THERMODYNAMICAL SYSTEM OF GIBBS 105 



Si only, it is necessary for equilibrium that f = t" and m' = 

 Hi", but not that ii2 = /X2", or that p' = p". The difference of 

 hydrostatic pressure on the two sides of the membrane which is 

 necessary to preserve equilibrium is the osmotic pressure of the 

 solution, and is that which is required to make the value of 

 potential of Si m the solution the same as its value in the 

 solvent. We shall calculate its value in simple cases in a later 

 section. 



V. Coexistent Phases 



13. The Phase Rule* The variation of the energy of a 

 homogeneous body, containing n independently variable com- 

 ponents, has been expressed by the equation : 



dt = tdr\ — pdv + indrtii + /X2c?m2 ... + HndiUn. (95) 



In this equation, there are altogether 2n + 5 variables, viz., 



mi, rrhj . . . w„, 



/Xi, /i2, ... Hn, 



and €, t, 77, p, V. 



These quantities are not all independent, for the n -\- 2 quanti- 

 ties, t, p, jjLi, M2, • • • Mn can be derived from the original equation by 

 differentiation. Thus, the equations 



\t) = ^' C/l = - V, 



y^V/v, Tni,...mn \^^/ V, nn,...mn 



i 





= Hi, etc. 



nil,., .mn 



give us n -(- 2 independent relations between the 2n -\~ 5 vari- 

 ables. The original equation (95) is an additional relation, so 

 that if € is known as a function of 77, v, rrii,. . .nin, there are 

 altogether n -f 3 known relations between the 2n -f 5 variables 

 and the remainder, n -(- 2 in number, are independent. 



The homogeneous body may thus undergo n + 2 independent 



Gibbs, I, 96-97. 



