F 



THE QUANTITIES x, ^, T, AND THE CRITERIA 

 OF EQUILIBRIUM 



[Gibbs, I, pp. 89-92] 

 E. A. MILNE 



The following notes amount to an independent treatment of 

 Gibbs' results in this section. They also iaclude an extension 

 of some of his calculations so as to take account of second order 

 terms where discussion of first order terms alone ("differen- 

 tials") is insufficient. Some of the later calculations are adapted 

 from Lewis and Randall's Thermodynamics. 



1. Stability Tests. At the beginning of his memoir, The 

 Equilibrium of Heterogeneous Substances, Gibbs establishes 

 criteria of stability which may be stated as follows : Let A denote 

 any increment of a quantity, not necessarily small. Let d denote 

 a "differential" of the quantity, which may (non-rigorously) be 

 identified approximately with a small increment. 



Then if e denotes the energy of a system, ?? its entropy, we 

 have: 



For stable equilibrium, 



(At;), < Oor (Ae), > 0. 



For neutral equilibrium, in general, 



(At,), ^ Oor(Ae), ^0, 



but there exist variations for which 



(Atj), = Oor (Ac), = 0. 



For unstable equilibrium, 



(rfT,), = Oor(d€), = 0, 

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