THERMODYNAMIC AL SYSTEM OF GIBBS 109 



Therefore, eliminating djj.' from (97), we have 



(vW - v"m')dv = Wm" - rj"m')dt, 



or 



dp r\'m" — r]"m' 



dt v'm ' — V m 



(98) [131] 



If we consider unit quantity of the substance in each of the two 

 phases, we may put m' = 1 and m" = 1, so that (98) becomes 



d'p 

 dt 

 Now, 



where Q is the heat absorbed when a unit of the substance 

 passes from one state to the other, at the same temperature and 

 pressure, and v" — v' is the corresponding change of volume. 

 Thus, we obtain the Clapeyron-Clausius equation :* 



dv Q 



dt t{v" - v'Y 



(99) 



Gibbs derives a general expression, similar to (98), for a 

 system of n independently variable components, >Si, . . . aS„, 

 in r = n + 1 coexistent phases. In this case there are n + 1 

 equations of the general form of (56), one for each of the 

 existent phases. But the values of dp and dt must be the same 

 for all phases and the same is true of djxi, c?^2, etc., so far as each 

 of these occurs in the different equations. Thus, if each phase 

 is regarded as being composed of some or all of the n independ- 

 ent components, a variation of the system must satisfy the 

 following equations: 



v' dp = T]' dt -{- nil dm + m2' c?/x2 . . . + w„' dfin, ' 



v" dp = ■(]" dt + rrix' dm + m^" d^ . . . + m„" dju„, 



v"'dp = v"'dt + mi'"dni + r)h"'dn2 . . . + mr/"diji„, 

 etc. 



(100) [127] 



* Clapeyron, J. de I'ecole polytechnique, Paris, 14, 173, (1834). Clau- 

 sius, Ann. Physik, 81, 168, (1850). Also obtained by W. Thomson, Phil. 

 Mag., 37, 123, (1850). 



