REPRESENTATION BY ZETA FUNCTION 323 



under like conditions. Hence, 



nidni -\- UidfXi = 0, or x j- -\- [l — x) — = 0. 



Differentiation of f = Xfxi + (1 — x)n2 with respect to x (at 

 constant temperature and pressure) gives 



d^ dfjLi diJL2 



Tx= ''d^ + ^' -^ ^'^ - ""^ dx - ^' = ^' - ^" 



from the preceding result. Thus at any x-point of the f-curve, 

 we can determine both ni and ^2 by means of the two equations 



f = a^Mi + (1 — x) fjL2, 



dX 



^ = ^^ - '^^' 



whence we deduce the results 



Ml = fi = r + (1 - x) -, 

 ^^ = ^^ = f-^^' 



Consider now the state of affairs for x = (pure W). From 

 the preceding results we have 



(mi)x = o= (f)i = + 



\dz/x^i 



It is clear that (r)x = o is the f (free energy) of 1 mol of pure W. 

 Now fxi is the increase of free energy of an inj&nite phase of 

 composition x on the addition (at constant pressure and tem- 

 perature) of one mol of X, whilst (jui)x = o is the limiting value 

 to which Ml approaches as x approaches zero. 



Let pi denote the partial vapor pressure of X in equilibrium 

 with the liquid phase of composition x at the given pressure 

 and temperature, and let (pi)o denote the vapor pressure of X 

 in equilibrium with pure liquid X at the same temperature 



