OSMOTIC AND MEMBRANE EQUILIBRIA 183 



of supporting an excess of pressure on either side and permeable 

 to some of the components Sh, Si, . . ., but impermeable to others 

 Sa, Sb, . . • , the conditions for equihbrium between the two 

 phases as regards the components Sk, *S., . . •, 



w' = w",l 



are of exactly the same form as (3) [21]. 



But the potentials of the components Sa, Sb, . . . , to which 

 the membrane is impermeable, will in general not be equal, 

 that is, 



Ha 7^ Ma",l 



,.'^,."} (5) [77] 



Moreover, in general the pressures of the two phases will not be 

 equal, that is, 



p' ^ V"- (6) [77] 



The pressure on each phase will be equal and opposite to the 

 pressure exerted by the phase on the membrane, and so the 

 resultant force per unit area on the membrane wiU be equal to 

 the difference between the pressures of the two phases. 



2. Proof of General Conditions of Membrane Equilibrium. 

 The derivation of the general conditions (4) [77] of membrane 

 equilibrium is given by Gibbs (I, 83). In this proof the 

 quantities chosen as independent variables are the entropy tj 

 of each phase, the volume v of each phase, and the quantities 

 Wi, W2, ... w„ of the various chemical species Si, Sz, ... Sn 

 in each phase. The corresponding characteristic function is 

 the energy c. The appropriate form for the general criterion 

 of the equilibrium is that expressed by [2] (Gibbs, I, 56) . 



In accordance with the footnote (Gibbs, I, 90) a somewhat 

 more familiar derivation of (4) [77] can be obtained by choosing 

 as independent variable the temperature t instead of the entropy 



