Phase equilibrium, 4.28

Phase equilibrium describes the way phases (such as solid, liquid and/or gas) co-exist at some temperatures and pressure, but interchange at others. 

the criteria for phase equilibria are discussed in terms of single-component systems. Then, when the ground rules are in place, multi-component systems are discussed in terms of partition, distillation and mixing. 

The chapter also outlines the criteria for equilibrium in terms of the Gibbs function and chemical potential, together with the criteria for spontaneity. 

A phase is defined as the part of the system that has uniquely distinguishing properties from the other part of the system. That property can be, for example, density (e.g., water-ice-water vapor) or different crystallographic forms (e.g., a — Pd//3 - Pd). The coexistence and number of different phases p depends on the number of components c, and on external physical parameters called degrees of freedom /. These are most typically pressure and temperature. The governing relationship is the Gibbs phase rule. 

For example for a two-phase (p = 2) water-ice (c = 1) system we have only one degree of freedom (/ = 1). Thus, we can change the temperature of water and still have coexisting ice, but only at one given pressure. This is the melting point (temperature) A (Fig. A.2) that lies on the coexistence line. If we move to point A we are in the water phase (p = 1) and according to (A.35) we now have two degrees of freedom (/ = 2), temperature and pressure. This triple point is where all three phases coexist (p = 3) it is uniquely defined (/ = 0). This temperature is the official zero of the Celsius scale. 

If phases 1 and 2 each contain only one species, then the above expression for this species may be used to obtain a relation between 

Equation (22) is the well-known Clapeyron equation for phase equilibrium. 

The enthalpy change of a mole of the species in passing from phase 1 to phase 2 is the heat of transition per mole, L12 = 2/ 2 In 

For transitions in which 1 is a condensed phase and 2 is a gas, 1 2/ 2 1/ 1 since the volume per mole of a gas is considerably larger (usually by a factor of about 10 ) than that of a liquid or solid. If it is also assumed that the gas obeys the ideal-gas equation of state, pV2 = N2R T, then equation (22) reduces to 

For chemical reactions in ideal gases, it is possible to express the equilibrium condition in more convenient forms by relating pi to other properties of the gas mixture. Substituting equation (15) into equation (21) yields 

If phases 1 and 2 each contain only one species, then the above expression = 2 for this species may be used to obtain a relation between p and T at phase equilibrium. Differentiating Pi = P2 yields dp. = dp2 for a small change which maintains equilibrium. Expressing dp and dp2 in terms of dT and dp by applying equation (8) to each phase separately, we then obtain  

The enthalpy change of a mole of the species in passing from phase 1 to phase 2 is the heat of transition per mole, Lj2 = H2/N2 — Hr/Ni- In equation (22) S2/N2 — Si/Ni is often replaced by L12/T their equivalence follows from the definition G — H — TS and the result Gi/Nj = G2IN2 [which is a direct consequence of equation (7) (applied to each phase) since = 7 2]. 

The term phase equilibrium, often used in the context of this discussion, refers to equilibrium as it applies to systems in which more than one phase may exist. Phase 

This sugar-syrup example illustrates the principle of phase equilibrium using a liquid-solid system. In many metallurgical and materials systems of interest, phase equilibrium involves just solid phases. In this regard the state of the system is reflected in the characteristics of the microstructure, which necessarily include not only the phases present and their compositions, but, in addition, the relative phase amounts and their spatial arrangement or distribution. 

Thus it is important to understand not only equilibrium states and structures, but also the speed or rate at which they are established and the factors that affect that rate. This chapter is devoted almost exclusively to equilibrium structures the treatment of reaction rates and nonequOibrium structm-es is deferred to Chapter 10 and Section 11.9. 

Concept Check 9.1 What is the difference between the states of phase equilibrium and metastability  

Much of the information about the control of the phase structure of a particular system is phase diagram conveniently and concisely displayed in what is called a phase diagram, also often termed an 

3 PHASE EQUILIBRIA, MOLECULAR DIFFUSION AND MASS TRANSFER 

In industrial PET synthesis, two or three phases are involved in every reaction step and mass transport within and between the phases plays a dominant role. The solubility of TPA in the complex mixture within the esterification reactor is critical. Esterification and melt-phase polycondensation take place in the liquid phase and volatile by-products have to be transferred to the gas phase. The effective removal of the volatile by-products from the reaction zone is essential to ensure high reaction rates and low concentrations of undesirable side products. This process includes diffusion of molecules through the bulk phase, as well as mass transfer through the liquid/gas interface. In solid-state polycondensation (SSP), the volatile by-products diffuse through the solid and traverse the solid/gas interface. The situation is further complicated by the co-existence of amorphous and crystalline phases within the solid particles. 

The phase equilibria of the most important compounds will be described in the following section. In the sections thereafter, we will treat mass transport in melt-phase polycondensation, as well as in solid-state polycondensation, and discuss the diffusion and mass transfer models that have been used for process simulation. 

In typical industrial operations, TPA is not dissolved in EG or BHET but in prepolymer. The latter contains PET oligomers with one to approximately six to eight repeat units and a significant concentration of carboxyl end groups of between 200 and llOOmmol/kg. It was found [94] that the solubility of TPA in prepolymer is much higher than indicated by the values given in the literature. Nevertheless, the esterification reactor still contains a three-phase system and only the dissolved TPA may react with EG in a homogenous liquid-phase 

Roult s law is known to fail for vapour-liquid equilibrium calculations in polymeric systems. The Flory-Huggins relationship is generally used for this purpose (for details, see mass-transfer models in Section 3.2.1). The polymer-solvent interaction parameter, xo of the Flory-Huggins equation is not known accurately for PET. Cheong and Choi used a value of 1.3 for the system PET/EG for modelling a rotating-disc reactor [113], For other polymer solvent systems, yj was found to be in the range between 0.3 and 0.5 [96], 

Several sources of liquid-liquid equilibria (LLE) are available, including the DECHEMA databank and others [22-26], Their use is recommended since vapor-liquid (VLE) data provides less accurate predictions. There are implemented in 

The ion exchanger is known to be in dimeric form in aliphatic diluents [30], and the stoichiometry in Eq. (7) was found with classical slope analysis at low concentrations and FTIR-analysis even at high concentrations [31, 32], A compilation of all thermodynamic parameters is given in http //dechema.de/Extraktion/, as this system is a recommended test system for reactive extraction studies by the European Federation of Chemical Engineering (EFCE). The predictability of the model is quite good, as is depicted in Fig. 10.10, where zinc extraction from chloride media is predicted from sulfate media [33], 

For molecules which differ in size or shape interactions between the surface of the molecules, different Gibbs excess models, such as NRTL [34] or UNIQUAC [35], are recommended, respectively. The predictive group contribution method UNIFAC [36] will fail if several polar groups compose a solvent or solute molecule. As a 

FIGURE 6.4 Schematic illustration of the phase diagram of (a) water compared to (b) most other substances indicating the way in which ice melts under pressure. The scale is exaggerated. 

The conditions for phase equilibria can also be extended to multicomponent systems. Consider an isolated system consisting of w components and two phases, which we refer to as A and B. The 

Inserting the constraint relations given in Eq. (3.5) into Eq. (3.6), we find 

This is equal to zero only it all the coeffieienis of the change terns are zero. As a consequence, we find 

This argument can be generalized to a system containing tt phases and w components. In this case, we have the temperature, pressure, and chemical potentials of each species are equal in each phase. 

The criterion for thermodynamic equilibrium between two phases of a multicomponent mixture is that for every component, i  

Substitution from equations 8.29 and 8.30 into equation 8.28, and rearranging gives  

4 i can be calculated from an appropriate equation of state (see Section 8.16.3). ffL can be computed from the following expression  

The exponential term in equation 8.32 is known as the Poynting correction, and corrects for the effects of pressure on the liquid-phase fugacity. 

is calculated using the same equation of state used to calculate 0,.  

0- = the fugacity coefficient of the pure component i at saturation yf = the liquid molar volume, m /mol 

Yet another important aspect of phase equilibria rests on tire fact that they determine certain maximum or minimum quantities associated with tire process. Suppose, for example, that a liquid evaporates into an enclosure. By allowing the process to proceed to equilibrium, i.e., to full saturation, we are able to determine the maximum amount of liquid that will have evaporated, or, conversely, the minimum mass of air tiiat can accommodate that amount of vapor. Suppose next that the same liquid, for example, water, adheres to a solid that is to be dried by passage of air over it. Then by using very low flow rates we can ensure that the air leaving the drying chamber is fully saturated and that consequently the air consumption is at a minimum. 

While some of the topics in the present chapter will be new to the reader, others may be known from previous courses in thermodynamics. They are repeated to provide a refresher and a link to subsequent chapters. 

Mass Transfer and Separation Processes Principles and Applications 

For a component i, each point of the curve representing equilibrium between the two phases I and II is characterized by pj(p, T) = pf(p, T). For an infinitesimally 

FIGURE 3.3 Schematic representation of a phase diagram for water. 

Equation 3.40 is known as the Clapeyron equation. When applied to vaporization (i.e., when one of the phases, say, phase II, is the gas phase and, moreover, if the gas phase behaves ideally v - 9lr/p), Equation 3.40 may be modified into 

A full treatment of the thermodynamics of phase equilibria may be found elsewhere. Only the terminology and some essential concepts will be reviewed in this section. 

1 K-values. The equilibrium compositions in a vapour-liquid system are related by equilibrium K-values  

Process Feed Phases Product Separating agenP  

The separability of two species i and j is indicated by their relative volatility. 

2 Ideal mixtures. Ideal mixing is a good approximation for mixtures of molecules of similar types (e.g. alkanes) at low to moderate pressures. 

Gas chromatography involves chemical equilibria between phases to bring about a particular separation. Thus, a brief discussion of phase equilibria is pertinent at this point. Phase equilibria separations can be understood with the use of the second law of thermodynamics. The phase rule states that if we have a system of C components which are distributed between. P phases, the composition of each of these phases will be completely defined by C-l concentration terms. Thus, to have the compositions of P phases defined it is necessary to have P(C-l) concentration terms. The temperature and pressure also are variables and are the same for all the phases. Assuming no other forces influence the equilibria it follows that. 

for P phases of C components we have C(P-l) independent equations. It follows then that C(P-l) variables are fixed, which leaves, 

In its strictest sense the phase rule assumes that the equilibrium between phases is not influenced by gravity, electrical or magnetic forces, or by surface action. Thus, the only variables are temperature, pressure, and concentration if two are fixed, then the third is easily determined (another reason for the constant 2 in Equation 2.3). 

By a phase we mean any homogeneous or physically distinct part of the system which is separated from other parts of the system by definite bounding surfaces. 

By a component we mean the smallest number of independently variable components from which the composition of each phase can be expressed (directly or in the form of a chemical equation). 

We assume the reader is familiar with the theory of phase equilibria in mixtures containing a few components. As the number of components increases, one again reaches a point where a continuous description might be preferable. It is relatively straightforward to extend the classical theory to a continuous description. 

Now suppose the system in fact exists as N separate phases, with K = 1,2,.. ., N the phase index let be the number of moles of phase K per mole of the system. The a f s must satisfy the mass balance condition 

Let X (x) be the mole fraction distribution in phase K. The X s must satisfy the mass balance condition 

In the multiphase condition, the total free energy of mixing per mole of system 

It follows that the system will in fact exist as a single phase if, for any choice of the scalars afc and the distributions X (x) satisfying Eqs. (24) and (25), one has 

For a closed multi-component system, eq 2.135 can be used to derive the equilibrium conditions between two or more phases in a system at constant temperature and pressure. If we indicate the various phases by a, P, y, , n, and the various species by 1,2,3, , , the following equilibrium conditions in terms of the chemical potential result  

Alternatively, it can be shown that phase equilibrium can also be defined in terms of the fugacity  

Substitution if eqs 2.111 and 2.123 into this equilibrium condition gives  

This formalism is known as the gamma-phi approach for calculating vapour-liquid equilibria. The fugacity coefficient of each component that accounts for the non-ideality of the vapour phase can be evaluated from an equation of state model, while the activity coefficient/)- to describe the non-ideal behaviour of the liquid phase can be obtained from an excess Gibbs function model. 

The fugacity p of pure species i can be obtained from the relation  

Another model for which (2.1), (2.2) may be applied is the two-phase fluid system without memory which models two-phase equilibria in pure fluid. It has one constituent in two phases which are uniform bodies where the masses and volumes of which are denoted by m and respectively. For the whole volume 

Therefore, we assume that energy and entropy are additive—each of them sums corresponding quantities of both phases taken as pure uniform bodies (i.e., we neglect surface energy or entropy on the phase contact). Memory is excluded because independent and dependent variables are taken in the same present instant. On the other hand, the pressure (2.109) and also temperature T (intensive quantities) are assumed to be the same in both phases (cf. discussion at the end of this Sect. 2.5). Using the deflnitions of free energies F for the whole system (2.12) and for both phases 

Because also the reduced inequality (2.13) for the two-phase system is valid, introducing here (2.111) we obtain with (2.105), (2.106)  

According to admissibility principle, the entropy inequality must be valid at any admissible thermodynamic process. The last one is again defined as time functions for y, y( ), m,T(a = 1,2), and (2.107)-(2.111), which fulfill the balances (2.1), (2.105), (2.106). Again, arbitrary time functions T t), yd)(f), mP- t) 

In a standard way (with Lemma A. 5.1) we obtain from (2.112) identities 

For the calculation of VLB it is necessary to relate the fugadty to measurable parameters, such as concentration, temperature and pressure. Therefore auxihary quantities are introduced the fugacity coefficient (p and the activity coefficient which are defined for Hquid and vapor phase as follows  

Experimental data necessary to describe this behavior are available in large computerized data bases (e.g. Dortmund Data Bank, DDB). A small part of the data is also published in data collections (Gmehhng et al., 1977 Sorensen et al., 1979 Gmehling et al, 1986 Gmehling et al., 1988 Gmehhng et al., 2004 ). Both routes allow the calculation of VLE (see Ghapter 3.2.2.1, Sections 3.2.2.1.1 and 3.2.2.1.2) for multicomponent systems when the behavior of the binary subsystems is known. 

Route A requires an equation of state and sophisticated mixing rules for calculating the fugacity coefficient for both the vapor and the liquid phase. The advantage of using equations of state is that other information (e.g. molar heat capacities, densities, enthalpies, heats of vaporization), which is necessary for designing and optimizing a sustainable distillation process, is also obtained at the same time. 

Besides the standard fugacity. Route B needs a model for calculating the activity coefficient. The fugacity of the pure liquid at system pressure and system temperature is usually chosen as the standard fugacity. Therefore, standard fugacity is defined as 

This requires a knowledge of the saturation vapor pressure, which is usually calculated from the Antoine equation with the Antoine constants A, B and G and the absolute temperature T  

In spite of much jnstifiable criticism, the Hory-Huggins theory can still generate considerable interest because of the limited success that can be claimed for it in relation to phase equilibria studies. 

The Hory-Huggins theory can be used to predict the equilibrium behavior of two liquid phases when both contain amorphous polymer and one or evrai two solvraits. 

As the temperature is increased, the hmits of this two-phase coexistence contract, until eventually they coalesce to produce a homogeneous, one-phase mixture at T, the critical solution temperature. This is sometimes referred to as the critical con-solute point. 

In general, we can say that if the free-energy-composition curve has a shape that allows a tangent to touch it at two points, phase separation will occur. 

The critical solution temperature is an important quantity and can be accurately defined in terms of the chemical potential. It represents the point at which the inflexion points on the curve merge, and so it is the temperature where the first, second, and third derivatives of the Gibbs free energy with respect to mole fraction are zero. 

To account for nonideal vapor-hquid equilibrium and possible vqxjr-liquid-liquid equilibrium (VLLE) for these quaternary systems, the NRTL model or UNIQUAC model is used for activity coefficients. Table 7.2 provides the model parameters for these five quaternary systems where the EtAc, IPAc, and AmAc systems are described by the NRTL model and MeAc and BuAc systems are represented using the UNIQUAC model. Because of the near atmosphaic pressure, the only vapor phase nonideality considered is the dimerization of acetic acid as described by the Hayden-O CoimeU second virial coefficient model. The Aspen Plus built-in association parameters are used to compute fiigacity coefficients. 

It should be emphasized that the quality of the model parameters (Table 7.2) is essential to generate a correct process fiowsheet. Two important steps to validate model parameters are good prediction of azeotropes and a reasonable description of the LL envelopes for VLLE systems. Correct description of the existence of azeotropes and the ranking of 

TABLE 7.2 Activity Coefficient Models Parameters for Five Esterification Systems 

STEADY-STATE DESIGN FOR ACETIC ACID ESTERIFICATION 

MeOH/MeAc EtOH/EtAc/HzO IPOH/IPAc/HjO BuOH/BuAc/HjO AmOH/AmAc/HjO 

Equilibrium data correlations can be extremely complex, especially when related to non-ideal multicomponent mixtures, and in order to handle such real life complex simulations, a commercial dynamic simulator with access to a physical property data-base often becomes essential. The approach in this text, is based, however, on the basic concepts of ideal behaviour, as expressed by Henry s law for gas absorption, the use of constant relative volatility values for distillation and constant distribution coeficients for solvent extraction. These have the advantage that they normally enable an explicit method of solution and avoid the more cumbersome iterative types of procedure, which would otherwise be required. Simulation examples in which more complex forms of equilibria are employed are STEAM and BUBBLE. 

Actual concentration profiles (Fig. 1.27) in the very near vicinity of a mass transfer interface are complex, since they result from an interaction between the mass transfer process and the local hydrodynamic conditions, which change gradually from stagnant flow, close to the interface, to more turbulent flow within the bulk phases. 

According to the Whitman Two-Film theory, the actual concentration profiles, as shown in Fig. 1.27, are approximated for the steady state with no chemical reaction, by that of Fig. 1.28. 

A thin film of fluid exists on either side of the interface. 

Each film is in stagnant or laminar flow, such that mass transfer across the films is by a process of molecular diffusion and can therefore be described by Ficks Law. 

It is understood that the (local) Gibbs energy depends on the local stress, and thus aH(NH) and (A/h) reflect the self- and coherency stresses in the Me-H system. In addition, if coherency is lost due to plastic deformation or cracking, the Me atoms in the deformation zone may well become mobile and Me then is well defined near the interface. This could explain the fact that aK(N (P)) (= aH(Ajj(a))) corresponds, in essence, to the value of the a/p equilibrium calculated using independent thermodynamic data. 

The pKs value can be calculated from the pH-solubility proLle on the basis of Equation 4.5, which can be rearranged to give Equation 4.32. 

Since this is zero, by equation (27.1), when the system is in equilibrium, it follows that 

Compared with an experimental value of 982.5 kPa, the best predictions are by the Soave equation and the Redlich-Kwong equation. 

The conditions for equilibrium discussed in Section 1.1.3 are applied here to the problem of phase equilibria. These conditions are that, in order for two or more phases to coexist at equilibrium, they must have the same temperature and pressure and the chemical potential of each component must be equal in all the phases. The chemical potential is not a measurable quantity and is not intuitively related to observable physical properties. Applying the conditions of equilibrium to real fluids involves a transformation to more practical terms and the utilization of fluid models such as equations of state. 

Department of Chemical and Environmental Engineering Rensselaer Polytechnic Institute, Troy, New York 

Most of the common separation methods used in the chemical industry rely on a well-known observation when a multicomponent two-phase system is given sufficient ttmu to attain a statioenry state called equilibrium, the composition of one phase is different from thet of the other. It is this property of nature which eenbles separation of fluid mixtures by distillation, extraction, and other diffusions operations. For rational design of such operations it is necessary to heve a quantitative description of how a component distributes itself between two contacting phases. Phase-equilibrium thermodynamics, summarized here, provides a framework for establishing that description. 

If experimental phase-equilibrium measurements were simple, fast, and inexpensive, chemical engineers would have little need for phase-equilibrium thermodyenmics because in thet happy event all component-distribution data required for design won Id be obtained readily in the laboratory. Unfortuentely, however, compunent-distributton data are not easily obtained because experimental studies require much patience and skill. As a result, required data are often not at hand but must be estimated using suitable physicochemical models whose parameters are obtained from correlations or from limited experimental data. 

It was Einstein wbo said that when God made tha world, he was subtle but not malicious, The subtlety of nature is evident by onr ienbility to construct models of mixtures which give directly to the chemical engineer the required information in the desired fores temparatute. pressure, phase compositions. Nature, it seems, does not choose to reveal secrets In the everyday language of chemical process design but prefers to use en abstract language—thermodyenmics. 

To achieve a quantitative description of phase equilibria, thermodynamics provides a usefbl theoretical framework. By itself, thermodynamics cennot provide all the numerical information we desire but, when coupled with concepts from molecular physics and physical chemistry, it can efficiently organize limited experimental information toward helpful interpolation and extrapolation. Thermodyenmics is not magic it cannot produce something for nothing some experimental information is always necessary. But when used with skill and courage, thermodynamics can squeeze the last drop not of a nearly dried-up lemon. 

Like harmony in music there is a dark inscrutable workmanship that reconciles discordant elements, makes them cling together in one society. 

The importance of knowing the phase diagram in a particular system cannot be overemphasized. It is the roadmap without which it is very difficult to interpret and predict microstructure distribution and evolution, which in turn have a profound effect on the ultimate properties of a material. 

In principle, phase diagrams provide the following information  

The composition of the phases present at any time during heating or cooling 

The range of solid solubility of one element or compound in another 

It is not always easy to acquire data on the concentration dependence of the separation factor because only one feed mixture is available for equilibrium measurements. Therefore, mixtures of other composition must be produced by mixing or from the products of experimental separations (which are also neeeded to determine the height of theoretical stages). 

Experimental determination of all the properties and their dependences on the operating conditions is tedious and expensive. Computational thermody- 

Solubilities and separation factors provide the necessary basic data for carrying out an analysis of the separation process on the basis of theoretical stages. This analysis is independent of the type of separation equipment used. 

The brief survey presented here rmst necessarily begin wife a discussion of thermodynamics as a langutq[ie most of Section 1.2 is concerned widi the definition of thermodynamic terms such as chemical potential, fogacity, and activiQf. At the end of Section 1.2, the phase-eqoilibrium probimn is dearly stated in several thettnodynamic forms each of these finms is particularly sitited for a particular situation, as indkaied in Sections 1.5, 1.6, and 1.7. 

Section 1.3 discusses fiigacities (through fiigacity coefRcienls) in the vapor phase. Illustrative examples are given using equations of state. 

Section 1.4 discusses fiigacities (thnough activi coefficients) in the liquid phase. Illustrative examples are given using semiempirical models for liquid mixtures of nonelectiolytes. 

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Selected References for Calculation of Multicomponent Fluid-Phase Equilibria... 

Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibria," Prentice-Hall, Englewood Cliffs, N.J. (1969)... 

An adequate prediction of multicomponent vapor-liquid equilibria requires an accurate description of the phase equilibria for the binary systems. We have reduced a large body of binary data including a variety of systems containing, for example, alcohols, ethers, ketones, organic acids, water, and hydrocarbons with the UNIQUAC equation. Experience has shown it to do as well as any of the other common models. V7hen all types of mixtures are considered, including partially miscible systems, the... 

Figure 4 shows experimental and predicted phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate positive deviations from Raoult s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

Null, H. R., "Phase Equilibria in Process Design," Wiley-Interscience (1970). 

PRAUSNITZ J.M. MOLECULAR THERMODYNAMICS OF FLUID PHASE EQUILIBRIA, PRENTICE-HALL. ENGLEWOOD CLIFFS. N.J.I19691. 

In the petroleum refining and natural gas treatment industries, mixtures of hydrocarbons are more often separated into their components or into narrower mixtures by chemical engineering operations that make use of phase equilibria between liquid and gas phases such as those mentioned below ... 

In the case of three-phase equilibria, it is also necessary to account for the solubility of hydrocarbon gases in water. This solubility is proportional to the partial pressure of the hydrocarbon or, more precisely, to its partial fugacity in the vapor phase. The relation which ties the solubility expressed in mole fraction to the fugacity is the following ... 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Levelt Sengers J M H 1999 Mean-field theories, their weaknesses and strength Fluid Phase Equilibria 158-160 3-17... 

Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 523-35... 

Scott R L 1965 Phase equilibria in solutions of liquid sulfur. I. Theory J. Phys. Chem. 69 261-70... 

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

For a multicomponent system, it is possible to simulate at constant pressure rather than constant volume, as separation into phases of different compositions is still allowed. The method allows one to study straightforwardly phase equilibria in confined systems such as pores [166]. Configuration-biased MC methods can be used in combination with the Gibbs ensemble. An impressive demonstration of this has been the detennination by Siepmaim et al [167] and Smit et al [168] of liquid-vapour coexistence curves for n-alkane chain molecules as long as 48 atoms. 

With the rapid development of computer power, and the continual hmovation of simulation methods, it is impossible to predict what may be achieved over the next few years, except to say that the outlook is very promising. The areas of rare events, phase equilibria, and quantum simulation continue to be active. 

Panagiotopoulos A Z, Quirke N, Stapleton M and Tildesley D J 1988 Phase equilibria by simulation in the Gibbs ensemble. Alternative derivation, generalization and applioation to mixture and membrane equilibria Mol. Phys. 63 527-45... 

Panagiotopoulos A Z 1992 Direot determination of fluid phase equilibria by simulation in the Gibbs ensemble a review Mol. SImul. 9 1 -23... 

Panagiotopoulos A Z 1994 Moleoular simulation of phase equilibria Supercritical Fluids—Fundamentals for Application NATO ASI Series ed E Kiran and J M H Levelt Sengers (Dordreoht Kluwer)... 

Smit B, Karaborni S and Siepmann J I 1995 Computer simulations of vapor-liquid phase equilibria of n-alkanesJ. Chem. Phys. 102 2126-40... 

Panagiotopoulos A Z 1989 Exaot oaloulations of fluid-phase equilibria by Monte Carlo simulation in a new statistioal ensemble Int. J. Thermophys. 10 447-57... 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Laso M, dePablo J J and Suter U W 1992 Simulation of phase equilibria for chain molecules J Chem. Phys. 97 2817... 

The use of group contribution methods for the estimation of properties of pure gases and Uquids [20, 21] and of phase equilibria [22] also has a long history in chemical engineering. 

Simulating Phase Equilibria by the Gibbs Ensemble Monte Carlo Method... 

Martin M G and J I Siepmann 1999. Novel Configurational-bias Monte Carlo Method for Blanche Molecules. Transferable Potentials for Phase Equilibria. 2. United-atom Description of Branchi Alkanes. Journal of Physical Chemistry 103 4508-4517. 

In Secs. 4.3 and 4.4 we discussed the thermodynamics of the crystal -> Uquid transition. This and other famiUar phase equilibria are examples of what are called first-order transitions. There are other less familiar but also well-known... 

A. Muan and E. F. Osborn, Phase Equilibria Among Oxides in Steelmaking, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. 

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