THERMODYNAMIC AL SYSTEM OF GIBBS 175 



thus represent the Hmits of stabihty with regard to continuous 

 changes. K the temperature is varied in the direction of the 

 critical point, the phases P and Q approach each other and at 

 the critical temperature become identical. If CD is the f-curve 

 at the critical temperature t", the point T representing the 

 critical phase, where the points P, Q, R, S, all coalesce, is a 

 point of undulation at which 



i(P^/dx-')p. t = and {d'^/dx')p. t = 0. 



Finally, at a temperature t'" beyond the critical point, the 

 f-curve is concave ever5nvhere. Now (d'^^/dx^) t, p is positive for 

 all homogeneous phases, which are stable with regard to both 

 continuous and discontinuous changes. 



It is evident that by a shght variation of the critical phase we 

 may obtain either (1), a phase which is unstable with regard 

 to both continuous or discontinuous changes, or (2), a phase 

 which is stable with regard to continuous changes but unstable 

 with regard to discontinuous changes, or (3), a phase which is 

 stable with regard to both continuous and discontinuous 

 changes. 



XIII. Equilibrium of Two Components in Two Phases 



39. The Equilibrium. We can now consider in more detail 

 the relation between temperature, pressure and composition in 

 systems of two components. Si and S2, in two phases. Let 

 the quantities referring to the first phase be distinguished by 

 single accents, and those referring to the second phase by double 

 accents. Then, for any change of state, while the phases remain 

 in equihbrium, we have 



v' dv = v' dt -\- mi dm + m^' c?^2,] 



(272) 

 v"dp = r}"dt + mi" dm + mi'dm-] 



If we consider quantities of the phases for which m^' = W/i' , 

 we have 



(v" - v')dv = (r;" - ■t\')dt + (mi" - miO^Mi. (273) 



