Energy , free minimization method

The study of chemical equilibrium can detect thermodynamic constraints on the achievable conversion and selectivity. In this section we make use of the Gibbs free-energy minimization method available in Aspen Plus [9], We assume that both cyclohexanone and cyclohexanol are products. The curves in Figure 5.2 show the evolution of the phenol equilibrium conversion, yield and selectivity with the ratio hydrogen/phenol at temperatures of 180, 200, 220 °C and a pressure of 3 bar. 

In typical environmental situations, the liquid phase may contain a number of ions which can form complexes with the multiple charge cations present. In order to be able to calculate the thermodynamic driving force for the precipitation of a particular solid phase, it is necessary to determine the concentrations of free lattice ions in the solutions. The commonly used procedures for the calculations of solute components in a homogeneous solution are the "equilibrium constant and the "free energy minimization" methods. The former utilizes an approach wherein first, a "basis" is selected from the species, usually that having the highest concentration at equilibriiam. 

The other solution species are formed from this "basis" by a series of chemical reactions, and their concentrations can be expressed through the use of equilibrium constants, in terms of the concentration of the chosen "basis". The resulting set of nonlinear simultaneous equations consists of as many unknowns as there are elements and can be solved by conventional niamerical methods. The "free energy minimization" method utilizes only free energy criteria for chemical equilibria making no distinction among the constituent species and is essentially a constraint non-linear minimization problem. A number of search methods have... 

The non-stoichiometric formulation, in which the stoichiometric equations are not used, instead the material balance constraints are treated by means of Lagrange multipliers. In these direct free energy minimization methods the problem is usually expressed as minimizing G, for fixed T and p, subject to the material balance constraint. 

Weare, 1980 Eugster et al., 1980 Harvie et al., 1982, 1984 Brantley et al., 1984 Weare, 1987). These calculations, coupled with the free energy minimization method for computing chemical equilibria (described in Chapter 19) successfully predicted the mineral assemblages formed during the evaporation of seawater to almost complete desiccation. It is difficult to conceive of a more complicated test of the Pitzer model. Remember that these calculations represent a prediction from experimental data on systems containing no more than two different salts at a time. 

In the equilibrium state, the total free energy is minimized. The director components n, (i = x, y, z) in space in the equilibrium state can also be numerically calculated by the relaxation method. At the lattice site (l, ly, 4) of the mesh the changes of the director components from step r to step (t + 1) are calculated from the values of the director components at step t,... 

A second approach incorporates free energy information into the branch and bound algorithm. Specifically, harmonic entropic contributions are calculated and included at each minima of the upper and lower bounding functions. In this way, the progression of lower and upper bounds includes a temperature-dependent entropic term. A similar modification to the Monte Carlo minimization method has also been proposed [87] and has been shown to be effective in locating low-energy conformers of peptides [88,89]. 

Results are presented here for halide ion - molecule complexes in the gas phase and for complexation of halide ions in solution by 1 and 2. The gas-phase interaction energies were obtained from energy minimizations and the relative binding affinities in solution were calculated with the free-energy perturbation (fep) method using Monte Carlo sampling. All calculations were carried out with the BOSS program. ... 

Stoichiometric method Gibbs free energy minimization method... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

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