EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 269 



of the gases specified in the gas-mixture, and p 2 , p s , etc., 



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



gases, we obtain 



72 



_ ?k> dp 2 - ^ dp, - etc. (585) 



72 73 



This equation affords values of the differential coefficients of a- with 

 respect to t, p 2 , p s , etc., which may be set equal to those obtained 

 by differentiating the equation between these variables. 



Thermal and Mechanical Relations pertaining to the Extension of a 



Surface of Discontinuity. 



The fundamental equation of a surface of discontinuity with one 

 or two component substances, besides 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 there is only a single 

 component substance. We may treat the surface as plane, and 

 place the dividing surface so that the surface density of the single 

 component vanishes. (See page 234.) 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 q sw . Therefore, if we denote by Q the quantity of 

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



Q = trj s(l)} ' (586) 



and by (514), 



- (587) 



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. 



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

 while the pressure in the liquid and vapor remains constant, the 

 temperature will evidently be maintained constant by condensation 

 of the vapor. If we denote by M the mass of vapor condensed per 

 unit of surface formed, and by 7/ M ' and 7/ M " the entropies of the liquid 

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

 will require that 



-V*) = * < 588 ) 



