286 Mr' Vegard, On some general Properties 



components will pass through, forming above the membrane a 

 solution, which we may call the secondary solution of the osmotic 

 system. By diminishing the pressure, the current through the 

 membrane will be less, until for a certain pressure an equilibrium 

 will set in. There will be (i) equations determining the equili- 

 brium state ; from these the (*' — 1) concentrations of the secondary 

 solution, as well as the osmotic pressure, will be determined. 

 When the osmotic pressures are derived in this way we shall say 

 that they are formed upon a given solution. 



The number of osmotic systems that can be formed upon a 

 solution is 



^^i\{r-i+l)\ ' 



For a binary solution iV = 2. 



For a solution containing three components. . . N =Q. 

 For a solution containing four components ... iV= 14. 

 For a solution containing five components ... iV^ = 30. 



The most important of these systems are those we get for ^ = 1 

 and i = r. 



§ 7. The partial osmotic pressure. 



When the membrane is permeable to all but one component, 

 we get an osmotic pressure which is analogous to the partial 

 pressure in a mixture of gases ; we may call it the partial osmotic 

 pressure for the component considered. 



Let the membrane be impermeable for the component (r), and 



let Ci' = — T, c/= — ^, ... Cr-i= — ^-r- be the concentrations at the 

 nio vIq mo 



secondary solution. The equilibrium conditions are expressed by 



the equations 



(28) /,=//', i=0,l,2...(r-l). 



fi = Fi {Ci, Ci ... Cr, P) = thermodynamic potential per unit mass 



of component (i) in primary solution. 



fi'=Fi{ci,Ci...c'r-i,0,pr)\= thermodynamic potential per unit mass 



of component (i) in secondary solution. 



P the pressure on the primary, p,. that on the secondary solution. 

 If Ci, Ca ... Cr, P are supposed to be given, the r equations (28) de- 

 termine the r unknown Ci,C2 ... c'r-i,Pr- What interests us is 

 the osmotic pressure Qr = {P —pr)- 



In order to get a nearly exact expression for this pressure we 

 shall assume that in the state of equilibrium the differences 

 Ci — Ci are small quantities; or, that the substances for which the 



