Phase equilibrium problem formation

This scenario is supported in a number of works on phase equilibrium problems near the phase boimdaiy (see reviews [214-218] and referenees in them). These works, using a large number of different objects (magnets, low moleeular weight liquids, solutions and polymer mixtures) as the examples, prove that the struetural reeonstructions in the volume are related to the processes near the phase boundary in a very eomplex way. Sometimes this relation results in the formation of anisotropic structures in the volume [217,218]. Out of the whole variety of theoretical and experimental results in this field, let us consider the numerical dependences of dispersion concentration as the funetion of the distance from the surface, obtained in the scaling approximation by de-Zhen [32]. When studying the behavior of the semidiluted solution in contact with the impenetrable wall, he showed that... 

STANJAN The Element Potential Method for Chemical Equilibrium Analysis Implementation in the Interactive Program STANJAN, W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the element potentials ) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates. 

Various approaches have been employed to tackle the problem of micelle formation. The most simple approach treats micelles as a single phase, and this is referred to as the phase-separation model. In this model, micelle formation is considered as a phase-separation phenomenon, and the cmc is then taken as the saturation concentration of the amphiphile in the monomeric state, whereas the micelles constitute the separated pseudophase. Above the cmc, a phase equilibrium exists with a constant activity of the surfactant in the micellar phase. The Krafft point is viewed as the temperature at which a solid-hydrated surfactant, the micelles, and a solution saturated with undissociated surfactant molecules are in equiUbrium at a given pressure. 

In synthetic methods, a mixture of known composition is prepared and the phase equilibrium is observed subsequently in an equilibrium cell (the problem of analyzing fluid mixtures is replaced by the problem of synthesizing them). After known amounts of the components have been placed into an equilibrium cell, pressme and temperature are adjusted so that the mixture is homogeneous. Then temperature or pressure is varied until formation of a new phase is observed. This is the common way to observe cloud points in demixing polymer systems. No sampling is necessary. Therefore, the experimental equipment is often relatively simple and inexpensive. For multicomponent systems, experiments with synthetic methods yield less information than with analytical methods, because the tie lines carmot be determined without additional experiments. This is specially trae for polymer solutions where fractionation accompanies demixing. 

The processes discussed in this chapter demonstrate the great variety of phase equilibrium that can arise beyond the basic vapor-liquid problems discussed in most of the previous chapters. Many other systems could be included The adsorption of gases onto solids (used in the removal of pollutants from air), the distribution of detergents in water/oil systems, the wetting of solid surface by a liquid, the formation of an electrochemical cell when two metals make contact are all examples of multiphase/multicomponent equilibrium. They all share one important common element their equilibrium state is determined by the requirement that the chemical potential of any species must be the same in any phase where the species can be found. These problems are beyond the scope of this book. The important point is this The mathematical development of equilibrium (Chapter 10) is extremely powerful and encompasses any system whose behavior is dominated by equilibrium. 

The model framework for describing the void problem is schematically shown in Figure 6.3. It is, of course, a part of the complete description of the entire processing sequence and, as such, depends on the same material properties and process parameters. It is therefore intimately tied to both kinetics and viscosity models, of which there are many [3]. It is convenient to consider three phases of the void model void formation and stability at equilibrium, void growth or dissolution via diffusion, and void transport. 

Zeolite crystallization represents one of the most complex structural chemical problems in crystallization phenomena. Formation under conditions of high metastability leads to a dependence of the specific zeolite phase crystallizing on a large number of variables in addition to the classical ones of reactant composition, temperature, and pressure found under equilibrium phase conditions. These variables (e.g., pH, nature of reactant materials, agitation during reaction, time of reaction, etc.) have been enumerated by previous reviewers (1,2, 22). Crystallization of admixtures of several zeolite phases is common. Reactions involved in zeolite crystallization include polymerization-depolymerization, solution-precipitation, nucleation-crystallization, and complex phenomena encountered in aqueous colloidal dispersions. The large number of known and hypo-... 

To develop the kinetic equations in condensed phases the master equation must be formulated. In Section 3 the master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The latter set of equations permits consideration of history of formation of the local solid structure as well as its influence on the subsequent elementary stages. The many-body problem and closing procedure for kinetic equations are discussed. The influence of fast and slow stages on a closed system of equations is demonstrated. The multistage character of the kinetic processes in condensed phase creates a problem of self-consistency in describing the dynamics of elementary stages and the equilibrium state of the condensed system. This problem is solved within the framework of a lattice-gas model description of the condensed phases. 

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