Chemical equilibrium phases

Fig. 4. Plots of initial mineralization rates (IMR) versus equilibrium aqueous phase concentrations for biphenyl-degrading bacteria. Cap B, Sch A, Cap A, and Col A indicate soils used in the study. Letters A and B indicate A and B soil horizon, respectively. Cap, Sch, and Col represent Capac, Schoolcraft, and Colwood soil, respectively. Reprinted with permission from Feng et al. (2000). Copyright (2000) American Chemical Society.

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

From the intersection of the two surfaces representing the hadron and the quark phase one can calculate the equilibrium chemical potentials of the mixed phase,... 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

A phase is a region of space in which the intensive properties vary continuously as a function of position. The intensive properties change abruptly across the boundary between phases. For equilibrium between phases, the chemical potential of any species is the same in all phases in which it exists. 

The last point we consider in this section is the question of whether micellization should be viewed in terms of chemical reaction equilibrium or phase equilibrium. If we think of micellization as a chemical reaction, then Reaction (A) should surely be written as a sequence of stepwise additions ... 

The condition for thermodynamic equilibrium between phases is that the species chemical potentials are equal in each of the phases. Thus, at equilibrium,... 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

In concentrated Al-Zn alloys, the kinetics of precipitation of the equilibrium 0 phase from a are too rapid to allow the study of spinodal decomposition. An Al-22 at. % Zn alloy, however, has decomposition temperatures low enough to permit spinodal decomposition to be studied. For Al-22 at. % Zn, the chemical spinodal temperature is 536 K and the coherent spinodal temperature is 510 K. The early stages of decomposition are described by the diffusion equation... 

So far it has not been possible to measure the chemical potentials of the components in the mesophases. This measurement is possible, however, in solutions which are in equilibrium with the mesophases. If pure water is taken as the standard state, the activity of water in equilibrium with the D and E phases in the system NaC8-decanol-water is more than 0.8 (4). From these activities in micellar solutions, the activity of the fatty acid salt has sometimes been calculated. The salt is incorrectly treated as a completely dissociated electrolyte. The activity of the fatty acid in solutions of short chain carboxylates has also been determined by gas chromatography from these determinations the carboxylate anion activity can be determined (18). Low CMC values for the carboxylate are obtained (15). The same method has shown that the activity of solubilized pentanol in octanoate solutions is still very low when the solution is in equilibrium with phase D (Figure 10) (15). 

For condensed phases, it is the fugacity in the equilibrium vapor phase (vapor fugacity or very nearly vapor pressure) that gives the fugacity of the condensed phase. Equation (11.39) applies to relate this vapor fugacity to the chemical potential in the condensed phase. 

Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares). 

Solubility normally is measured by bringing an excess amount of a pure chemical phase into contact with water at a specified temperature, so that equilibrium is achieved and the aqueous phase concentration reaches a maximum value. It follows that the fugacities or partial pressures exerted by the chemical in these phases are equal. Assuming that the pure chemical phase properties are unaffected by water, the pure phase will exert its vapor pressure Ps (Pa) corresponding to the prevailing temperature. The superscript S denotes saturation. In the aqueous phase, the fugacity can be expressed using Raoult s Law with an activity coefficient y ... 

Four possible equilibrium situations may exist, depending on the nature of the chemical phase, each of which requires separate theoretical treatment and leads to different equations for expressing solubility. These equations form the basis of the correlations discussed later. 

The fundamental condition for equilibrium between phases, the equality of the chemical potential of a component in every phase in which the component is present still applies. We then substitute for the chemical potential of a component the equivalent chemical potential of the same substance in terms of species or appropriate sums of chemical potentials of the species, as determined by the methods of Section 8.15 and used in the preceding sections. Several examples are discussed in the following paragraphs. 

Most speciation modelling is based on the assumption of thermodynamic equilibrium between phases, so it is necessary to describe the various equations that are used to quantify these chemical reactions. 

The chemical potential of a homogeneous material (a phase) is a function of two intensive variables, usually chosen as temperature and pressure. We say that such a material has two degrees of freedom (i.e., we are free to set two intensive variables). (Note that only intensive variables count as degrees of freedom.) In addition to being able to specify a number of intensive variables equal to the number of degrees of freedom of a system, we are also at liberty to specify the size of the phase with one extensive variable. The chemical potential can be represented as a surface on a plot of p versus P and T. The condition for equilibrium between phase a and phase p is, according to Eq. (24),... 

The influence of equilibrium chemical reactions on vapor-liquid phase diagrams. Chem. Engng. Sci.,... 

A comprehensive discussion of the most important model parameters covers phase equilibrium, chemical equilibrium, physical properties (e.g., diffusion coefficients and viscosities), hydrodynamic and mass transport properties, and reaction kinetics. The relevant calculation methods for these parameters are explained, and a determination technique for the reaction kinetics parameters is represented. The reaction kinetics of the monoethanolamine carbamate synthesis is obtained via measurements in a stirred-cell reactor. Furthermore, the importance of the reaction kinetics with regard to axial column profiles is demonstrated using a blend of aqueous MEA and MDEA as absorbent. 

The driving force of diffusion is the gradient of the chemical potential. In the case of singlecomponent diffusion in a zeolite, the chemical potential can be related to the concentration by considering the equilibrium vapor phase [12,20]... 

Yield-Based Reactor Fractional Conversion Reactor Combined Specification Model Well-Stirred Reactor Model Plug Flow Reactor Model Two Phase Chemical Equilibrium General Phase and Chemical Equilibrium... 

The classification of separations should reflect the patterns of component transport and equilibrium that develop in the physical space of the system. The transport equations show that we have two broad manipulative controls that can be structured variously in space to affect separative transport. First is the chemical potential which controls both relative transport and the state of equilibrium. Chemical potential, of course, can be varied as desired in space by placing different phases, membrane barriers, and applied fields in appropriate locations. A second means of transport control is flow, which can be variously oriented with respect to the phase boundaries, membranes, and applied fields—that is, with respect to the structure of the chemical potential profile. 

The phase rule of Josiah Willard Gibbs (1839-1903) gives the general conditions for chemical equilibrium between phases in a system. At equilibrium, AG = 0, there is no further change with time in any of the system s macroscopic properties. It is assumed that surface, magnetic, and electrical forces may be neglected. In this case, the phase rule can be written as... 

When a piece of aluminum is added to clear, transparent molten cryolite, foglike streamers spread out from the metal and they soon render the melt completely opaque [188-195], It is clear that the metal fog is not a stable chemical phase in equilibrium with the electrolyte, since it dissipates when it rises from the molten... 

One of the most fundamental problems of chemical physics is the study of the forces between atoms and molecules. We have seen in many preceding chapters that these forces are essential to the explanation of equations of state, specific heats, the equilibrium of phases, chemical equilibrium, and in fact all the problems we have taken up. The exact evaluation of these forces from atomic theory is one of the most difficult branches of quantum theory and wave mechanics. The general principles on which the evaluation is based, however, are relatively simple, and in this chapter we shall learn what these general principles are, and see at least qualitatively the sort of results they lead to. 

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