X, "A, r, AND THE CRITERIA OF EQUILIBRIUM 229 



Hence a necessary condition that such a system shall be in 

 stable equilibrium under the stated conditions is 



(Ar)p. t>0, 



for it then cannot undergo any change of itself. This estab- 

 lishes Gibbs' criterion concerning Af by an independent method. 



10. Further Illustration. The following original example 

 illustrates further the properties of the ^-function. 



"A system, which can perform external work in any manner, 

 is brought reversibly from a temperature ti to a temperature 

 <2( < ^i) in such a way that it only gives up heat at the tempera- 

 ture ti. Prove that the external work performed, AW, is given by 



ATF = A-A + mik - k) 



where Ai^ is the decrease in the work function \p between the 

 temperatures ^i and ^2, and tji is the entropy at <i." (This is a 

 generalisation of the similar result in the particular case ti = t^ 

 estabhshed above.) 

 We have 



^1 — '/'2 = Cl — €2 — (flt/i — ^2^72) 



ci - €2 = Ae = AQ + ATF, 



where now AQ denotes the heat given up at ^2- Since the 

 process is reversible and the heat is given up at tz 



m — m = AQ/ti. 

 Hence 



AW = A\p -\- (tiTji - tiVi) — t2(vi - V2) 

 = Alp + r]i{ti — ti). 



This result is, of course, somewhat trivial. We may, however, 

 extend it to include irreversible processes. The following 

 theorem may be established. 



"If the system is brought by any process, reversible or 

 irreversible, from the state at ti to the state at t^, and not neces- 

 sarily subject to the condition of only giving up heat when at 



