Excess Helmholtz free energy

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

If the quantity of interest is the excess Helmholtz free energy, as is almost always... 

Thus the excess Helmholtz free energy of mixing may be calculated from the equation... 

The Helmholtz free energy A(T,V) in excess of that of the medium is... 

In general, the excess Helmholtz free energy AF of a hard-particle solution over that of the solvent is written in the form... 

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 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. 

By relaxing mathematical rigor in establishing the connection between excess free-energy models and EOS, several successful approximate models have been developed in the limit of infinite pressure. One such model that uses excess Helmholtz free energy was introduced by Orbey and Sandler (1995c) and is as follows ... 

In these equations, G , the molar excess Gibbs free-energy obtained from any excess free-energy model, is a function of temperature and composition only. Even though the Wong-Sandler derivation involves the Helmholtz free-energy of the mixture, this substitution is due to the assumption that G (7 , Zi) = P, z,). See... 

Because the excess Helmholtz free energy is closely related to this departure term (see Wong and Sandler 1992 for details) the same C term appears in the excess Helmholtz free-energy term obtained from the PR EOS. 

It has proven to be useful to decompose the Helmholtz free energy functional A[p(N)], where p(N) is the N-body particle density, into an ideal contribution from non-interacting particles and an excess contribution,... 

The HMSA/MC equation of state was used to describe the excess Helmholtz free energy. An extended Lorentz-Berthelot approximation was used to generate the interaction between unlike species. 

Figure A2.2.5 shows a sketch off(r) for Lennard-Jones pair potential. Now if AA is the excess Helmholtz free energy relative to its ideal gas value, then (-PAT) = log(2/2ideal AU/N= [5(PAzl/A)/(5p)]. Then, integrating with respect to P, one obtains...

The temperature at which B iT) is zero is the Boyle temperature T-q. The excess Helmholtz free energy follows from the thermodynamic relation... 

This is Carnahan and Starling s (CS) equation of state for hard spheres it agrees well with the computer simulations of hard spheres in the fluid region. The excess Helmholtz free energy... 

Integration with respect to P, from p = 0 to finite P, leads to the excess Helmholtz free energy ... 

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