Phase equilibrium ideal mixtures

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

Effect of Pressure on Solid + Liquid Equilibrium Equation (6.84) is the starting point for deriving an equation that gives the effect of pressure on (solid + liquid) phase equilibria for an ideal mixture in equilibrium with a pure... 

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. 

Volumetric equations of state (EoS) are employed for the calculation offluid phase equilibrium and thermo-physical properties required in the design of processes involving non-ideal fluid mixtures in the oil, gas and chemical industries. Mathematically, a volumetric EoS expresses the relationship among pressure, volume, temperature, and composition for a fluid mixture. The next equation gives the Peng-Robinson equation of state, which is perhaps the most widely used EoS in industrial practice (Peng and Robinson, 1976). 

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

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. 

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. 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

Thermodynamic predictions of the solid-phase composition have been very successful for the growth by MOCVD of group III-V compound semiconductors (e.g., InAs Sb and GaAs SbJ even though the gas-phase reactions are far from equilibrium (88-91). The procedure is also useful for estimating solid-vapor distribution coefficients of group II-VI compound semiconductors (e.g., Cd Hg e and ZnSe SJ grown by MOCVD (92). In the analysis, the gas phase is considered to be an ideal mixture, that is... 

Retrograde Condensation For near-ideal mixtures, the intersection of the (p, X2 or y ) isotherm with the critical locus occurs at the maximum in the (p, y2) equilibrium line. For example, an enlargement of the two-phase (p, xi or yi) section for (xi oryi)Ar + (x2 or j2)Kr at T— 177.38 K is shown in Figure 14.12. The point of intersection with the critical locus at point (c) gives rise to an... 

Chapters 17 and 18 use thermodynamics to describe solutions, with nonelectrolyte solutions described in Chapter 17 and electrolyte solutions described in Chapter 18. Chapter 17 focuses on the excess thermodynamic properties, with the properties of the ideal and regular solution compared with the real solution. Deviations from ideal solution behavior are correlated with the type of interactions in the liquid mixture, and extensions are made to systems with (liquid + liquid) phase equilibrium, and (fluid -I- fluid) phase equilibrium when the mixture involves supercritical fluids. 

Figure 35.6 (a) Ideal liquid mixture, at equilibrium at temperature, 7, containing a component, i, which exerts a partial pressure. Pi, in the vapour phase and which, in the liquid phase behaves ideally, so that Raoult s law holds p = x,.P. (b) Shows the system (a) under conditions where x) = 1 (i.e. i is the sole component). The vapour pressure is then equal to P. ... 

If the liquid mixture is extremely non-ideal, liquid phase splitting will occur. Here, we first consider the hypothetical ternary system. The physical properties are adopted from Ung and Doherty [17] and Qi et al. [10]. The catalyst is assumed to have equal activity in the two liquid phases. The corresponding PSPS is depicted in Fig. 4.5, together with the liquid-liquid envelope and the chemical equilibrium surface. The PSPS passes through the vertices of pure A, B, C, and the stoichiometric pole Jt. The shape of the PSPS is affected significantly by the liquid phase non-idealities. As a result, there are three binary nonreactive azeotropes located on... 

Mujtaba (1989) and Mujtaba and Macchietto (1992) considered a ternary separation using Butane-Pentane-Hexane mixture. Only the optimal operation for the first main-cut and the first off-cut was considered. Table 8.7 lists a variety of separation specification (on CUT 1) and column configurations in each case. A fresh feed of 6 kmol at a composition of <0.15, 0.35, 0.50> (mole fraction) is used in all cases. Also in each case a constant condenser vapour load of 3 kmol/hr is used. For convenience Type IV-CMH model was used with ideal phase equilibrium. 

Figure A.2 (right) emphasizes a particular position where phase equilibrium and stoichiometric lines are collinear. In other words the liquid composition remains unchanged because the resulting vapor, after condensation, is converted into the original composition. This point is a potential reactive azeotrope, but when the composition satisfies chemical equilibrium too it becomes a true reactive azeotrope. Some examples of residue curve maps are presented below. Ideal mixtures are used to illustrate the basic features, which may be applied to some important industrial applications.

When liquid and gas phases are both present in an equilibrium mixture of reacting species, Eq. (11.30), a criterion of vapor/liquid equilibrium, must be satisfied along with the equation of chemical-reaction equilibrium. There is considerable choice in the method of treatment of such cases. For example, consider a reaction of gas A and water B to form an aqueous solution C. The reaction may be assumed to occur entirely in the gas phase with simultaneous transfer of material between phases to maintain phase equilibrium. In this case, the equilibrium constant is evaluated from AG° data based on standard states for the species as gases, i.e., the ideal-gas states at 1 bar and the reaction temperature. On the other hand, the reaction may be assumed to occur in the liquid phase, in which case AG° is based on standard states for the species as liquids. Alternatively, the reaction may be written... 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

A modified local composition (LC) expression is suggested, which accounts for the recent finding that the LC in an ideal binary mixture should be equal to the bulk composition only when the molar volumes of the two pure components are equal. However, the expressions available in the literature for the LCs in binary mixtures do not satisfy this requirement. Some LCs are examined including the popular LC-based NRTL model, to show how the above inconsistency can be eliminated. Further, the emphasis is on the modified NRTL model. The newly derived activity coefficient expressions have three adjustable parameters as the NRTL equations do, but contain, in addition, the ratio of the molar volumes of the pure components, a quantity that is usually available. The correlation capability of the modified activity coefficients was compared to the traditional NRTL equations for 42 vapor—liquid equilibrium data sets from two different kinds of binary mixtures (i) highly nonideal alcohol/water mixtures (33 sets), and (ii) mixtures formed of weakly interacting components, such as benzene, hexafiuorobenzene, toluene, and cyclohexane (9 sets). The new equations provided better performances in correlating the vapor pressure than the NRTL for 36 data sets, less well for 4 data sets, and equal performances for 2 data sets. Similar modifications can be applied to any phase equilibrium model based on the LC concept. 

Raoult s law can be used to calculate the partial pressure of a component in the vapour phase above a liquid under conditions of equilibrium if the composition of the liquid is known and if the system is assumed to be ideal. Mixtures of the hydro-fluoroalkane propellants HFA 134a and HFA 227 obey Raoult s law over a wide... 

In a two-phase vapor-liquid mixture at equilibrium, if all the components can vapor- ize and condense, a component in one phase is in equilibrium with the same compo-nent in the other phase. The equilibrium relationship depends on the temperature and pressure, and perhaps composition, of the mixture. Figure 3.15 illustrates two cases, one at constant pressure and the other at constant temperature. At the pairs of points A and B, and C and D, the respective pure components exert their respective vapor pressures at the equilibrium temperature. In between the pairs of points, as the overall composition of the mixture changes, two phases exist, each having a different composition for the same component as indicated by the dashed lines. Two useful linear ( ideal ) equations exist to relate the mole fraction of one component in the vapor phase to the mole fraction of the same component in the liquid phase. 

Next the equations that we can write are for calculating system properties. Because equilibrium is assumed, the rate equations and, therefore, the transport and transfer properties are of no concern. In general, the thermodynamic properties of mixtures will depend on temperature, pressure, and composition, we will assume that the mixture is an ideal solution to simplify the computation of thermodynamic properties. Thus, we can write the enthalpies of the mixtures as mole fraction averages of the pure component enthalpies, without an enthalpy of mixing term. We can also write the phase equilibrium relations as functions of temperature and pressure only and not composition. The pure component enthalpies of liquids generally do not depend strongly on pressure, but there may be some effect of pressure on the vapor-phase enthalpy. We will neglect this effect for simplicity. 

In order to better demonstrate whether a system follows Raoult s law, a diagram of the phase equilibrium called T-x-y should be plotted. This plot (Figure 2) shows the equilibrium temperatures at which either a liquid solution will start bubbling (bubble curve) or a vapor mixture starts condensing (dew curve). The two systems with their experimental data and the calculation curve of the ideal solution is shown in Figure 2. In Figure 2, the system of hexane-benzene at the pressure of 101.33 kPa [10] and the system of ethylacetate-benzene [11] show negative deviations from RaoulTs law. 

The Equilibrium Constant in Terms of Molar Concentrations of Gases Gas-phase equilibria (ideal gas reaction mixture)... 

The resistance to mass transfer according to (1.221) and (1.223) is made up of the individual resistances of the gas and liquid phases. Both equations show how the resistance is distributed among the phases. This can be used to decide whether one of the resistances in comparison to the others can be neglected, so that it is only necessary to investigate mass transfer in one of the phases. Overall mass transfer coefficients can only be developed from the mass transfer coefficients if the phase equilibrium can be described by a linear function of the type shown in eq. (1.217). This is normally only relevant to processes of absorption of gases by liquids, because the solubility of gases in liquids is generally low and can be described by Henry s law (1.217). So called ideal liquid mixtures can also be described by the linear expression, known as Raoult s law. However these seldom appear in practice. As a result of all this, the calculation of overall mass transfer coefficients in mass transfer play a far smaller role than their equivalent overall heat transfer coefficients in the study of heat transfer. 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

To solve equations of state, you must solve algebraic equations as described in this chapter. Later chapters cover other topics governed by algebraic equations, such as phase equilibrium, chemical reaction equilibrium, and processes with recycle streams. This chapter introduces the ideal gas equation of state, then describes how computer programs such as Excel , MATLAB , and Aspen Plus use modified equations of state to easily and accurately solve problems involving gaseous mixtures. 

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