OSMOTIC AND MEMBRANE EQUILIBRIA 185 



The same form, (11) [77], for the conditions of membrane equi- 

 Hbrium is thus obtained whether entropy or temperature be 

 chosen as one of the independent variables. In fact, whatever 

 choice one makes of independent variables an analogous treat- 

 ment will lead to the same result, (11) [77]. 



3. Choice of Independent Comyonents. If the various quantities 

 mh, rrii, ... are not independently variable but are subject to cer- 

 tain restrictions expressible in the form of linear relations between 

 dnih, dnii, . . ., then (10) holds not for any values of dnih, dtUi, 

 . . . but only for such sets of values of dnih, dnii, ... as conform 

 with the linear restrictions. Instead of the conditions (11) [77] 

 one then obtains a smaller number of independent conditions 

 of the type [78] (Gibbs, I, 83) . The physical meaning of this is 

 quite simple. The condition for membrane equilibrium is equality 

 of the potential for those components to which the membrane is 

 permeable, provided the species chosen as independent compo- 

 nents include all those which are able to pass the membrane inde- 

 pendently. An example will make this clear. Suppose the mem- 

 brane is permeable to methyl alcohol CH4O but not to water H2O. 

 Then the corresponding condition of membrane equilibrium is 



MCH4O = MCH.O- (12) 



But from a purely thermodjoiamic standpoint it would be 

 allowable to choose as independent components methylene 

 CH2 and water H2O, since these will serve just as well as methyl 

 alcohol CH4O and water H2O to define the composition of each 

 phase. With this choice of components both methylene and 

 water are able to pass through the membrane, not independ- 

 ently but only in the proportions in which they form methyl 

 alcohol. Formula (10) in this case is 



(mch, - MCH,) dm'cR, + (mhjO - MH20) drn'^^o = 0. (13) 

 But diucKj and dm^^o ^^^ subject to the restriction 



p q ' 



