Alternative Expressions of the Gibbs Criterion

Second-derivative conditions such as (5.17) are known as stability conditions, expressing the self-restorative property of thermal equilibrium. 

The positivity of Cv, i.e., the fact that temperature must rise when heat is added, is a well-known property of thermally stable systems. Like the uniformity of temperature (5.13), the positivity of Cv (5.19) is a deduced (rather than assumed) feature of thermal equilibrium in the Gibbs formulation. 

Other stability conditions are obtained from the negativity of second derivatives with respect to V or N. (More generally, determinants of such second derivatives must also be negative in order to guarantee stability with respect to arbitrary combinations of energy, volume, and mass changes.) In summary, we can say that the Gibbs criterion of equilibrium for a closed system is equivalent to conditions of uniform intensive properties 7, P, 

Let us now attempt to re-express the Gibbs criterion of equilibrium in alternative analytical and graphical forms that are more closely related to Clausius-like statements of the second law. For this purpose, we write the constrained entropy function S in terms of its 

Using the definition (5.22), a Taylor series expansion (Section 1.4) of S(X + 8X), and the expression (5.23) for the Lagrange multiplier Xx, we obtain 

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