11.1. VAPOR–LIQUID EQUILIBRIUM RELATIONS

11.1A. Phase Rule and Raoult's Law

As in the gas–liquid systems, the equilibrium in vapor–liquid systems is restricted by the phase rule, Eq. (10.2-1). As an example we shall use the ammonia–water, vapor–liquid system. For two components and two phases, F from Eq. (10.2-1) is 2 degrees of freedom. The four variables are temperature, pressure, and the composition yA of NH3 in the vapor phase and xA in the liquid phase. The composition of water (B) is fixed if yA or xA is specified, since yA + yB = 1.0 and xA + xB = 1.0. If the pressure is fixed, only one more variable can be set. If we set the liquid composition, the temperature and vapor composition are automatically set.

An ideal law, Raoult's law, can be defined for vapor–liquid phases in equilibrium:

Equation 11.1-1


where pA is the partial pressure of component A in the vapor in Pa (atm), PA is the vapor pressure of pure A in Pa (atm), and xA is the mole fraction of A in the liquid. This law holds only for ideal solutions, such as benzene–toluene, hexane–heptane, and methyl alcohol– ethyl alcohol, which are usually substances very similar to each other. Many systems that are ideal or nonideal solutions follow Henry's law in dilute solutions.

11.1B. Boiling-Point Diagrams and x-y Plots

Often the vapor–liquid equilibrium relations for a binary mixture of A and B are given as a boiling-point diagram, shown in Fig. 11.1-1 for the system benzene (A)–toluene (B) at a total pressure of 101.32 kPa. The upper line is the saturated vapor line (the dew-point line) and the lower line is the saturated liquid line (the bubble-point line). The two-phase region is in the region between these two lines.

Figure 11.1-1. Boiling point diagram for benzene (A)–toluene (B) at 101.325 kPa (1 atm) total pressure.


In Fig. 11.1-1, if we start with a cold liquid mixture of xA1 = 0.318 and heat the mixture, it will start to boil at 98°C (371.2 K), and the composition of the first vapor in equilibrium is yA1 = 0.532. As we continue boiling, the composition xA will move to the left since yA is richer in A.

The system benzene–toluene follows Raoult's law, so the boiling-point diagram can be calculated from the pure vapor-pressure data in Table 11.1-1 and the following equations:

Equation 11.1-2


Equation 11.1-3


Equation 11.1-4


Table 11.1-1. Vapor-Pressure and Equilibrium-Mole-Fraction Data for Benzene–Toluene System
Vapor Pressure
TemperatureBenzeneTolueneMole Fraction Benzene at 101.325 kPa
K°CkPamm HgkPamm HgxAyA
353.380.1101.32760  1.0001.000
358.285116.987746.03450.7800.900
363.290135.5101654.04050.5810.777
368.295155.7116863.34750.4110.632
373.2100179.2134474.35570.2580.456
378.2105204.2153286.06450.1300.261
383.8110.6240.01800101.3276000

EXAMPLE 11.1-1. Use of Raoult's Law for Boiling-Point Diagram

Calculate the vapor and liquid compositions in equilibrium at 95°C (368.2 K) for benzene–toluene using the vapor pressure from Table 11.1-1 at 101.32 kPa.

Solution: At 95°C from Table 11.1-1 for benzene, PA = 155.7 kPa and PB = 63.3 kPa. Substituting into Eq. (11.1-3) and solving,


Hence, xA = 0.411 and xB = 1 − xA = 1 − 0.411 = 0.589. Substituting into Eq. (11.1-4),



A common method of plotting the equilibrium data is shown in Fig. 11.1-2, where yA is plotted versus xA for the benzene–toluene system. The 45° line is given to show that yA is richer in component A than is xA.

Figure 11.1-2. Equilibrium diagram for system benzene (A)–toluene (B) at 101.32 kPa (1 atm).


The boiling-point diagram in Fig. 11.1-1 is typical of an ideal system following Raoult's law. Nonideal systems differ considerably. In Fig. 11.1-3a the boiling-point diagram is shown for a maximum-boiling azeotrope. The maximum temperature Tmax corresponds to a concentration xAz and xAz = yAz at this point. The plot of yA versus xA would show the curve crossing the 45° line at this point. Acetone–chloroform is an example of such a system. In Fig. 11.1-3b a minimum-boiling azeotrope is shown with yAz = xAz at Tmin. Ethanol–water is such a system.

Figure 11.1-3. Equilibrium boiling-point diagrams: (a) maximum-boiling azeotrope, (b) minimum-boiling azeotrope.


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