Other Thermodynamic Potentials Gibbs and Helmholtz Free Energy

5 OTHER THERMODYNAMIC POTENTIALS GIBBS AND HELMHOLTZ FREE ENERGY 

The variations of entropy S and energy U express the content of the second law in a deep way, whether as displacements (8) from equilibrium (e.g., for a closed system), 

However, these potentials do not yet express the second law in the form most convenient for chemical applications. Open laboratory vessels exposed to the temperature and pressure of the surroundings are subject neither to constraints of isolation (as required for entropy maximization) nor to adiabatic constant-volume conditions (as required for energy minimization). Hence, we seek alternative thermodynamic potentials that express the criteria for equilibrium under more general conditions. 

Let us first introduce a useful short-cut to the constrained optimization procedure employed in Section 5.2, based on the general Clausius inequality [cf. (4.43)] for spontaneous changes toward equilibrium  

If we write the first law (dU = dq + dw) for the case of PV work only, we obtain from (5.33), under conditions of constant S and V, 

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Equations (4.39), (4.40), (4.41), (4.42), (4.43), (4.44), (4.45), (4.46), and (4.47) allow to calculate all material coefficients related to the Helmholtz free energy from coefficients derived from the Gibbs function and vice versa. In the example worked out above the two thermodynamic potentials differ in two of the independent variables. The same procedure may, in principle, be followed if in the transition from one potential to the other only one variable is exchanged. 

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