98 



EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



It will be observed that yu a , for example, in (124) denotes the potential 

 in the mass considered for a substance S a which may or may not 

 be identical with any of the substances 8 V $ 2 , etc., to which the 

 potentials in (125) relate. Now as the equations between the 

 potentials by means of which the elimination is performed are similar 

 to those which subsist between the units of the corresponding sub- 

 stances (compare equations (38), (43), and (51)), if we denote these 

 units by @ a , <S&, etc., & v @ 2 , etc., we must also have 



m a / @ a +m 6 '@ 6 +etc. = ^ 1 / @ 1 + ^ 2 / (S 2 ... +4 '<. (126) 



But the first member of this equation denotes (in kind and quantity) 

 the matter in the body to which equations (124) and (125) relate. 

 As the same must be true of the second member, we may regard this 

 same body as composed of the quantity A^ of the substance 8 V with 

 the quantity A^ of the substance $ 2 , etc. We will therefore, in 

 accordance with our general usage, write m/, m 2 ', etc., for A^ t A^, 

 etc., in (125), which will then become 



v'dp = ri'dt+m l 'dfj. l + m 2 'd]UL 2 ... +m n 'djj. n . (127) 



But we must remember that the components to which the m/, m 2 ', 

 etc., of this equation relate are not necessarily independently variable, 

 as are the components to which the similar expressions in (97) and 

 (124) relate. The rest of the n + l equations may be reduced to a 

 similar form, viz., 



v"dp = rj"dt+m l "d/uL l +m 2 "diuL 2 ... +m n "dfjL n , (128) 



etc. 



By elimination of dp^ djj. 2 , ... dfjL n from these equations we obtain 



...m^ 



, // 



7) W 1 



V llt-t li 



11" m " w " vn ' 



(/ //I/-! I'vn . . . ilvfl 



v'" >m '" m '" m ' 



(/ //(/i //f/9 ... 1'V'H 



dp 



...m 



n 



' 



ri m m 2 ...m w 

 if" m/" m 9 '"...mn'" 



dt. 



(129) 



In this equation we may make i>', i>", etc., equal to unity. Then 

 m/, m 2 ', w,/', etc., will denote the separate densities of the components 

 in the different phases, and rf, rf f , etc., the densities of entropy. 



When n=l, 



(mV - mV)dp = (m'V - mV')rf, (130) 



or, if we make m x = 1 and m"= 1, we have the usual formula 



dp_r\-rj' _Q nqn 



7T ~^ 7 77 ^~ j~7 77 7T i \ / 



in which Q denotes the heat absorbed by a unit of the substance in 

 passing from one state to the other without change of temperature or 

 pressure. 



