X, i^, r, AND THE CRITERIA OF EQUILIBRIUM 217 



We will first show that (5) implies (6). To do this we will 

 show that if there exists a state violating (6) then there exists a 

 state violating (5). If then (5) is known to hold, there can be 

 no state violating (6), and so (6) holds. 



Let us then suppose that a state exists for which 



(Ar)«. p < 0. 



Now 



f = ^ + py, 



and so 



Af = A^ + pAv + vAp + AvAp. 



Here Ap = 0, and hence 



Af = Ai/' + pAv < 0. 



Therefore 



AiA < -pAv. (7) 



Now change the volume and pressure reversibly at constant 

 temperature. For these changes the infinitesimal increments 

 are given by 



d'e = i d'r} — p d'v 



by the first and second laws of thermodynamics. Hence 



dV = d'(€ - tri) = -pd'v, 



since d't = 0. It follows that 



AV = - \ P d'v, 



whence 



p. 



At/' -\- A'^p < - pAv + / P d'y. 



