Thermodynamic Gibbs free energy , minimization

Typically, solving (5.151) to find fc(oo ) is not the best approach. For example, in combusting systems Srp(0 4)1 < 1 so that convergence to the equilibrium state will be very slow. Thus, equilibrium thermodynamic methods based on Gibbs free-energy minimization are preferable for most applications. 

Thermodynamic calculations presented here are based on Gibbs free energy minimization and were carried out using HSC Chemistry. The equilibrium amount of each species that is formed is normalized on the basis of one mole of n-Ci6, a model compound for diesel fuel, fed to the reactor. Carbon formation is a function of both the S/C ratio and reforming temperature. Figure 17 shows the minimum amount of S/C ratio thermodynamically required for carbon-free SR of n-Ci6 at a given temperature. Carbon-free operation of n-Cig is thermodynamically possible above the curve. Higher temperatures and S/C ratios inhibit carbon formation. 

Thermodynamic software packages may be used to find equilibrium compositions at prescribed temperatures and pressures. Such calculations require knowledge of feed components and products and their thermodynamic properties and are based on Gibbs free energy minimization techniques. Examples of thermodynamic packages may be found in Smith and Missen (Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982) and in Walas (Phase Equilibria in Chemical Engineering, Butterworths, 1985). 

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 this way, the routine Gibbs free energy minimization program can still be used, but thermodynamic data of ordinary graphite should be substituted by thermodynamic data of activated graphite, as listed in Table 1 for the carbon-hydrogen system. The derivation of these data cem be found in our previous papers and books [13-19], and will not be repeated here again. 

A complete description of any groundwater system necessitates consideration of reactions between rock forming minerals and the aqueous phase. This cannot be achieved without accurate thermodynamic properties of both the participating aluminosilicate minerals and aqueous aluminum species. Most computer codes used to calculate the distribution of species in the aqueous phase utilize the "reaction constant" approach as opposed to the "Gibbs free energy minimization" approach (3). In the former, aluminosilicate dissolution constants are usually written in terms of the aqueous aluminum species, Al, which is related to other aqueous aluminum species by appropriate dissociation reactions. 

The thermodynamic equilibrium compositions can be determined by using an algorithm based on Gibbs free energy minimization, only taking into account the chemical species, that is, reactants and products. Results obtained with a water/ethanol inlet molar ratio of 3, corresponding to the stoichiometry of reaction (24.4), are shown m Figure 24.1. Except carbon, all compounds present in Eqs (24.3)-(24.13) are Included m the thermodynamic calculations. 

In other words, the system Gibbs free energy is optimized in the equilibrium state. 1 leave for the reader to prove that, as a matter of fact, the Gibbs free energy is minimized in the equilibrium. This last result is a consequence of the second law of thermodynamics, which states that the Gibbs free energy minimizes in the equilibrium for systems held at constant pressure and temperature (Planck 1945). 

Tang H, Kitagawa K. 2005. Supercritical water gasification of biomass Thermodynamic analysis with direct Gibbs free energy minimization. Chem Eng J 106 261—267. 

Equilibrium combustion product compositions and properties may be readily calculated using thermochemical computer codes which minimize the Gibbs free energy and use thermodynamic databases... 

The calculation is based on the rule of thermodynamics, which states that a system will be in equilibrium when the Gibbs free energy is at a minimum. Cl The objective then is the minimization of the total free energy of the system and the calculation of equilibria at constant temperature and volume or at constant pressure. It is a complicated and lengthy calculation but, fortunately, several computer programs are now available that considerably simplify the task. PI... 

Naslain, R., Thebault, J., Hagenmuller, P., and Bernard, C., The Thermodynamic Approach to Boron CVD Based on the Minimization of the Total Gibbs Free Energy, J. Less Common Metals, 67(1) 85-100 (1979)... 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Equilibrium combustion product compositions and properties may be readily calculated using thermochemical computer codes which minimize the Gibbs free energy and use thermodynamic databases containing polynomial curve-fits of physical properties. Two widely used versions are those developed at NASA Lewis (Gordon and McBride, NASA SP-273, 1971) and at Stanford University (Reynolds, STANJAN Chemical Equilibrium Solver, Stanford University, 1987). 

Stripping of chlorine from hydroxides such as Cl2Sn(OH)2 could eventually lead to gas-phase SnO or Sn02. However, at the relatively low temperatures typical of tin oxide CVD ( 873-973 K), we do not expect these oxides to form, based on the equilibrium calculations described above. Thus, the formation of tin hydroxides is not only thermodynamically favored (i.e., based on minimization of the Gibbs free energy), but there are also exothermic reaction pathways that we expect to be kinetically favorable. The primary tin carrier in the CVD process could therefore be a tin hydroxide. Complete conversion to Sn02 would most likely occur via reactions on the surface. 

As is well known from thermodynamics,6 a system will be in equilibrium when the Gibbs Free Energy is at a minimum. Therefore, all that is necessary is to express the Gibbs Free Energy in terms of the degree of completion of the reaction, and then minimize that function. The Gibbs Free Energy can be expressed as follows,7... 

At constant external pressure the enthalpy is the relevant thermodynamic potential for the Boltzmann distribution. A large area A implies that the distribution of the thicknesses is confined near the minimum of the enthalpy, while a small value of A corresponds to large thickness fluctuations. The first case corresponds to a low enthalpy, but a large entropy, whereas the second to a large enthalpy, but a low entropy. The real distribution is provided by the minimization of the Gibbs free energy with respect to A at constant external pressure II. 

Figure 2. The enthalpy H and its harmonic approximation U, in the vicinity of the minimum, for planar, rigid surfaces separated by a distance d (CE = 0.1 M, 11= 1 x 104 N/m2, T = 3.3 x 10 mol/m2, KD = 0.5 M, b, = 3.08 x 10 22J,b2= 6.28 x 10 14 J/m, b3 = 8.28 x 107 J/m, 64 = 6.13 x 1016 J/m2, br, = -9.00 x 10 23 J, and T = 300 Kj. pi is the distribution of the intersurface distances for interfaces with bending modulus Kc = 2 kT interacting via the potential C/h. This distribution coincides with the Boltzmann distribution of finite pieces of area A. p2 is the Boltzmann distribution of the pieces of area A, but now the enthalpy H and not its harmonic approximation U, is the thermodynamic potential. p3 is calculated using for A the value obtained from the minimization of the Gibbs free energy (eq 11a).

The translationally ordered state characteristic of crystallites that are large with respect to the wavelengths of X-rays represents perfect phases in the sense of thermodynamics, to which phase diagrams apply. Even this idealized state of matter cannot exist without deviations from perfect ordering, however, because of a requirement of thermodynamics (the entropy of the material at equilibrium must be nonzero for the Gibbs free energy to be minimized). Thus, the material will contain a number of deviations from the ideal arrangement, called defects."... 

NPT systems that minimize Gibbs free energy maximize Z. Therefore G is a useful thermodynamic potential for NPT systems NPT systems spontaneously move down gradients in Gibbs free energy. 

CVD normally involves a multi-component and a multi-phase system. There are various ways to calculate thermodynamic equilibrium in multicomponent systems. The following is a brief discussion of the optimization method where the minimization of Gibbs free energy can be achieved. The free energy G of a system consisting of m gaseous species and s solid phases can be described by. 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

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