Nonideal solutions phase diagrams

Thus, usiag these techniques and a nonideal solution model that is capable of predictiag multiple Hquid phases, it is possible to produce phase diagrams comparable to those of Eigure 15. These predictions are not, however, always quantitatively accurate (2,6,8,91,100). 

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. 

In this section we provide the basis for later discussions pertaining to phase diagrams the results are also of intrinsic interest. We consider the process of generating a nonideal solution from its pure constituents the formulation obtained here should be contrasted with that of Section 2.5. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

The cell theory plus fluid phase equation of state has been extensively applied by Cottin and Monson [101,108] to all types of solid-fluid phase behavior in hard-sphere mixtures. This approach seems to give the best overall quantitative agreement with the computer simulation results. Cottin and Monson [225] have also used this approach to make an analysis of the relative importance of departures from ideal solution behavior in the solid and fluid phases of hard-sphere mixtures. They showed that for size ratios between 1.0 and 0.7 the solid phase nonideality is much more important and that using the ideal solution approximation in the fluid phase does not change the calculated phase diagrams significantly. 

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.. ... 

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,... 

Calculate the phase diagram of nonideal solutions using Raoult s law. 

The ideal solution assumes equal strength of self- and cross-interactions between components. When this is not the case, the solution deviates from ideal behavior. Deviations are simple to detect upon mixing, nonideal solutions exhibit volume changes (expansion or contraction) and exhibit heat effects that can be measured. Such deviations are quantified via the excess properties. An important new property that we encounter in this chapter is the activity coefficient. It is related to the excess Gibbs free energy and is central to the calculation of the phase diagram. 

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. 

FIGURE 7.14 Temperature-composition phase diagram for a nonideal solution showing a positive deviation from Raoult s law. Notice the appearance of a point at which liquid and vapor have the same composition. 

FIGURE 7.15 Temperature-composition phase diagram for a nonideal solution showing negative deviation from Raoulfs law. The azeotrope is maximum-boiling, rather than minimum-boiling as shown in Figure 7.14. 

Nonideal solid-liquid TX diagram at 1 atm for Cu and Al (only about the left half of the diagram is shown). The two-phase regions are indicated. There is a very limited solubility of Cu in Al this is phase a. There is similarly a limited solubility of Cu in the stoichiometric phase or intermetallic compound CuAI2 (called the 6 phase). The liquid solution of Al in Cu freezes at the lowest possible temperature ( 540°C) for 32 mass % Cu this is the eutectic point (which is technologically useful in solders). 

In 8.4.5 we described the stability conditions that, when violated, can cause a one-phase liquid mixture to separate into two liquid phases. We also showed in Figure 8.20 an isobaric, liquid-liquid, Txx diagram on which one-phase states divide into stable, metastable, and unstable states. Liquid-liquid separations occur in nonideal mixtures that have strong positive deviations from ideal-solution behavior in such mixtures the activity coefficients become much greater than unity. This occurs when attractive forces between molecules of the same species are stronger than those between molecules of different species. Liquid-liquid separations have never been observed in mixtures that are negative deviants over the entire composition range. 

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