The 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 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... 

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. 

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,... 

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

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... 

This effect can be examined further by studying the reversible adiabatic extension of an elastomer. Although this experiment is more easily carried out imder conditions of constant pressure rather than constant volume, it is best to derive the relevant equations for constant voliune. For this reason, we will consider first the Helmholtz function for the system, i.e.. 

The strain-energy function is actually equal to the change in the Helmholtz function A for the rubber, since A 4 = — T A5 for an isothermal change for a medium like rubber in which there is no change in the internal energy at constant temperature. 

The 1981 article summarizes the development of a thermodynamic surface for water with which all thermodynamic properties for the fluid states can be calculated from the freezing line to 1000 K and up to 1 GPa in pressure. The discussion is very brief, but gives references to earlier work and indicates that a more detailed publication Is forthcoming. Given are coefficients of the Helmholtz function which define the surface. Plots of heat capacity, enthalpy, and speed of sound are included. 

The Helmholtz function can thus only decrease or remain constant. The criterion of equilibrium of a system enclosed in a rigid container and held at constant temperature is therefore that A has reached its minimum possible value. 

Like the Helmholtz function, O can be used to provide a criterion of equilibrium. For an infinitesimal change at constant temperature and pressure, we have from (2 22)... 

Since most equations of state have temperature, molar volume, and composition as independent variables, while the Gibbs function is explicit in temperature, pressure, and composition a formulation of the critical conditions in terms of the Helmholtz function is required. The following equations allow a transformation between G T, p, x) and A T, V,... 

The Helmholtz function AiJ, V, x) and the derivatives required for eqs 2.165 and 2.166 may be obtained from any equation of state that gives the pressure through... 

For higher-order critical points, the transformation equations from the Gibbs into the Helmholtz function become extremely complex. 

Wong and Sandler followed a different approach by matching the Helmholtz function at infinite pressure from the cubic equation of state and from an activity coefficient model. This approach ensures consistency with statistical mechanics requirements that the second virial coefficient of a mixture has a quadratic dependence on composition. For the case of the Peng-Robinson equation of state, the Wong-Sandler mixing rules are ... 

Chapter 12 describes Helmholtz-function based multi-parameter equations of state and provides the mixing and combining rules used to extend the equations for pure fluids to mixtures. In this approach, the Helmholtz function equations of state for the components in a mixture are combined with an excess function to effectively account for the interactions between unlike molecules.Of particular importance to this field is the work of Kunz et that provided... 

In the SAFT model, a molecule is considered to consist of a collection of segments, and the Helmholtz function is written as ... 

---