J. W. Gibbs — B,quilibriu)n of Heterogeneous Substances. 211 



1 £ — Em 7; ^_ , m , , 



c loar = — — H + aXocf—. (255) 



em in v 



This is a fundamental equation (see pages 140-144) for an ideal gas of 

 invariable composition. It Avill be observed that if we do not have 

 to consider the properties of the matter which forms the gas as ap- 

 pearing in any other form or combination, but solely as constituting 

 the gas in question (in a state of jjurity), we may without loss of 

 generality give to E and H the value zero, or any other arbitrary 

 values. But when the scope of our investigations is not thus limited, 

 we may have determined the states of the substance of the gas for 

 which ez=:Q and ;/=:0 with reference to some other form in which the 

 substance appears, or, if the substance is compound, the states of its 

 components for which ez=.0 and ;/=0 may be already determined ; so 

 that the constants E and H cannot in general be treated as arbitrary. 

 We obtain from (255) by differentiation 



; , 1 , <x , / cE c+a f/ \ y 



f^ de= -dt/ dv + ( ^r- + — -2 (^ni, 256) 



f^m m V \e — Jlini m m^/ ' 



8-E'. 



whence, in virtue of the general relation expressed by (86), 



e — Em 



c m 



(257) 



8 — Em , ^ , 



p = a , (258) 



cv 



u = E+ —-^\c m. + a m - ?/). (259) 



We may obtain the fundamental equation between //•, t, i\ and ?n 

 from equations (87), (255), and (257). Eliminating £ we have 

 if' =z Em + c m t — ^ //, 



and c losr t=: ^ + « log - ; 



and eliminating //, we have the fundamental equation 



/ m\ 



= Em ^ mty<- — H - c log t + (/ log - J. (260) 



Differentiating this equation, we obtain 



/ 1 y \ T amt 

 dip =- m\H+ cXo^t-^ «log -J dt ^- dv 



j.Ie + t Ic + <i - H - c\ogt + a log '-^1 jdm • (261 ) 



