322 SCHREINEMAKERS art. h 



V. Note by F. G. Donnan. (Analytical Addendum to the 



Geometry) 



It can be proved in the following manner that the f-curve 

 touches the lines WW and XX' at the points W and X' 

 respectively (see page 296 of Professor Schreinemakers' article). 

 Denoting by f„ the zeta function (free energy) for a liquid 

 phase containing ni mols of X and 712 mols of W, where 

 rii -{- rii = n, then it follows from Euler's theorem that 



/afA , /afn\ 



tn = ni[-—] + ^2 I r~ I , 



since f „ is a homogeneous function of the first degree in rii and 

 712. This expression may be written in the convenient form 

 tn = W]fi + 722^2, when f 1 and ^2 are termed the partial molar 

 free energies of X and W respectively. Since fi = ni, ^2 = M2, 

 we shall follow the notation of Gibbs and write f„ = n^ui + 

 n2iU2, where /xi and 1x2 are the 'potentials (per mol) of the com- 

 ponents A" and W respectively. For unit (molar) phase we 

 must divide by rii + n2, and write therefore 



— — — = f = a:/ii + (1 - x) 112, 



Hi ~X~ 102 



where 



X = ; ' 1 — a; = 



ni + W2 ni -j- 712 



This expresses the f of unit phase in terms of the composition 

 parameter x and the potentials. At constant temperature and 

 pressure jui and ju2 are functions of x only. 



Differentiating the expression f „ = 7i\ni -\- 7121x2 for a change 

 of rii and 712 at constant temperature and pressure (change of 

 composition), 



d^n = Uidni + /i]fZn] + 'n2C?yU2 + ii2d7i2. 

 But 



d^n — (JildTli + IJi2d7l2 



