thus, suppose the two components are miscible only over 

 restricted ranges of concentrations, and that we are dealing 

 with a solution and its conjugate ; if so, we have the most 

 complicated sets of equilibria possible for binary mixtures. 

 The notation can be extended at once to cover such a case 

 by assigning dashes to the suffixes a and b to represent the 

 conjugate. It should be noted that a and /3 need not be 

 altered, for both solutions have the same vapour-pressures 

 (and osmotic pressures). Other special cases, such as solu- 

 tions of non-volatile components whether solid or liquid, are 

 obviously easily dealt with. 



* Incidentally, this affords a simple proof that tt^ — ^^ or 7r a0 — 7J" a7r . 

 Further, if we keep p & constant and change the concentration in (5), the 

 method will show that the change in p necessary to retain osmotic equi- 

 librium will always be such that the vapour-pressure of the solution, 

 remains constant and equal to that of the solvent. 



