Helmholtz free energy density

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

Having the hole energies, the Helmholtz free energy density FC[M] can be evaluated according to the standard procedure for the Fermi gas. By minimizing F[Af] = Fs[Af] + FC[M] with respect to M at given T, H, and hole concentration p, one obtains M(T, H) as a solution of the mean-field equation,... 

Analytical solutions for the RPM are conveniently given in terms of the excess part ex of the reduced free energy density = Ao3/1cbTV — 3>ld+ ex, where A is the Helmholtz free energy and ld the ideal gas contribution. For the MSA one finds for the ion-ion contribution [189]... 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

Application of DFT as a general methodology to classical systems was introduced by Ebner et al. (1976) in modeling the interfacial properties of a Lennard-Jones (LJ) fluid. The basis of all DFTs is that the Helmholtz free energy of an open system can be expressed as a unique functional of the density distribution of the constituent molecules. The equilibrium density distribution of the molecules is obtained by minimizing the appropriate free energy. 

In the last equations A(n) is the Helmholtz free energy of the total NVT system fully defined by the vector n = nm, n 2,... providing the molecular number in each tiny volume and iq) is the chemical potential at a given iq position, i.e., within the corresponding tiny volume. Note that the molecular number can be used as a continuous variable, given the fact that for any thermodynamic property in a macroscopic system the variation due to a single molecule is virtually equivalent to a differential. From the definition of the chemical potential and probability density in the r) space p(t, ti), we readily have... 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

The first requirement is met by defining a Helmholtz free energy density f, such that Fg - Jh f(z)Adz then... 

Next, define a Helmholtz free energy density f such that F - fd3rf. As long as the volume remains fixed we may set dF - Jd3rdf. Comparison with (5.6.5) establishes that... 

Note that in (5.6.7) P - 0 and E - D - Ea if the material medium is absent. Therefore the quantity E2/8n may be considered to represent the free energy density arising from the presence of the electric field in vacuum. The term - P E0/2 then represents the Helmholtz free energy density due to the interaction of the system with the medium we write... 

Consider an inhomogeneous fluid mixture of polyatomic species with the density distributions Pa(r) of interaction sites a in an external field with the site potential M (r). Minimization of the Helmholtz free energy with respect to variations performed analogously to Refs. [34,35] yields the relation... 

The functional Eq. (6.33) suggests that it is most natural to consider the Helmholtz free energy as a functional of the external potential energy field A = A[(Pa(r)]. Our interest will he in density functionals. To pursue this we introduce the Legendre transform (Gallon, 1985), A — fv a(f)Pa(f ) = [Pa(f)]-... 

The form of the Helmholtz free energy F depends on the type of the origin of surface charges on the interacting particles. The following two types of interaction, that is (i) interaction at constant surface charge density and (ii) interaction at constants surface potential are most frequently considered. We denote the free energy F for the constant surface potential case by F and that for the constant surface... 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

If we take account of the thermodynamic properties of the material the energy density iPE should, more properly, be described as the density of Helmholtz free energy. 

Equilibrium conditions are enforced by subjecting the subsystem to a virtual displacement for which 5 Eg 17 y =0, subject to the requirement that the number of moles of the various species i in the entire cylinder remain fixed 8ni = 0. For this purpose we introduce a Helmholtz free energy density fg by the relation Fg — Jq fg(z)Adz. We then use the properties introduced in Eq. (5.1.1) and consider a fixed element of volume, to rewrite Eq. (5.1.4) as... 

We next define the Helmholtz free energy density a through the relation A = fy d ra-, comparison with (5.7.4) establishes that... 

As deduced from (5.7.1), the first term in the middle represents the Helmholtz free energy density of the free space associated with the field q, the remaining term exhibits the response of the matter to the field, in terms of its Helmholtz free energy density. 

---