THERMODYNAMIC AL SYSTEM OF GIBBS 121 



Let f ", f ' be the values of f for two states of the gas at the 

 same temperature t. By (26) we have 



r - r = e" - t' - tin" - v') + P"v" - pV 



= - t W -7?'), (123) 



since the energy of a perfect gas at constant temperature is 

 independent of its volume, and the product pv is also constant. 

 In order to find the entropy change of the gas when its volume 

 changes from v' to v" at constant temperature, we have by (3) 



idr] = pdv 



and, introducing the value of p/t given by (122), 



Am dv 

 dv = ^-- (124) 



Integrating this from y' to v", we thus have 



,, , Am , v" Am , v' , ^ 



,"-V = ^log---^log - (125) 



or, inserting these values in (123), 



Amt^ v" , Amt , v' 

 ^ +l^'°8,I = f +-M '°«» = ™'^. 



where C is a constant, which is a function of the temperature. 

 The value of ^ for any volume v is thus given by the expression 



Amt m 

 r = mC + — log-. (126) 



and the potential of the gas is therefore 



At m 

 or, by (122), 



M = C + - log - (127) 



At 

 M = m + - log p. (128) 



