SECTION 4 Phase Equilibrium

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

This database provides thermophysical property data (phase equilibrium data, critical data, transport properties, surface tensions, electrolyte data) for about 21 000 pure compounds and 101 000 mixtures. DETHERM, with its 4.2 million data sets, is produced by Dechema, FIZ Chcmic (Berlin, Germany) and DDBST GmhH (Oldenburg. Germany). Definitions of the more than SOO properties available in the database can be found in NUMERIGUIDE (sec Section 5.18). 

Fig. 7. Isothermal cross section of the system H20-CH4-CsH8 on a water-free basis at —3° C. The points represent experimental results and the curves have been obtained from a theoretical analysis. The line AB represents the four-phase equilibrium HiHn ice G the gas G consists of almost pure methane, Hj contains only methane. Consequently, the composition of the latter two phases almost coincide in the figure, and the situation around point A has therefore been drawn separately on an enlarged scale.

Each of these processes is characterised by a transference of material across an interface. Because no material accumulates there, the rate of transfer on each side of the interface must be the same, and therefore the concentration gradients automatically adjust themselves so that they are proportional to the resistance to transfer in the particular phase. In addition, if there is no resistance to transfer at the interface, the concentrations on each side will be related to each other by the phase equilibrium relationship. Whilst the existence or otherwise of a resistance to transfer at the phase boundary is the subject of conflicting views"8 , it appears likely that any resistance is not high, except in the case of crystallisation, and in the following discussion equilibrium between the phases will be assumed to exist at the interface. Interfacial resistance may occur, however, if a surfactant is present as it may accumulate at the interface (Section 10.5.5). 

It is known that in five of the six principal types of binary fluid phase equilibrium diagrams, data other than VLE may also be available for a particular binary (van Konynenburg and Scott, 1980). Thus, the entire database may also contain VL2E, VL E, VL]L2E, and L,L2E data. In this section, a systematic approach to utilize the entire phase equilibrium database is presented. The material is based on the work of Englezos et al. (1990b 1998)... 

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

GPA research is proposed, financed, and conducted on an individual project basis. The GPA research strategy is normally to measure only sufficient data to allow parameter determination in models. GPA seldom attempts definitive system studies. GPA research is directed by an Enthalpy Steering Committee and a Phase Equilibrium Committee. Dr. L. D. Wiener chairs the Enthalpy Committee. Dr. K. H. Kiigren chairs the Equilibrium Committee. Both committees are divisions of Technical Section F chaired by M. A. 

In a ternary isothermal section a similar procedure is used where an alloy is stepped such that its composition remains in a two-phase field. The three-phase field is now exactly defined by the composition of the phases in equilibrium and this also provides the limiting binary tie-lines which can used as start points for calculating the next two-phase equilibrium. 

Thermodynamic activity coefficients can be determined from the phase equilibrium measurements, and they are a measure of deviations from Raoult s law. Data of the activity coefficients covering the whole range of liquid composition of IL + molecular solvent mixtures have been reported in the literature and discussed in sections 1.6,1.7, and 1.8 as the values obtained from the SLE, LLE, and VLE data. When a strong interaction between the IL and the solvent exists, negative deviations from ideality should be expected with the activity coefficients lower than one. 

The structures of protonated azoles in the gas phase (equilibrium ) can be determined by mass spectrometry in a chemical ionization experiment followed by collision-induced dissociation. The method has been used to study the protonation of benzimidazole (5), indazole (7), and 1-ethylimidazole (179) (all, as expected, on the pyridinelike nitrogen atoms) (80OMSI44) of 1-ethylpyrrole (probably at the -position) (80OMS144) and indole (at the -position) (85IJM49) (see Section IV.A). 

Since charcoal is such a good sorbent and is readily available, the solution to some sampling problems is to find a way to increase the recovery of that compound from charcoal. One way is by increasing the solvent/sorbent ratio as discussed in the phase equilibrium section. Two other approaches are the use of mixed solvents and the two-phase solvent system. 

Following this, the thermodynamic arguments needed for determining CMC are discussed (Section 8.5). Here, we describe two approaches, namely, the mass action model (based on treating micellization as a chemical reaction ) and the phase equilibrium model (which treats micellization as a phase separation phenomenon). The entropy change due to micellization and the concept of hydrophobic effect are also described, along with the definition of thermodynamic standard states. 

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 ... 

In summary, whether a reaction equilibrium or a phase equilibrium approach is adopted depends on the size of the micelles formed. In aqueous systems the phase equilibrium model is generally used. In Section 8.5 we see that thermodynamic analyses based on either model merge as n increases. Since a degree of approximation is introduced by using the phase equilibrium model to describe micellization, micelles are sometimes called pseudophases. 

In this section we consider the thermodynamics of micellization from two points of view the law of mass action and phase equilibrium. This will reveal the equivalency of the two approaches and the conditions under which this equivalence applies. In addition, we define the thermodynamic standard state, which must be understood if derived parameters are to be meaningful. 

We start out by considering a very simple example, the partitioning of a compound i between two bulk phases 1 and 2 exhibiting the volumes Vx and V2. As discussed in the previous section, at equilibrium the molar concentrations Cn and Ci2 of i in the two phases are related by the corresponding equilibrium partition constant/ coefficients ... 

Figure 2.2-2 Possible locations of one- and two-phase equilibria around a three-phase equilibrium in a Pjc- section.

In a PyX- or 7 -section a three-phase equilibrium is represented by three points, which give the compositions of the three coexisting phases. All mixtures with composition on the line through these points will split into three phases. Figure 2.2-2 gives a schematic example of a three-phase equilibrium aPy in a P c-section. Around the three-phase equilibrium three two-... 

As a major deficit, in both DH and MSA theory the Mayer functions fxfi = exp —f q>ap(r) — 1 are linearized in ft. This approximation becomes unreasonable at low T and near criticality. Pairing theories discussed in the next section try to remedy this deficit. Attempts were also made to solve the PB equation numerically without recourse to linearization [202-204]. Such PB theories were also applied in phase equilibrium calculations [204-206]. 

Here spr is the projected entropy of an ideal mixture. The first term appearing in it, p0 = J dop a), is the zeroth moment, which is identical to the overall particle density p defined previously. If this is among the moment densities used for the projection (or more generally, if it is a linear combination of them), then the term — Tp0 is simply a linear contribution to the projected free energy/pr(p,) and can be dropped because it does not affect phase equilibrium calculations. Otherwise, p0 needs to be expressed—via the A —as a function of the pit and its contribution cannot be ignored. We will see an example of this in Section V. 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

The calculation of two-phase (hydrate and one other fluid phase) equilibrium is discussed in Section 4.5. The question, To what degree should hydrocarbon gas or liquid be dried in order to prevent hydrate formation is addressed through these equilibria. Another question addressed in Section 4.5 is, What mixture solubility in water is needed to form hydrates ... 

Finally, Section 4.6 concerns the relationship of phase equilibrium to other hydrate properties. The hydrate application of the Clapeyron equation is discussed... 

The intention of this section is to relate these enthalpies both to the phase equilibrium values and to show how these values relate to microscopic structure and to hydration numbers at the ice point. 

Because there is a very large phase equilibrium data base, existing over 70 years as shown in Chapter 6, and because recent spectroscopic tools (e.g., Raman, NMR, and diffraction) have provided microscopic hydrate data, the latter approach was chosen in this monograph and the accompanying computer programs. While the latter method used in this book represents a theoretical advance, it is shown to compare favorably with the existing commercial hydrate programs in Section 5.1.8. 

In a thorough review of calorimetric studies of clathrates and inclusion compounds, Parsonage and Staveley (1984) presented no direct calorimetric methods used for natural gas hydrate measurements. Instead, the heat of dissociation has been indirectly determined via the Clapeyron equation by differentiation of three-phase equilibrium pressure-temperature data. This technique is presented in detail in Section 4.6.1. 

Henry s law constant for solute in feed liquid phase Henry s law constant for solute in solvent liquid phase equilibrium constant distribution coefficient molecular weight of feed without solute molecular weight of solvent without solute interfacial surface tension from Fig. 7.12, dyn/cm partial pressure of solute, atm raffinate density, column section 1, lb/ft3 entering solvent, lb... 

Section 4.2 is focused on phase equilibrium-controlled vapor-liquid systems with kinetically or equihbrium-controlled chemical reactions. The feasible products are kinetic azeotropes or reactive azeotropes, respectively. 

Thermodynamic non-idealities are taken into account while calculating necessary physical properties such as densities, viscosities, and diffusion coefficients. In addition, non-ideal phase equilibrium behavior is accounted for. In this respect, the Elec-trolyte-NRTL model (see Section 9.4.1) is used and supplied with the relevant parameters from Ref. [50]. The mass transport properties of the packing are described via the correlations from Refs. [59, 61]. This allows the mass transfer coefficients, specific contact area, hold-up and pressure drop as functions of physical properties and hydrodynamic conditions inside the column to be determined. 

Vapour phase enthalpies were calculated using ideal gas heat capacity values and the liquid phase enthalpies were calculated by subtracting heat of vaporisation from the vapour enthalpies. The input data required to evaluate these thermodynamic properties were taken from Reid et al. (1977). Initialisation of the plate and condenser compositions (differential variables) was done using the fresh feed composition according to the policy described in section 4.1.1.(a). The simulation results are presented in Table 4.8. It shows that the product composition obtained by both ideal and nonideal phase equilibrium models are very close those obtained experimentally. However, the computation times for the two cases are considerably different. As can be seen from Table 4.8 about 67% time saving (compared to nonideal case) is possible when ideal equilibrium is used. 

These are described in the next section. Note that when atom balances are used, Dluzniewski and Adler (17) show that "fictitious elements" prevent reaction. Consider a reactor that produces ethylbenzene by reaction of benzene and ethylchloride in the presence of AICI3 catalyst. For calculation of phase equilibrium, downstream of the reactor, fictitious element A replaces a hydrogen atom in benzene (C0H5A) and the moles of each species remain unchanged. 

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