Equations for Closed Systems in Equilibrium

We have demonstrated from the second law of thermodynamics that a system is in a state of equilibrium if 

The first law of thermodynamics for closed systems can be written in the form 

The restriction to a closed system is necessary since, in the first law, energy changes due to transfer of mass between the system and the surroundings are not considered. It is possible to extend the first law to open systems by enlarging the definition of work to include energy increments resulting from the addition of matter. We note that changes 

Equation (6-4) is the general criterion of equilibrium for closed systems. It applies to heterogeneous as well as homogeneous systems. Special forms of this criterion, applicable to a system subject to additional restraints, are of interest. 

In order to express Eq. (6-7) in a more concise fashion, it is useful to define an extensive function of state. A, the work function or Helmholtz free energy. The Helmholtz free energy is defined by the relation 

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