Linear Dependence, Dimensionality, and Gibbs-Duhem Equations

The singular limit (S 10.2-6), in which one of the / independent R-) is a zero vector (implying that the original basis vectors R,) were linearly dependent), occurs at a critical state Sc, where the number of phases p and dimension/ are changing. Critical state limits will be examined in Chapter 11. 

3 LINEAR DEPENDENCE, DIMENSIONALITY, AND GIBBS-DUHEM EQUATIONS 

The starting point for thermodynamic description, whether in the calculus-based or the geometry-based formalism, is the Gibbs fundamental equation for a given equilibrium state S. In the energy representation, this is expressed as 

Either could be taken (with equal justification) as the starting point for a full thermodynamic description. We have explicitly chosen the 1/-based starting point (10.11) for the geometrical description, but it is important to realize that an entirely equivalent description might have been constructed from the S-based starting point (10.12), based on 

From the chosen (/-based starting point (M = M(t/)), each possible state S = S(Q of a single-phase system of c independent chemical components can be parametrized by the numerical values ( ) of the c + 2 extensive variables XL in S  

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