434 J. W. Gihbs — Equilibrium, of Heterogeneous SSuhstances. 

 clG = - 7/s(i^ dt - Tg, ,^ d)j^ + 7^3(1) c?//3 + etc., 



<7^3 = 7/v3 rft -f ;(/3 (^Afg, 

 etc., 

 where the suffix ^ relates to the component of which the surface- 

 density has been made to vanish, and y^, y n, etc. denote the densities 

 of the gases specified in the gas mixture, and PziPzt ^^^'•t Vv2? ^/v3? 

 etc. the pressures and the densities of entropy due to these several 

 gases, we obtain 



da= — ^//s(j^ ^^^^ /;v2 ;- 7v3 — etc. j dt 



_ La{1} dp^ - ^^' dp^ - etc. (585) 

 K2 Yz 



This equation aftbrds values of the diiFerential coefficients of a with 

 respect to ^, ji9„, jOg, etc., which may be set equal to those obtained 

 by differentiating the equation between these variables. 



Thermal and Mechanical lielations ^yertaining to the Extension of a 

 Surface of Discontinuity. 



The fundamental equation of a surface of discontinuity with one 

 or two component substances, beside its statical applications, is of 

 use to determine the heat absorbed when the surface is extended 

 under certain conditions. 



Let us first consider the case in which thei-e is only a single com- 

 ponent substance. We may treat the surface as plane, and place 

 the dividing surface so that the surface density of the single com- 

 ponent vanishes. (See page 397.) If we suppose the area of the 

 surface to be increased by unity without change of temperature or 

 of the quantities of liquid and vapor, the entropy of the whole will 

 be increased by ?/s(j). Therefore, if we denote by Q the quantity of 

 heat which must be added to satisfy the conditions, we shall have 



^ = «%(,„ (586) 

 and by (514), 



^ d<3 da ,~r,H\ 



^ dt dlogt ^ ' 



It will be observed that the condition of constant quantities of 

 liquid and vapor as determined by the dividing surface which we 

 have adopted is equivalent to the condition that the total volume 

 shall remain constant. 



