Entropy as a thermodynamic potential

To begin the development of other directionality parameters, we must describe our entropy function a little more completely. This statement about the entropy can also serve as another statement of the Second Law, and is paraphrased from Callen (1960). It contains some aspects of the entropy that we have not explicitly stated before, and reflects Callen s postulational approach to thermodynamics. 

There exists a homogeneous first order state variable, the entropy, which for isolated systems (those having constant U and V) achieves a maximum when the system is at stable equilibrium. Entropy and its derivatives are single-valued, continuous and differentiable functions of the other state variables. Entropy is a monotonically increasing function of the energy U. 

For those who skipped Euler s theorem in Chapter 2, the expression homogeneous first order in this definition simply means that we are defining S rather than S (i.e., total entropy rather than molar entropy). The fact that the entropy achieves a maximum rather than a minimum is the result of a sort of historical accident, as discussed in 5.3.1. It disturbs the symmetiy of the thermodynamic potentials, since all the others achieve minima, but this creates no serious problems. 

We define the new function S and its derivatives as continuous and differentiable to avoid the possibility that even though maximized, it might be A-shaped rather than n-shaped. This would be mathematically inconvenient. 

The mathematical statement that S achieves a maximum for given, U,V, is 

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Fundamental equation 2.2-8 has been presented as the equation resulting from the first and second laws, but thermodynamic treatments can also be based on the entropy as a thermodynamic potential. Equation 2.2-8 can alternatively be written as... 

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