Phase equilibria nonideal liquid solutions

Figure 3.9 Conceptualization of the fugacity of a compound i (a) in an it/ea/ gas (h) in a pure liquid compound i (c) in an Wen/ liquid mixture and (d) in a nonideal liquid mixture (e.g., in aqueous solution). Note that in (b), (c), and (d), the gas and liquid phases are in equilibrium with one another.

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

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 case of binary solid-liquid equilibrium permits one to focus on liquid-phase nonidealities because the activity coefficient of solid component ij, Yjj, equals unity. Aselage et al. (148) investigated the liquid-solution behavior in the well-characterized Ga-Sb and In-Sb systems. The availability of a thermodynamically consistent data base (measurements of liquidus, component activity, and enthalpy of mixing) provided the opportunity to examine a variety of solution models. Little difference was found among seven models in their ability to fit the combined data base, although asymmetric models are expected to perform better in some systems. 

Few liquid mixtures are ideal, so vapor-liquid equilibrium calculations can be more complicated than is the case for the hexane-triethylamine system, and the system phase diagrams can be more structured than Fig. 10.1-6. These complications arise from the (nonlinear) composition dependence of the species activity coefficients. For example, as a result of the composition dependence of y, the equilibrium pressure in a fixed-temperature experiment will no longer be a linear function of mole fraction. Thus nonideal solutions exhibit deviations from Raoult s law. We will discuss this in detail in the following sections of this chapter. However, first, to illustrate the concepts and some of the types of calculations that arise in vapor-liquid equilibrium in the simplest way, we will assume ideal vapor and liquid solutions (Raoult s law) here, and then in Sec. 10.2 consider the calculations for the more difficult case of nonideal solutions.. ... 

From this equation it is dear that liquid-liquid phase separation is a result of solution nonideality. If the solutions were ideal, that is, if yl = yP = 1, then xj = x for all species i and there would be only a single phase, not two liquid phases of different composition, present at equilibrium. 

The discussion so far has been restricted to gas-phase reactions. One reason for this is that a large number of reactions, including many high-temperature reactions (except metallurgical reactions), occur in the gas phase. Also, the identification of the equilibrium state is easiest for gases, as nonidealities are,.generally less important than in liquid-phase reactions. However, many reactions of interest to engineers occur in the liquid phase.- The prediction of the equilibrium state in such cases can be complicated if the only information available is iifa or since liquid solutions are rarely ideal,... 

In Fig. 9.26, the thermodynamic equilibrium, solid-liquid phase diagram of a binary (species A and B) system is shown for a nonideal solid solution (i.e., miscible liquid but immiscible solid phase). The melting temperatures of pure substances are shown with Tm A and Tm B. At the eutectic-point mole fraction, designated by the subscript e, both solid and liquid can coexist at equilibrium. In this diagram the liquidus and solidus lines are approximated as straight lines. A dendritic temperature T and the dendritic mass fractions of species (p)7(p)s and (p)equilibrium partition ratio kp is used to relate the solid- and liquid-phase mass fractions of species (p)7(p)J and (p)f/(p)f on the liquidus and solidus lines at a given temperature and pressure, that is,... 

In 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in 10.2 they include pure-component standard states and dilute-solution standard states. 

The gas-phase mixture is considered an ideal gas, and in this case Dalton s law states that concentrations are equal to partial pressures divided by the overall pressure p (N m" ). According to HEA, these partial pressures are equal to the saturation pressures of the liquid aerosols. The appropriate description of such saturation pressures depends on the circumstances (see Table 18.2). A hydrocarbon gas does not readily dissolve in water, and therefore two sets of immiscible aerosols will exist in independent equilibrium with the gas phase. Raoult s law describes equilibrium over dilute mixtures, whereas equilibrium over nonideal binary solution requires contaminant-specific empirical models. An example of the latter is Wheatley s model, which states that ... 

Figure 7.12 shows a liquid-vapor phase diagram for positive deviations from Raoult s law. Each component has a higher-than-expected vapor pressure, so the total pressure in equilibrium with the liquid solution is also higher than expected. Ethanol/benzene, ethanol/chloroform, and ethanol/water are systems that show a positive deviation from Raoult s law. Figure 7.13 shows a similar diagram, but for a solution that shows a negative deviation from Raoult s law. The acetone/chloroform system is one example that exhibits such nonideal behavior. 

In low-pressure VLB satisfying ideal gas and ideal solution behavior, Raoult s law can be used very effectively to obtain the values of the intensive variables in a closed system at equilibrium. Unfortunately, there are many VLB systems where Raoult s law is not obeyed. Specifically, in many cases of low-pressure VLB, tbe liquid phase behaves nonideally y, varies with composition and is y l. The nature of the deviation of y, from 1 often determines the nature of the VLB. Systems where y, > 1 are said to display positive deviations from Raoult s law when yu < 1, the systems show a negative deviation from Raoult s law. 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

Physical Equilibria and Solvent Selection. In nrder lor two separale liquid phases to exist in equilibrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy. G. nf a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in iwo phases. For the binary system containing only components A and B. the condition for the formation of two phases is... 

Nitric acid is a strong electrolyte. Therefore, the solubilities of nitrogen oxides in water given in Ref. 191 and based on Henry s law are utilized and further corrected by using the method of van Krevelen and Hofhjzer (77) for electrolyte solutions. The chemical equilibrium is calculated in terms of liquid-phase activities. The local composition model of Engels (192), based on the UNIQUAC model, is used for the calculation of vapor pressures and activity coefficients of water and nitric acid. Multicomponent diffusion coefficients in the liquid phase are corrected for the nonideality, as suggested in Ref. 57. 

Thermodynamic nonidealities are considered both in the transport equations (A10) and in the equilibrium relationships at the phase interface. Because electrolytes are present in the system, the liquid-phase diffusion coefficients should be corrected to account for the specific transport properties of electrolyte solutions. 

Experimental vapor-liquid-equilibrium data for benzene(l)/n-heptane(2) system at 80°C (176°F) are given in Table 1.8. Calculate the vapor compositions in equilibrium with the corresponding liquid compositions, using the Scatchard-Hildebrand regular-solution model for the liquid-phase activity coefficient, and compare the calculated results with the experimentally determined composition. Ignore the nonideality in the vapor phase. Also calculate the solubility parameters for benzene and n-heptane using heat-of-vaporization data. 

Compute the Gfj parameters for the Wilson equation. General engineering practice is to establish liquid-phase nonideality through experimental measurement of vapor-liquid equilibrium. Models with adjustable parameters exist for adequately representing most nonideal-solution behavior. Because of these models, the amount of experimental information needed is not excessive (see Example 3.9, which shows procedures for calculating such parameters from experimental data). 

The two liquid phases are necessarily nonideal solutions and their component equilibrium coefficients are best calculated from the activity coefficients in each phase (Section 1.3.5) ... 

The sodium-lithium phase system has been studied by thermal analysis in the liquid and solid regions to temperatures in excess of 400°C. Two liquid phases separate at 170.6°C. with compositions of 3.4 and 91.6 atom % sodium. The critical solution temperature is 442° zt 10°C. at a composition of 40.3 atom % sodium. The freezing point of pure lithium is depressed from 180.5°C. to 170.6°C. by the addition of 3.4 atom % sodium, and the freezing point of pure sodium is depressed from 97.8° to 92.2°C. by the addition of 3.8 atom % lithium. From 170.6° to 92.2°C. one liquid phase exists in equilibrium with pure lithium. Regardless of the similarity in the properties of the pure liquid metals, the binary system deviates markedly from simple nonideal behavior even in the very dilute solutions. Correlation of the experimentally observed data with the Scatchard-Hildebrand regular solution model using the Flory-Huggins entropy correction is discussed. 

Nonideal Solutions, The final level of complexity for modeling the relationship between vapor and liquid compositions accounts for nonideal interactions in the liquid phase. The equilibrium ratio is still ased for such systems, but in this instance it is defined as... 

For liquids in near-critical fluids the analysis is generally more complicated than the above development for solids because the equilibrium liquid phase is not pure. To the degree that the solvent dissolves in the liquid, the solubility of the solute can decrease. But such solutions may be nonideal, and if the corresponding activity coefficients are greater than unity, then they may partially compensate for the decrease in solubility. 

If the top temperature is too cold and the bottom tenperature is too hot to allow sandwich conponents to exit at the rate they enter the column, they become trapped in the center of the column and accumulate there fKister. 20041. This accumulation can be quite large for trace conponents in the feed and can cause column flooding and development of a second liquid phase. The problem can be identified from the simulation if the engineer knows all the trace conponents that occur in the feed, accurate vapor-liquid equilibrium (VLE) correlations are available, and the simulator allows two liquid phases and one vapor phase. Unfortunately, the VLE may be very nonideal and trace conponents may not accumulate where we think they will. For example, when ethanol and water are distilled, there often are traces of heavier alcohols present. Alcohols with four or more carbons (butanol and heavier) are only partially miscible in water. They are easily stripped from a water phase (relative volatility 1), but when there is litde water present they are less volatile than ethanol. Thus, they collect somewhere in the middle of the column where they may form a second liquid phase in which the heavy alcohols have low volatility. The usual solution to this problem is to install a side withdrawal line, separate the intermediate component from the other components, and return the other components to the column. These heterogeneous systems are discussed in more detail in Chapter 8. 

An important first step in any model-based calculation procedure is the analysis and type of data used. Here, the accuracy and reliability of the measured data sets to be used in regression of model parameters is a very important issue. It is clear that reliable parameters for any model cannot be obtained from low-quality or inconsistent data. However, for many published experimentally measured solid solubility data, information on measurement uncertainties or quality estimates are unavailable. Also, pure component temperature limits and the excess GE models typically used for nonideality in vapor-liquid equilibrium (VLE) may not be rehable for SEE (or solid solubility). To address this situation, an alternative set of consistency tests [3] have been developed, including a new approach for modehng dilute solution SEE, which combines solute infinite dilution activity coefficients in the hquid phase with a theoretically based term to account for the nonideality for dilute solutions relative to infinite dilution. This model has been found to give noticeably better descriptions of experimental data than traditional thermodynamic models (nonrandom two liquid (NRTE) [4], UNIQUAC [5], and original UNIversal Eunctional group Activity Coefficient (UNIEAC) [6]) for the studied systems. 

VAPOR-LIQUID EQUILIBRIUM RATIOS FOR IDEAL-SOLUTION BEHAVIOR 3.1 FUGACITY OF PURE LIQUID 3.3 NONIDEAL GAS-PHASE MIXTURES 3.4... 

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