Helmholtz energy hard spheres

A particularly simple form of Pr(V ) is obtained for the solvation Helmholtz energy of a in a solvent consisting of N hard-sphere solvent particles of diameter a in a volume V. If the density N/V is very small, so that one can neglect solvent-solvent interactions, the probabihty density Po(R ) is simply and the integral on the rhs of Eq. (9.4.7) reduces to... 

For hard spheres the Helmholtz free energy is purely entropic. Thus Table I shows the difference between the entropy per particle of the two phases, [5) . - Stlcp /N. 

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 Helmholtz free energy of the reference hard sphere system, in turn, can be expressed as two terms. One is due to the ideal gas contribution (accounting for the momenta of all particles) while the other term is the repulsive interaction among the hard spheres. It is... 

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

Estimate the internal energy, Helmholtz energy, and entropy of a hard-sphere fluid in units of Nk Tassuming a concentration of 10 M, a diameter of 300 pm, a molecular mass of 30 g, and a temperature of 25°C. 

Fig. 2.11 Plots of the Helmholtz energy for a hard-sphere fluid (a = 0.37 nm) against the logarithm of the pressure at constant temperature as indicated.

Values of the Helmholtz energy estimated as a function of pressure at constant temperature on the basis of equation (2.9.22) are shown in fig. 2.11. These plots are reasonably linear in the logarithm of the pressure at low pressures. This is to be expected, since the density is proportional to the pressure under these conditions, and the effects of non-ideality are relatively unimportant. However, at higher pressures the value of A, starts to rise sharply due to non-ideality. Eventually, one reaches positive values of A, indicating that the fluid is not stable. It has been shown that the hard-sphere system undergoes a phase transition from fluid to solid when = 0.943. For the system considered in fig. 2.11, this... 

The statistical-associated fluid theory (SAFT) of Chapman et al. [25, 26] is based on the perturbation theory of Wertheim [27]. The model molecule is a chain of hard spheres that is perturbed with a dispersion attractive potential and association potential. The residual Helmholtz energy of the fluid is given by the sum of the Helmholtz energies of the initially free hard spheres bonding the hard spheres to form a chain the dispersion attractive potential and the association potential,... 

A worthwhile question to ask is why there should be a phase transition from a fluid to a solid in the hard-sphere model. We can formulate an initial perspective by considering the configurational Helmholtz free energy of the hard-sphere system, which is related to the configurational partition function via... 

Baus [10] has presented an interesting argument about how the hard-sphere phase transition can be understood from the perspective of DFT. The argument is based on an analysis of the difference between the Helmholtz energies of the solid and fluid phases as a function of the density, p. This can be written exactly within the context of DFT as... 

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... 

A = Helmholtz free energy A<> — Helmholtz free energy of a hard-sphere fluid An — nth order term in the temperature expansion of the free energy... 

Space not permit us to review the extensive literature of these theories, including the many recent developments. Instead, we shall try to apply the free volume concept to the hard sphere system in the simplest fashion. Imagine that the diameter a of the N hard spheres in the volume V of our system is shrunk till the molecules are elastic point centers. We now have an ideal gas whose Helmholtz free energy is - (w = p )... 

The two models (SAFT and PHSC) differ substantially in the way they represent the different contributions to the expression of the residual Helmholtz free energy of a system. In the SAFT model, is the sum of different contributions due to hard spheres, dispersion, chain, and association, respectively ... 

In the PHSC EoS, the Helmholtz free energy is expressed as the sum of two different terms, one is a reference term accounting for chain connectivity and hard sphere interactions, and the other is a perturbation term, which represents the contributions of mean-field forces ... 

To describe the measured cloud-points, the SAFT equation of state (eos) has been used. The SAFT eos [6] is based on the perturbation theory (see Chapter 3), and, in spite of the rather complex derivation of the model equations, the basic idea and the application of the model is less complex. The SAFT eos can be written as a sum of Helmholtz energies. The first contribution is the Helmholtz energy of an ideal gas, followed by a correction for a mixture of hard spheres, a correction for chain formation, and a correction for the dispersion and association forces ... 

Here we have introduced the normalized Helmholtz energy F = Fvo/kTV. The first term on the right-hand side of (3.8) is the ideal contribution, while the second hard-sphere interaction term is the Camahan-Starling equation of state [12]. 

In (5.1) is the number of large hard spheres, 2 is the chemieal potential of the small hard spheres imposed by the (hypothetical) reservoir, Fq is the (Helmholtz) free energy of the (large) hard sphere system without added small spheres, is the pressure of the small hard spheres in the reservoir, and (Vfree)o is the free volume of an added small hard sphere in the system of A i large hard spheres in a volume V. The quantity (Vfree)o is given by the same expression as the free volume of an added penetrable hard sphere,... 

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