I 



[ 31 ] 

 II. On the Theory of Osmotic Equilibrium. 

 To the Editors of the Philosophical Magazine, 

 Gentlemen, — 



N the March number of the Philosophical Magazine 

 appears a paper by the Earl of Berkeley relating to 

 osmotic equilibrium in binary mixtures. The object of the 

 paper is to obtain relations connecting the conditions of the 

 various conceivable cases of equilibrium. The method 

 adopted in the paper is, however, fundamentally erroneous. 



The fundamental error is contained in the following 

 proposition * : — 



" It is possible to change the pressures on the solution and 

 its mixed vapours (separated from one another by a membrane 

 permeable to both components) in such a manner as to keep 

 osmotic equilibrium between them without any change in 

 concentration taking place." 



This proposition is easily seen to be false. The change 

 contemplated involves the constancy of three variables 

 (temperature and two concentrations) and the variation of 

 two (the two pressures). Now these five variables are con- 

 nected by two relations (the conditions of osmotic equilibrium 

 of the two components) so that only three are independent. 

 Hence the change contemplated is impossible. AVe may 

 prove the same thing in a slightly different manner. In the 

 absence of the osmotic membrane the system would, by the 

 Phase Rule, be bivariant. The membrane merely increases 

 by one the number of degrees of freedom, by removing the 

 condition that the pressure must be the same in both phases. 

 The system is therefore tri variant. Hence etc. 



It is instructive to examine more closely the fallacy of 

 this proposition. Suppose a mixture of two liquids A andB 

 to be in equilibrium with the mixed vapours through a 

 membrane permeable to A only, under conditions of pressure 

 and concentration, such that they would also be in equili- 

 brium, if placed in communication through a membrane 

 permeable to B. Letp and yfr be the pressures of the liquid 

 and vapour respectively. Let U3 increase p to p + Sp and 

 yjr to -v/r + frvjr, adjusting the increments so that there is no 

 disturbance of the equilibrium. Let s a and cr a denote the 

 " apparent specific volumes " of A in the liquid and vapour 

 respectively. Now the increment of pressure 8p will increase 

 the chemical potential of A in the liquid by an amount s a Sp, 



* P. 267. 



