Gibbs energy minimization

G = free energy of the system, n = the molal quantity of components, fij = the chemical potential of species j, jjiw = the chemical potential of water, 

This approach finds the minimum in the Gibbs energy of the system by removing and adding phases (Fig. 3.15) subject to mass balance constraints, 

In Section 12.2, expressions for the calculation of the chemical equilibrium constant were derived (Eqs. (12.18-12.23)), starting from the knowledge that the Gibbs energy shows a minimum. So, it should also be possible to skip the auxiliary chemical equilibrium constant K and to determine the minimum of the Gibbs energy directly 

Using the standard Gibbs energies in the ideal gas state, this means = l atm and assuming ideal gas behavior, the required fugacities (atm) can be expressed by the partial pressures p, (atm)  

This means that Eq. (12.32) can be replaced by the following relation to calculate the total Gibbs energy at a given temperature T  

In chemical equilibrium, the total Gibbs energy should show a minimum. For a simple reversible reaction, this can be shown graphically (see Example 12.7). 

In the case of multiple reaction equilibria, the number of moles of all reactive compounds in chemical equilibrium can be determined with the help of nonlinear regression methods [11]. But at the same time, the element balance has to be satisfied this means the amount of carbon, hydrogen, oxygen, nitrogen has to be the same before and after the reaction. This can either be taken into account with the help of Lagrange multipliers or using penalty functions [11], as shown in Example 12.8. 

Most of the mercury is therefore in the form of chloride and hydroxide complexes. 

The three residual components of /(x) with the largest deviation are 

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When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

This is a 4 2 reaction, and is thus pressure dependent. However, it is necessary to compute the equilibrium partial pressure of some alternative gaseous species, such as SiCls, and other hydrocarbons such as C2H2 and for this a Gibbs energy minimization calculation should be made. 

Kulik, D. A., 2002, Gibbs energy minimization approach to model sorption equilibria at the mineral-water interface Thermodynamic relations for multi-site-surface complexation. American Journal of Science 302,227-279. ... 

Gibbs energy minimization has also predicted negative isobaric expansion coefficients for certain crystalline zeolite framework structures, which subsequently were confirmed experimentally [6], Many solids show negative thermal expansion at very low temperatures, including even some alkali halides (Barron and White (Further reading)). Many other solids on heating expand in some directions and contract in others. 

III. Gibbs energy minimization formalism. Fluid phase equilibria, 218 (1), 15-31. 

Various authors have modified this classical procedure by introducing the so-called stability test based on the Gibbs energy minimization [19-23]. 

In this section two prediction techniques are discussed, namely, the gas gravity method and the Kvsi method. While both techniques enable the user to determine the pressure and temperature of hydrate formation from a gas, only the KVSI method allows the hydrate composition calculation. Calculations via the statistical thermodynamics method combined with Gibbs energy minimization (Chapter 5) provide access to the hydrate composition and other hydrate properties, such as the fraction of each cavity filled by various molecule types and the phase amounts. 

The thesis of Ballard (2002) details this calculation method, which includes multiphase systems, solid phases including ice and salts, and thermodynamic inhibition. The CSMGem (the last three initials are the first letters of Gibbs energy minimization ) User s Manual, included in the CD in the endpapers, and the examples of hydrate calculation shown in the Appendix A, enable the reader to use the CD programs. 

The Gibbs energy minimization method allows for calculations of the formation conditions for any phase (including the hydrate). It also allows for the calculation of phases present at any T and P (whether hydrates are present or not). Therefore, included are the options to perform all thermodynamic calculations with every phase and not just the hydrate. The types of calculations, combined with plotting capability, included in CSMGem are... 

The statistical thermodynamic method and the Gibbs energy minimization presented in this chapter represents the state-of-the-art for the prediction of the following types of phase equilibria ... 

Gibbs Energy Minimizer (GEMINI2), THERMODATA, Grenoble, Erance, 1994. 

Two approaches to the treatment of large systems have been developed the Gibbs energy minimization method and the equilibrium constant approach. Both are based on a knowledge of the chemical potentials of species under standard conditions and under actual conditions of composition of conditions in the chemical system. In the Gibbs energy minimization approach (Clason,... 

Today, all modem hydrate prediction programs are carried out using this new technique, which is based on Gibbs energy minimization. It is extremely useful to determine phase firaction in the aforementioned cases of two-phase hydrate formation (without a free water phase), one can use the Gibbs energy minimization to estimate whether there will be a sufficient hydrate amount to form a blockage in a process. 

The mathematical formulation leads to a constrained Gibbs energy minimization problem, subject to conservation of the total amounts of the individual chemical elements that make up the chemical species This constraint is incorporated into the problem via the method of Lagrange multipliers Details of the procedure are given elsewhere W ... 

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