Principles of phase equilibria

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

The basic principles of phase equilibria are discussed in the chapter covering thermodynamics (Chapter 4). In general, two or more phases are in equilibrium when they have the same temperatme, pressure, and fugacity (or, equivalently, chemical potential) for each species. 

There are two types of lipid-water phase diagrams. The first type, discussed above, is obtained from polar lipids, which are insoluble in water (i.e. the solubility is quite small, monolaurin for example has a solubility of about 10 m). Fig. 8.12 illustrates the principles of phase equilibria in this type of lipid-water system. The second type of binary system is obtained when the lipid is soluble as micelles in water. Examples of such lipids are fatty acid salts and lysolecithin. An aqueous soap system is illustrated in Fig. 8.13. When the lipid concentration in the micellar solution is increased, the spherical micelles are transformed into rod-shaped micelles. At still higher lipid concentrations the lipid cylinders are hexagonally arranged and the liquid-crystalline phase Hi is formed. The lamellar liquid-crystalline phase is usually formed in the region between Hi and the anhydrous lipid. Excellent reviews of the association behaviour of amphiphiles of this type have been published (Wennerstrom and Lindman, 1979 Lindman and Wennerstrom, 1980). 

Principles of phase equilibria after Gibbs (5) and Van der Waals (6) supply the ordering procedure for these phenomena. The application of these principles has been greatly stimulated by the work of Bakhuis Roozeboom (7). 

F. E. W. Wetmore and D. J. LeRoy, Principles of Phase Equilibria , McGraw-Hill, New York, 1950. 

The Kelvin equation is derived from the Young-Laplace equation and the principles of phase equilibria. It gives the vapour pressure, P, of a droplet (curved surface) over the ordinary vapour pressure (P ) for a flat surface (see Appendix 4.2 for the derivation) ... 

The history of CALPHAD is a chronology of what can he achieved in the field of phase equilibria by combining basic thermodynamic principles with mathematical formulations to describe the various thermodynamic properties of phases. The models and formulations have gone through a series of continuous improvements and, what has become known as the CALPHAD approach, is a good example of what can be seen as a somewhat difficult and academic subject area put into real practice. It is indeed the art of the possible in action and its applications are wide and numerous. 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

For the understanding of these processes and for the design and evaluation of new separation processes, it is crucial to get a better insight in the underlying principles and phase equilibria. Especially in the investigation of complex systems such as polydisperse polymers with additives or drugs with impurities, it is necessary to get information on the composition of the coexisting phases. Up to now, there is almost no such information available. 

In principle, mixtures containing a very large number of components behave in a way described by the same general laws that regulate the behavior of mixtures containing only a comparatively small number of components. In practice, however, the procedures for the description of the thermodynamic and kinetic behavior of mixtures that are usually adopted for mixtures of a few components rapidly become cumbersome in the extreme as the number of components grows. As a result, alternate procedures have been developed for multicomponent mixtures. Particularly in the field of kinetics, and to a lesser extent in the field of phase equilibria thermodynamics, there has been a flurry of activity in the last several years, which has resulted in a variety of new results. This article attempts to give a reasoned review of the whole area, with particular emphasis on recent developments. 

This chapter, then, deals primarily with the directed metal oxidation process, although selected examples of stability in metal matrix composites are also discussed briefly. The focus is, of course, on the applications of phase equilibria, and more generally, thermodynamic principles that are applicable to the formation of composites in the presence of molten metals. Because these general principles are the same regardless of whether the end product is an MMC or a CMC, little generality is lost by focusing the discussion on CMCs formed by directed metal oxidation. 

This handbook begins with coverage of principles that intersect most separation processes phase equilibria, mass transfer, and phase segregation. Separation processes cannot be understood without at least some acquaintance with the driving force for such processes—hence the discussion of phase equilibria—and with factors influencing... 

The objective of this chapter and the two that follow is to illustrate how the principles introduced in Chapter 8 for the thermodynamic description of mixtures together with the calculational procedures of Chapter 9 can be used to study many different types of phase equilibria important in chemical engineering practice. In particular, the following are considered ... 

The prerequisite of all types of extraction processes is the existence of a large miscibility gap between raffinate and extract. The thermodynamic principles of phase equilibrium are dealt with in Chap. 2. An extensive collection of liquid-liquid equilibria is given in the Dechema Data Collection (Sorensen and Arlt 1980ff). Volume 1 contains data of miscibility gaps of binary systems. Phase equilibrium data (miscibility gaps and distribution equilibrium) of ternary and quaternary mixtures are listed in volumes 2-7. 

The basic principles of mass transfer are discussed in detail in [1.95-1.97]. Thermal separation processes are actually mass transfer processes matter is transported between phases and across phase interfaces. Mass transfer is caused by differences in concentration within a phase and by disturbances of the phase equilibrium. The time taken to return to the phase equilibrium depends mainly on mass transfer, but also on heat transfer (heat is transported not only by convection and radiation at higher temperature, but also by mass). For the design of thermal separation processes, along with a knowledge of phase equilibria, it is also important to have a detailed understanding of how equilibrium is reached and the time required, taking into account restrictions in the mass transfer rate. 

For petroleum fractions or similar systems the treatment of phase equilibria will now be discussed briefly. The basic principles are the same as those outlined for polymer systems without recourse to segment-molar quantities. For petroleum fractions the phase-equilibrium problem of importance is the so-called flash calculation that is analogous to the calculation of coexistence curves for a polydisperse polymer solution and in the simplest case a single distribution function is required. For example, the system may contain many alkanes characterized by their normal boiling-point temperatures Tb that in this work will be denoted by x. At moderate pressures the equilibrium condition is given by the continuous thermodynamics form of Raoult s law ... 

One of the specific features of these polymer systems is a low mobility of the macromolecules and correspondingly slow phase transition rates. This enables one to use, in analyzing such systems, composite phase diagrams showing all the types of phase equilibria inherent in a given system. Extension of this principle to the systems "rigid-chain- polymer-solvent" makes it possible to construct a phase diagram which combines (a) equilibrium with the formation of a crystalline phase, (b) equilibrium with the formation of liquid-crystalline phases, and (c) equilibrium with the formation of amorphous (liquid) phases. 

The process of establishment of phase equilibrium is sometimes lengthy. For this reason, the successive attainment of some thermodynamic equilibrium state, whose phases are metastable with respect to the next equUibrium state with a lower level of free energy of the system, is possible. This new equilibrium can be replaced by another equilibrium (attainment of the next free energy minimum). The principle of the mutual independence of the individual types of phase equilibria, i.e., the possibility of the existence of equilibrium phases in the system independently of the next transition to a new, energetically more advantageous phase equilibrium, is especially characteristic of polymer systems (see, e.g., [50]), particularly due to the relatively low kinetic mobility of bulky macromolecules. 

Solving Equilibrium Problems with Applications to Qualitative Analysis, by Steven S. Zumdahl. Successfully used by thousands of students, this book offers thorough, step-by-step procedures for solving problems related to equilibria taking place both in the gas phase and in solution. Containing hundreds of sample exercises, test exercises with complete solutions, and end-of-chapter exercises with answers, the text utilizes the same problem-solving methods found in Chemistry and is an excellent source of additional drill-type problems. The last chapter presents an exploratory qualitative analysis experiment with explanations based on the principles of aqueous equilibria. 

In this section the theory of phase equilibria of polymer solutions is discussed as it is a simple practical illustration of the Flory-Huggins theory and can be extended to explain the principles behind the fractionation techniques which are used with polymer solutions. Finally the technique of gel-permeation chromatography, which is now widely used in polymer laboratories, is described in detail. 

A dilute solution of a weakly complexing ligand is the most frequently used mobile phase for this form of SCE, and these conditions generally favor the formation of one or more metal ligand species for each metal cation in the sample solution. The influence of these metal species on the liquid chromatographic retention of a metal cation is discussed in the following subsection. Some fundamental principles of complex equilibria are first reviewed in order to familiarize the reader with the t3q)es of reactions that may occur, along with the products which are likely to form, as a result of SCE reactions. 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

This technique provides quantitative information about tautomeric equilibria in the gas phase. The results are often complementary to those obtained by mass spectrometry (Section VII,E). In principle, gas-phase proton affinities, as determined by ICR, should provide quantitative data on tautomeric equilibria. The problem is the need to correct the measured values for the model compounds, generally methyl derivatives, by the so-called N-, 0-, or S-methylation effect. Since the difference in stability between tautomers is generally not too large (otherwise determination of the most stable tautomer is trivial) and since the methylation effects are difficult to calculate, the result is that proton affinity measurements allow only semi-quantitative estimates of individual tautomer stabilities. This is a problem similar to but more severe than that encountered in the method using solution basicities (76AHCS1, p. 20). 

A detailed explanation of the construction of thermodynamic phase stability diagrams may be found in References 22-25. In this section the basic principles of construction and interpretation for the specific situation of gas-metal equilibria will be addressed using a hypothetical system. 

In the sections below, we describe several studies in which flat-histogram methods were used to examine phase equilibria in model systems. The discussion assumes the reader is familiar with this general family of techniques and the theory behind them, so it may be useful to consult the material in Chap. 3 for background reference. Although the examples provided here entail specific studies, their general form and the principles behind them serve as useful templates for using flat-histogram methods in novel phase equilibria calculations. 

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