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10.1 Vapor Liquid Equilibrium in Ideal Mixtures 511 Equations 10.1.1 and 10.1-2 or their simplifications bere, together with the restric- tions that (10.1-5) and (10.1.6) and the mass and energy balance equations, are the basic relations for all vapor-liquid equilibrium calculations we consider in this section As the first illustration of the use of these equations, consider vapor-liquid equilib- rium in the hexane-triethylamine system at 60 C. These species form an essentially ideal mixture. The vapor pressure of hexane at this temperature is 0.7583 bar and that of triethylamine is 0.3843 bar, these are so low that the fugacity coefficients at satura- tion and for the vapor phase can be neglected. Consequently, Egs. 10.1-3 and 10.1-4 should be applicable to this system. The three solid lines in Fig. 10.1.1 represent the two species partial pressures and the total pressure, which were calculated using these equations and all are linear functions of the of liquid-phase mole fraction, the points are the experimental results. The close agreement between the computations and the labo- ratory data indicates that the hexane-tricthylamine mixture is ideal at these conditions. Note that this linear dependence of the partial and total pressures on mole fractions predicted by Eqs. 10.1-2 and 10.1-3 is true only for ideal mixtures; it is not true for nonideal mixtures. as we shall sec in Sec. 10.2 Once the equilibrium total pressure has been computed for a given liquid composi- tion using Egs. 10.1-2 or 10.1-4. the equilibrium composition of the vapor can be cal- culated using Eqs. 10.1-1 or 10.1-3, as appropriate. Indeed, we can prepare a complete vapor-liquid equilibrium composition diagram, orz-y diagram, at constant tempera- ture by choosing a collection of values for the composition of one of the phases, say the liquid-phase composition 3, and then using the vapor pressure data to compute the 08 T-arc 06 P, bar 04 02 0 0 04 02 06 OR 10 Heune mole fraction is liquid Figure 10.1-1 Equilibrium total pressure and species liquid-phase fugacities (PY) versus mole fraction for the essentially ideal hexane-triethylamine system at 60°C. [Based on data of J. L. Humphrey and M. Van Winkle. J. Chem. Eng. Data, 12, 526 (1967).
prap du Papp Ww= a fixed tem 512 Chapter 10: Vapor Liquid Equilibrium in Mixtures total pressure and value of n corresponding to each ;. For the hexane-tricthylamint system, to calculate the composition of the vapor in equilibrium with a 50 mol brate mixture at T = 60C. Eq. 10.1-4 is first used to compute the equilibrium pressure. Construction of any Ρ- Σπριν = αα Ρεν - ΣΤΡΕΣ = diagram =0.5 x 0.7583 +0.5 x 0.3843 = 0.5713 bar and then Eq. 10.1-3 is used to calculate the vapor-phase mole fraction of hexane: 0.5 x 0.7583 0.6637 P 0.5713 By choosing other liquid compositions and repeating the calculation perature, the complete constant-temperature vapor-liquid equilibrium composition di agram, or 2-y diagram, can be constructed. The results are shown in Fig. 10.1-2 along with points representing the experimental data. The second line in this figure is the line 1 = y; the greater the difference between the D-y curve of the mixture and the < =) line, the greater the difference in composition between the liquid and vapor phases, and the easier it is to separate the two components by distillation, as will be discussed later in this section. (Since -y diagrams are most often used in the study of distillation, it is common practice to include the x = y or 45° line.) An alternative way of presenting vapor-liquid equilibrium data is to plot, on a single figure, the equilibrium pressure and the compositions for both phases at fixed temper ature. This has been done for the hexane-triethylamine system in Fig. 10.1-3. In this figure the equilibrium compositions of the vapor and liquid as a function of pressure are given by the curves labeled "vapor" and "liquid." respectively, the compositions of the two coexisting phases at each pressure are given by the intersection of a horizontal line (i.c., a line of constant pressure) with the vapor and liquid curves. The term tie line is used here, and generally in this chapter, to indicate a line connecting the equilibrium compositions in two coexisting phases. The tic line drawn in Fig. 10.1-3 shows that at LO 08 T-60°C 0.6 Yhexane mole fraction in vapor phase 04 0.2 0.4 0.8 1.0 02 0.6 Xyhexane mole fraction in liquid phase Figure 10.1-2 The s-y diagram for the hexane-triethylamine system at T = 60°C. (Based on data of J.L. Humphrey and M. Van Winkle. J. Chem Eng. Data, 12, 526 (1967).)
10.1 Vapor Liquid Equilibrium in Ideal Mixtures 513 ON 07 TEC Liquid Vapor 06 The line P.bar 05 0.4 02 04 0.6 10 Mole fraction here Figure 10.1-3 Pressure composition diagram for the bestane tricthylamine system at fixed temperature. 60°C, a liquid containing 50 mol Shexanc is in equilibrium with a vapor containing 66.37 mol % hexane at 0.5713 bar. So far the discussion has been specific to systems at constant temperature: equiva- lently, pressure could be fixed and temperature and liquid phase composition taken as the variables. Although much experimental vapor-liquid equilibrium data are obtained in constant temperature experiments, distillation columns and other vapor-liquid seps- rations equipment in the chemical process industry are operated more nearly at constant pressure. Therefore, it is important that chemical engineers be familiar with both types of calculations. The vapor-liquid equilibrium temperature for specified pressure and liquid compo- sition is found as the solution to Eqs. 10.1-2 or, if the system is ideal, as the solu- tion 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 relation- ship between the vapor pressure and temperature (cf. the Clausius-Clapeyron equa- tion, Eq, 7.7-5 a), these equations are usually solved by iteration. That is, one guesses a value of the equilibrium temperature, compules the value of the vapor pressure of each species at this temperature, and then tests whether the pressure computed from Egs. 10.1-2 (or Eq. 10.1-4 if the system is ideal) equals the fixed pressure. If the two are equal, the guessed equilibrium temperature is correct, and the vapor-phase mole fractions can be computed from Eq. 10.1-1 (or if the system is ideal from Eq. 10.1-3)? If the two pressures do not agree, a new trial temperature is chosen and the calculation repeated. Figure 10.1-4 is a plot, on a single graph, of the equilibrium temperature and mole fractions for the hexane-triethylamine system at 0.7 har calculated in this way, and In fact, the species activity coefficients also depend on temperature: see 1993-22. However, since this temper ature dependence is usually small compared with the temperature variation of the vapor pressure, it is neglected here of the vapor phase mole fractions calcolated in this way do not sum to 1.only a single phase, vaper or sud, is present at equilibrio
Constant Value Temperature 323.14 K Data Table P [kPa] X1 [mol/mol] v1 [mol/mol] 7.386 0.00000 0.00000 7.893 0.00390 0.04310 9.066 0.01150 0.10900 23.371 0.22650 0.65680 25.265 0.29000 0.70710 29.904 0.49810 0.79850 31.464 0.58110 0.83370 32.691 0.69360 0.86780 34.530 0.82800 0.91430 35.250 0.93540 0.95940 35.450 0.97800 0.98120 35.610 1.00000 1.00000 (P - pressure, X-liquid mole fraction, y - vapor mole fraction) Reference Source von Zawidzki J.: Über die Dampfdrucke binärer Flüssigkeitsgemische. Z. Phys. Chem. (Leipzig) 35 (1900) 129- 203