Vapor-liquid equilibrium ideal solutions

Although Raoult s law is an important starting point in understanding vapor-liquid equilibrium in solutions, most real solutions deviate from ideal behavior. The nature and magnitude of these deviations can be understood in terms of the intermoiecuiar interactions between two volatile substances A and B. Consider the following two cases ... 

Raoult s Law. The molar composition of a liquid phase (ideal solution) in equilibrium with its vapor at any temperature T is given by... 

Silverman [2] has presented an analogous analysis of the vapor-liquid equilibrium for an ideal solution. 

When combined with the ideal-gas and ideal-solution models of phase behavi the criterion of vapor/liquid equilibrium produces a simple and useful equati known as Raoult s law. Consider a liquid phase and a vapor phase, both compris of N chemical species, coexisting in equilibrium at temperature T and pressil P, a condition of vapor/liquid equilibrium for which Eq. (10.3) becomes... 

The properties of mixtures of ideal gases and of ideal solutions depend solely on the properties of the pure constituent species, and are calculated from them by simple equations, as illustrated in Chap. 10. Although these models approximate the behavior of certain fluid mixtures, they do not adequately represent the -behavior of most solutions of interest to chemical engineers, and Raoult s law is not in general a realistic relation for vapor/liquid equilibrium. However, these models of ideal behavior—the ideal gas, the ideal solution, and Raoult s law— provide convenient references to which the behavior of nonideal solutions may be compared. 

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... 

In Chap. 6 we treated the thermodynamic properties of constant-composition fluids. However, many applications of chemical-engineering thermodynamics are to systems wherein multicomponent mixtures of gases or liquids undergo composition changes as the result of mixing or separation processes, the transfer of species from one phase to another, or chemical reaction. The properties of such systems depend on composition as well as on temperature and pressure. Our first task in this chapter is therefore to develop a fundamental property relation for homogeneous fluid mixtures of variable composition. We then derive equations applicable to mixtures of ideal gases and ideal solutions. Finally, we treat in detail a particularly simple description of multicomponent vapor/liquid equilibrium known as Raoult s law. 

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. 

VAPOR-LIQUID EQUILIBRIUM RATIOS FOR IDEAL-SOLUTION BEHAVIOR 3.1... 

A gas-liquid system in which the vapor-liquid equilibrium relationship for every volatile species is either Raoult s law or Henry s law is said to exhibit ideal solution behavior. An ideal liquid solution is a mixture of liquids that exhibits ideal solution behavior at equilibrium. 

We conclude this discussion with one final reminder. The vapor-liquid equilibrium calculations we have shown in Section 6.4c are based on the ideal-solution assumption and the corresponding use of Raoult s law. Many commercially important systems involve nonideal solutions, or systems of immiscible or partially miscible liquids, for which Raoult s law is inapplicable and the Txy diagram looks nothing like the one shown for benzene and toluene. 

Raoult s law is the simplest quantitative expression for vapor-liquid equilibrium. This law is based on the vapor phase being an ideal gas and the liquid phase being an ideal solution. Therefore, the vapor-phase fugacity is given by Eq. (2). The effect of pressure on the liquid phase is very small, and since the vapor is ideal, the liquid fugacity may be written as in Eq. (4). Here, is the vapor pressure of component i at the temperature of the solution. 

Fig. 3 Isobaric vapor-liquid equilibrium of acetone and chloroform at 0.9839 Bars. The solution is strongly hydrogen bonded, is non-ideal and does not obey Raoult s law. Solid lines are calculated from Wilson s equation.

Now we will use the ideal solution model to develop a mathematical description of vapor-liquid equilibrium in a multicomponent solution. We will make the assumption that we have a system that is separated into a coexisting vapor and liquid phase. The vapor phase will be assumed to behave like an ideal gas, while the liquid phase will be assumed to behave as an ideal solution. 

As discussed in Chapter 4, the vapor-liquid equilibrium between an ideal gas and an ideal liquid solution is governed by Raoult s law ... 

The graphical method assumes an idealized absorption/stripping model that is defined in terms of essentially three components or groups of components a liquid, a gas, and a distributed component or solute. The liquid is assumed not to vaporize (i.e., it has a very low A -value), and the gas is assumed not to dissolve in the liquid (i.e., it has a very high A -value). The distributed component is the key component to be absorbed by the liquid or stripped by the gas. It distributes itself between the two phases to satisfy vapor-liquid equilibrium criteria (Chapter 1). 

The vapor-liquid equilibrium temperature for specified pressure and liquid composition is found as the solution to Eqs. 10.1-2 or, if the system is ideal, as the solution to Eq..10.1-4. However, since the temperature appears only implicitly in these equations through the species vapor pressures, and since there is a nonlinear relationship between the vapor pressure and temperature (cf. the Clausius-Clapeyron equation, Eq. 

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.. ... 

The dashed lines in the figure are the predictions, at all temperatures for the acetone-water system that result from setting the binary parameter k 2 equal to zero. Note that very nonideal behavior is predicted, which shows that setting ]c 2 = 0 is not equivalent to assuming ideal solution behavior. In fact, such extreme nonideal behavior is. predicted that vapor-liquid equilibrium calculations made with the program VLMU do not even converge for acetone mole fractions less than about 0.15. 

Raoult s Law for Vapor-Liquid Equilibrium of Ideal Solutions... 

Idea] solution thermodynamics is most frequently applied to mixtures of nonpolar compounds, particularly hydrocarbons such as paraffins and olefins. Figure 4.5 shows experimental K-value curves for a light hydrocarbon, ethane, in various binary mixtures with other less volatile hydrocarbons at 100 F (310.93°K) at pressures from 100 psia (689.5 kPa) to convergence pressures between 720 and 780 psia (4.964 MPa to 5.378 MPa). At the convergence pressure, X-values of all species in a mixture become equal to a value of one, making separation by operations involving vapor-liquid equilibrium impossible. The temperature of lOO F is close to the critical temperature of 550.0°R (305.56°K) for ethane. Figure 4.5 shows that ethane does not form ideal solutions with all the other components because the X-values depend on the other component. 

We next consider the properties of the fugacities of the vapor phase in equilibrium with the moderately dilute ideal solution. For vapor-liquid equilibrium,... 

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