./. W. Gibhs — Eqidlibrium of Heterogeneous Substances. 435 



Again, if the surface is extended without application of heat, while 

 the pressure in the liquid and vapor remains constant, the tempera- 

 ture will evidently be maintained constant by condensation of the 

 vapor. If we denote by J/ the mass of vapor condensed per imit of 

 surface formed, and by rj^^ and i]^' the entropies of the liquid and 

 vapor per unit of mass, the condition of no addition of heat will 

 require that 



^('/m"-'/m') = %(,). (588) 



The increase of the volume of liquid will be 



r'(W-W)' 



and the diminution of the volume of vapor 



(589) 



/'(W-'/mT 



(590) 



Hence, for the work done (per unit of surface formed) by the exter- 

 nal bodies which maintain the pressure, we shall have 



Tf=-4'^"^,(i,-i). (591) 



and, by (514) and (131), 



da dt da da 



dt dp dp d log p' \ ' ) 



The work expended directly in extending the film will of course be 

 equal to g. 



Let us now consider the case in which there are two component 

 substances, neither of which is confined to the surface. Since we can- 

 not make the superficial density of both these substances vanish by 

 any dividing surface, it will be best to regard the surface of tension 

 as the dividing surface. We may, however, simplify the formula by 

 choosing such substances for components that each homogeneous 

 mass shall consist of a single component. Quantities relating to 

 these components will be distinguished as on page 431. If the sur- 

 face is extended until its area is increased by unity, while heat is 

 added at the surface so as to keep the temperature constant, and the 

 pressure of the homogeneous masses is also kept constant, the phase 

 of these masses will necessarily remain unchanged, but the quantity 

 of one will be diminished by l\^ and that of the other by F ^^. Their 



r r 



entropies will therefore be diminished by — ; ?/v' and -^ //y", respect- 



y y 



