The residual Helmholtz free energy

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as 

Given and equation of state, we can determine an explicit expression for the Helmholtz free energy from the fundamental equation of thermodynamics. At constant temperature and mole numbers, we have 

By integrating both sides of this equation from a total volume Vi to a total volume V2 

As the density of the system approaches zero, or the molar volume of the system approaches infinity, the properties of a system approach those of an ideal gas. In particular. 

Given an expression for the compressibility factor Z as a function of temperature and volume for a fluid (an equation of state), we can determine the residual Helmholtz free energy of the system. 

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Substituting this expression into Eq. (11.8), we can obtain the residual Helmholtz free energy for the van der Waals equation of state ... 

The residual chemical potential of species a can be obtained from the residual Helmholtz free energy by differentiating by the number of moles of a (see Eq. (11.1)) ... 

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 ... 

The coexistence of these two effects makes dilute near-critical solutions challenging to model properly. For modeling purposes, this characterization into solute-induced and compressibility-driven effects usually takes the form of asymptotic expressions for the mixture s Helmholtz free energy and its temperature, volume, and composition derivatives around the solvent s critical point. The usefulness of the resulting asymptotic expressions (from the residual Helmholtz free energy expansion, A"(T,p,x)y where T is temperature, p is density and jc is composition) resides in their mathematical simplicity They only require the pure solvent thermophysical properties and the critical value of the derivative dP/dXij) ... 

For mixtures containing polar molecules the most effective route to thermodynamic properties is to determine first the molal residual Helmholtz free-energy function A, where ... 

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,... 

Solution The system of two equations (Eqs. (4.214) and (4.215)) and two unknowns (that is and P ) for the Baker and Luks formulation can be solved via the secant method. This is just a Newton-Raphson method on numerical derivatives. Computation of the residual in Eq. (4.214) is fairly straight forward, because it only requires expressions for the second derivatives of the Helmholtz free energy/I in terms of V and These derivatives arc provided in Example 4.8. Equation (4.215) is somewhat more complicated its determinant requires derivatives of Eq. (4.214) with respect to V and N. The procedure presented in Problem 4.14 can be used to evaluate the determinant derivatives. After both residuals are computed, the system of two equations and two unknowns are solved by the Newton-Raphson method to convergence. 

The first term on the r ht-hand side in above equation corresponds to the electrostatic part of excess Helmholtz free energy for a bulk electrolyte solution and it can be estimated with EOS derived from MSA (Palmer and Weeks, 1973 Waisman and Lebowitz, 1972), and similar to Eqs. (31) and (32), the residual excess chemical potential A// j and residual two-body DCF Ar . ( r — r l) can be obtained by taking functional derivatives with respect to the local density. 

Develop expressions for the residual pressure (r). internal energy and Helmholtz free energy in terms of the configuration integral. 

It is convenient to write the thermodynamic mixing properties as the sum of two parts (1) a combinatorial or configurational contribution that appears in the entropy (and therefore in the Gibbs energy and in the Helmholtz energy) but not in the enthalpy of mixing and (2) a residual contribution, determined by differences in intermolecular forces and in free volumes between the components. For the entropy of mixing, we write... 

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