X, \^, r, AND THE CRITERIA OF EQUILIBRIUM 227 

 and so here, the temperature being constant, 



Ae - AiA = tAv ^ A'Q. 

 In any change whatever, whether reversible or irreversible, 



A'Q = Ae + A'W, 

 whence here 



Ae - ArA ^ Ae + A'W 

 or 



A'W ^ -A^. 



Thus the actual amount of external work performed, A'W, 

 cannot exceed — Aip. 



Now suppose a system enclosed in a fixed volume. If it 

 undergoes of itself any process whatever, at constant tempera- 

 ture, then necessarily 



A'W = 0, 



whence 



Axl^ ^ 0. 



Hence a necessary condition of equihbrium, subject to the 

 condition of constant temperature and constant volume, is 



(AiP)t.v > 0. 



A state for which all possible changes satisfy this relation will 

 be in stable equilibrium, for it cannot undergo any change of 

 itself. This estabhshes Gibbs' criterion concerning A\f/ by an 

 alternative method. 



9. The Free Energy Function f at Constant Temperature and 

 Constant Pressure. Let the system undergo a change at 

 constant temperature and constant pressure, doing any external 

 work whatever in the process. Then we have 



Af = A(e - trj -I- pv) 

 = Ae — tArj + pAv. 



