Multicomponent systems Gibbs energy

In the case of a siagle-phase, multicomponent system undergoiag just a single reaction, the total Gibbs energy is as foUows ... 

A simple example of the analysis of multicomponent systems will suffice for the present consideration, such as the calculation of the components in a gaseous mixture of oxygen, hydrogen and sulphur. As a first step, the Gibbs energy of formation of each potential compound, e.g. S2, H2S, SO, SO2, H2O etc. can be used to calculate the equilibrium constant for the formation of each compound from the atomic species of the elements. The total number of atoms of each element will therefore be distributed in the equilibrium mixture in proportion to these constants. Thus for hydrogen with a starting number of atoms and the final number of each species... 

Brinkley (1947) published the first algorithm to solve numerically for the equilibrium state of a multicomponent system. His method, intended for a desk calculator, was soon applied on digital computers. The method was based on evaluating equations for equilibrium constants, which, of course, are the mathematical expression of the minimum point in Gibbs free energy for a reaction. 

The Kohler model is a general model based on linear combination of the binary interactions among the components in a mixture, calculated as if they were present in binary combination (relative proportions) and then normalized to the actual molar concentrations in the multicomponent system. The generalized expression for the excess Gibbs free energy is... 

Chemical Equilibrium The chemical equilibrium approach is more complex computationally than applying the assumption of an infinitely fast reaction. The equilibrium composition of a multicomponent system is estimated by minimizing the Gibbs free energy of the system. For a gas-phase system with K chemical species, the total Gibbs free energy may be written as... 

Liquid-Solution Models. The simple-solution model has been used most extensively to describe the dependence of the excess integral molar Gibbs energy, Gxs, on temperature and composition in binary (142-144, 149-155), quasi binary (156-160), ternary (156, 160-174), and quaternary (175-181) compound-semiconductor phase diagram calculations. For a simple multicomponent system, the excess integral molar Gibbs energy of solution is expressed by... 

The thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed. 

When the state of a system is defined by assigning values to the necessary independent variables, the values of all of the thermodynamic functions are fixed. For a single-phase, multicomponent system the independent variables are usually the temperature, pressure, and mole numbers of the components. The Gibbs energy of such a system at a given temperature and pressure is additive in the chemical potentials of the components by Equation (5.62),... 

The equations are derived from the differential of the Gibbs energy for a one-phase, multicomponent system Equation (2.33). The differential is exact, and therefore the condition of exactness must be satisfied. Two equations... 

The second subject is the effect of the surface on the chemical potential of a component contained in a small drop. We consider a multicomponent system in which one phase is a bulk phase and the second phase is kept constant with the conditions that the interface between the two phases is contained wholly within the bulk phase and does not affect the external pressure. The differential of the Gibbs energy of a two-phase system may be written as... 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

Chen, C-C, et al., A Local Composition Model for the Excess Gibbs Energy of Multicomponent Aqueous Systems, AlChE Annual Meeting, San Francisco, 1984. 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

Here, we only present the simplest thermodynamic expressions used in the CALPHAD method for the major phase classes observed in multicomponent systems namely, disordered miscible and immiscible phases and ordered sublattice phases. The reader is referred to specialized textbooks for further discussion. The Gibbs energies for disordered two-component solid and liquid solution phases are most easily represented by the regular solution model (Eq. 2.10) or one of its variants ... 

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. 

In the theoretical treatment of diffusive reactions, one usually works with diffusion coefficients, which are evaluated from experimental measurements. In a multicomponent system, a large number of diffusion coefficients must be evaluated, and are generally interrelated functions of alloy composition. A database would, thus, be very complex. A superior alternative is to store atomic mobilities in the database, rather than diffusion coefficients. The number of parameters which need to be stored in a multicomponent system will then be substantially reduced, as the parameters are independent. The diffusion coefficients, which are used in the simulations, can then be obtained as a product of a thermodynamic and a kinetic factor. The thermodynamic factor is essentially the second derivative of the molar Gibbs energy with respect to the concentrations, and is known if the system has been assessed thermodynamically. The kinetic factor contains the atomic mobilities, which are stored in the kinetic database. 

The fundamental thermodynamic expression for the infinitesimal increase in Gibbs energy for a multicomponent system which can exchange matter with its surroundings was derived earlier on the basis of the first and second laws of thermodynamics (equation (1.3.21)) ... 

Thus, for phase equilibrium to exist in a closed, nonreacting multicomponent system at constant energy and volume, the pressure must be the same in both phases (so that mechanical equilibrium exists), the temperature must be the same in both phases (so that thermal equilibrium exists), and the partial molar Gibbs energy of each species must be the same in each phase fso that equilibrium with respect to species diffusion exists). ... 

Wilson" has proposed that the excess Gibbs energy of a multicomponent system is given by... 

In single-component systems (or pure substances), the chemical composition in all phases is the same. In multicomponent systems, the chemical composition of a given phase changes in response to pressure and temperature changes and these compositions are not the same in all phases. For single-component systems, first-order phase transitions occur with a discontinuity in the first derivative of the Gibbs free energy. In the transitions, T and p remain constant. 

Thus the chemical potential corresponds to the change in Gibbs energy of a homogeneous multicomponent system on the introduction of an infinitesimal amount of a component into the mixture at constant p, T and constant amounts of the other eomponents. 

For each phase in a multiphase, multicomponent system, the Gibbs free energy is given functionally as... 

Consider the system of the type shown in Fig. 18.5(a) two phases with a plane interface between them. Since the interface is plane, we have Pi = P2 = P and the Gibbs energy becomes a convenient function. If we have a multicomponent system the chemical potential of each component must have the same value in each phase and at the interface. The variation in total Gibbs energy of the system is given by... 

The thermodynamic potential that is used to analyze the phase behavior of multicomponent systems at constant temperatnie and pressure is the Gibbs free energy. C, A spontaneous change will take place any time it is associated with a reduction in free energy (AG < 0). 

When a multicomponent system consists of multiple coexisting phases, the compositions in each phase are such that the Gibbs energy is the lowest possible. This is the basis for the calculation of multicomponent phase equilibrium and is discussed in Chapter lo. 

This is a statement of fundamental importance in thermodynamics and the starting point for the discussion of phase equilibrium. To apply it we must first develop expressions for the Gibbs free energy of multicomponent systems. We begin by writing the differential of G in the standard form,... 

Thermodynamic calculations were performed using THERMO software package to calculate the adiabatic combustion temperatures and product phase distribution. This program is based on optimization of Gibbs free energy of multiphase and multicomponent systems. The gases are assumed to be ideal and condensed phases completely immiscible [19],... 

Wilson equation Interaction method for the excess Gibb s energy suitable for totally miscible systems, not applicable for systems with limited miscibility since only the binary parameters are used, applicable to multicomponent systems only valid for small and medium operating pressures Wilson, G.M., J. Am. Chem. Soc. 86 (1964) 127. 

Calculation of phase equilibria in a multicomponent system is obtained by the mmimizaticm of foe total Oibbs mergy, where G is a summation of the Gibbs energy of all phases that take part in each equilibrium as is eiqiressed by equation (1) ... 

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