Multicomponent systems energy

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

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

Distillation Columns. Distillation is by far the most common separation technique in the chemical process industries. Tray and packed columns are employed as strippers, absorbers, and their combinations in a wide range of diverse appHcations. Although the components to be separated and distillation equipment may be different, the mathematical model of the material and energy balances and of the vapor—Hquid equiUbria are similar and equally appHcable to all distillation operations. Computation of multicomponent systems are extremely complex. Computers, right from their eadiest avadabihties, have been used for making plate-to-plate calculations. 

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... 

During this period of intensive development of unit operations, other classical tools of chemical engineering analysis were introduced or were extensively developed. These included studies of the material and energy balance of processes and fundamental thermodynamic studies of multicomponent systems. 

Another very important technique for fundamental consideration of multicomponent systems is low energy ion scattering (LEIS) [Taglauer and Heiland, 1980 Brongersma et al., 2007]. This is a unique tool in surface analysis, since it provides the ability to define the atomic composition of the topmost surface layer under UHV conditions. The signal does not interfere with the subsurface atomic layers, and therefore the results of LEIS analysis represent exclusively the response from the outer surface. In LEIS, a surface is used as a target that scatters a noble gas ion beam (He, Ne, ... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

The histogram reweighting methodology for multicomponent systems [52-54] closely follows the one-component version described above. The probability distribution function for observing Ni particles of component 1 and No particles of component 2 with configurational energy in the vicinity of E for a GCMC simulation at imposed chemical potentials /. i and //,2, respectively, at inverse temperature ft in a box of volume V is... 

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 extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

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... 

There is no well developed theory which describes the sputtering of multicomponent systems (e.g., oxides), but some authors (sec, for example, Ref. 88) have assumed that the results of Sigmund s theory are valid and have applied reasonable values for the stopping power and binding energy to obtain results for various multi-component systems. 

The objective of this review is to characterize the excimer formation and energy migration processes in aryl vinyl polymers sufficiently well that the excimer probe may be used quantitatively to study polymer structure. One such area of application in which some measure of success has already been achieved is in the analysis of the thermodynamics of multicomponent systems and the kinetics of phase separation. In the future, it is likely that the technique will also prove fruitful in the study of structural order in liquid crystalline polymers. 

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... 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

Let us first recap the general criteria for spinodals and critical points in multicomponent systems. In order to treat the criteria derived from the exact and moment free energies simultaneously, we use the common notation p for the vector of densities specifying the system For the exact free energy, the components of p are the values p ) for the moment free energy, they are the reduced set pt. We write the corresponding vector of chemical potentials 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... 

Originally, extractive distillation was limited to two-component problems. However, recent developments in solvent technology enabled applications of this hybrid separation in multicomponent systems as well. An example of such application is the BTX process of the GTC Technology Corp., shown in Figure 6, in which extractive distillation replaced the conventional liquid-liquid extraction to separate aromatics from catalytic reformate or pyrolysis gasoline. This led to a ca. 25% lower capital cost and a ca. 15% decrease in energy consumption (170). Some other examples of existing and potential applications of the extractive distillations are listed in Table 6. 

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... 

We consider a two-phase, multicomponent system in which there is one planar surface. The state of the system is defined by assigning values to the entropy and volume of the system, the area of the surface, and the mole numbers of the components. The differential of the energy of the system is... 

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... 

We choose the total system to be the condenser and the entire dielectric medium. The condenser is immersed in the medium which, for purposes of this discussion, is taken to be a single-phase, multicomponent system. The pressure on the system is the pressure exerted by the surroundings on a surface of the dielectric. In setting up the thermodynamic equations we omit the properties of the metal plates, because these remain constant except for a change of temperature. The differential change of energy of the system is expressed as a function of the entropy, volume, and mole numbers, but with the addition of the new work term. Thus,... 

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