268 The Earl of Berkeley : Notes 



pressures on solution and mixed vapours, we can close either 

 half of the membrane and apply proposition (a). 



It can be deduced easily from proposition (b) that one 

 concentration of mixed vapour? only can be in osmotic equi- 

 librium with the solution through the double membrane ; 

 in other words, given a solution of definite concentration, 

 the mixed vapour in osmotic equilibrium with it through a 

 membrane permeable to both components can have one 

 concentration and one only, and this concentration is inde- 

 pendent of the equilibrating pressures. 



Moreover, it is clear that mixed vapours in all possible 

 relative concentrations can also be in equilibrium with the 

 solution (and also with its mixed vapour) through a mem- 

 brane permeable to one component only, provided the correct 

 pressures be put upon them. 



We can now proceed to the general method. 



Consider compartments (1), (2), (4), and (5) of fig. 1. 

 Over a large range of pressures any two of these can be 

 in osmotic equilibrium through a membrane permeable to 

 substance A; we can thereiore, in general, perform an 

 isothermal reversible cycle between two compartments. 



Take compartments (1) and (2) as an example (in all 

 the cycles the two compartments under consideration are 

 supposed to be shuttered off from the remainder of the 

 system). Let the pressure in (1) be 7T a when in equilibrium 

 with (2) under pressure i/r, and let them also be in equi- 

 librium at some other two pressures ttJ and yfr' ; then, pass 

 one gramme of substance A through the membrane from 

 (1) to (2) under the original pressures, and then change 

 the pressures to 7rJ and -\jr' (in such a manner that none of A 

 passes through the membrane) ; then return the one gramme 

 from (2) to (1) and restore the original pressures. The 

 application of the laws of thermodynamics shows that \ pdv 

 taken round the cycle is zero ; hence it follows at once that 



^dp = \ ad V . ..... (1) 



Carrying out similar cycles for the other pairs of com- 

 partments, six equations will result, as there are four 

 compartments which can be taken two at a time. These six 

 equations, if appropriate values for the limits be imposed, 

 can be combined so as to connect any three or all four of the 

 compartments. 



In the same way we can get another set of six equations 



