Helmholtz function

Note that eq. (3.16) involves the Helmholtz function, A, rather than the Gibbs function G, but the difference between A A and AG is negligible in solutions and we will use AG in this book. 

As U, T, and S are state functions, the quantity (U— TS) also must be a state function. This quantity is sufficiently important that it is given the name Helmholtz function, or Helmholtz free energy, which is defined as... 

The letter F previously has been associated with the Gibbs function, particularly in the United States. Some older tabulations of chemical thermodynamic data use F for the function of Equation (7.17). The term free energy is most commonly assigned to G, but we have adopted the name Gibbs function to be consistent with Helmholtz function for A and Planck function for Y. 

As the procedure is the same for both the Gibbs function and the Helmholtz function, we shall consider in detail only the derivation for the Gibbs function G. After Equation (7.14), we obtained the differential of the function, which was later to be defined as G, as... 

By an analogous procedure, it can be shown that the total differential of the Helmholtz function is given by the expression... 

If the same operations are carried out using the Helmholtz function, with namral variables T and V, the results are... 

USEFUL WORK AND mE GIBBS AND HELMHOLTZ FUNCTIONS 175 over the range of interest. The approximate result if both are assumed constant is... 

Thus far we have observed that the Gibbs and Planck functions provide the criteria of spontaneity and equilibrium in isothermal changes of state at constant pressure. If we extend our analysis to systems in which other constraints are placed on the system, and therefore work other than mechanical work can be performed, we find that the Gibbs and Helmholtz functions also supply a means for calculating the maximum magnitude of work obtainable from an isothermal change. 

The change in the Helmholtz function thus provides a limiting value for the magnitude of the total work (including work against the pressure of the atmosphere) obtainable in any spontaneous, isothermal process. That is. 

USEFUL WORK AND THE GIBBS AND HELMHOLTZ FUNCTIONS But at constant pressure, from Equation (4.3),... 

USEFUL WORK AND -mE GIBBS AND HELMHOLTZ FUNCTIONS 181 From Equation (7.49)... 

In the free-volume theory of hquids, the molar Helmholtz function is dehned by the equation... 

Note that in this entry, A is the affinity and A, the Helmholtz function (work function). 

The most important of the auxiliary functions just mentioned are the Helmholtz Function-. 

Finally, we have the expression for the Helmholtz function of mixing AmixF,... 

As previously described, there is the partial consistency (e.g., the pressure consistency), and there are the Helmholtz free energy functionals. Unfortunately, the exact Helmholtz functional suffers from intractability. The evaluation of such functionals is equivalent to thermodynamic integration, where the integrands are not exact. New efficient methods in the integration of functionals should be... 

Helmholtz energy (sometimes also called Helmholtz free energy, or Helmholtz function) is the thermodynamic state function equal to the maximum possible nonexpansion work output, which can be done by a closed system in an isothermal isochoric process (T = const, V = const). In terms of the -> internal energy and -> entropy... 

Since we will ultimately deal with equilibrium in constant volume systems it is convenient to define a modified Helmholtz free energy function Fg(z) = U(z) - T(z)S(z) which differs from the normal Helmholtz function by inclusion of the gravitational potential energy term. We now obtain... 

According to the Helmholtz function, (T)=g I)-TdgJT)/dTwhere g (7) is the volume Gibbs free energy and gJT) = and thus... 

Electrostriction. By the thermodynamics of dielectrics, the thermodynamic potential (Helmholtz function 10 of a didectric in a uniform external fidd Eo at constant pressure and temperature is ... 

For a reversible change at constant volume, the work done is equal to the change in the Helmholtz function, F ... 

Similar arguments may be advanced for virtual changes involving the other thermodynamic functions of state. However, one must be careful for example, virtual changes in the Helmholtz function assume the form... 

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