Vapor-Liquid Equilibrium Relations

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 of NH3 in the vapor phase and in the liquid phase. The composition of water (B) is fixed ify or. 4 is specified, since + ys = TO and, x + Xg = 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. 

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. 

In Fig. 11.1-1, if we start with a cold liquid mixture ofx, = 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 

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 27 is similar to the solid-liquid equilibrium relation used for non-electrolytes. As in the case of the vapor-liquid equilibrium relation for HC1, the solid-liquid equilibrium expression for NaCl is simple since the electrolyte is treated thermodynamically the same in both phases. 

A method of prediction of the salt effect of vapor-liquid equilibrium relationships in the methanol-ethyl acetate-calcium chloride system at atmospheric pressure is described. From the determined solubilities it is assumed that methanol forms a preferential solvate of CaCl296CH OH. The preferential solvation number was calculated from the observed values of the salt effect in 14 systems, as a result of which the solvation number showed a linear relationship with respect to the concentration of solvent. With the use of the linear relation the salt effect can be determined from the solvation number of pure solvent and the vapor-liquid equilibrium relations obtained without adding a salt. 

Therefore, the solvation number can be calculated by determining xia from the measured values using the vapor-liquid equilibrium relation obtained without... 

In the usual distillation problem, the operating pressure, the feed composition and thermal condition, and the desired product compositions are specified. Then the relations between the reflux rates and the number of trays above and below the feed can be found by solution of the material and energy balance equations together with a vapor-liquid equilibrium relation, which may be written in the general form... 

The performance of a given column or the equipment requirements for a given separation are established by solution of certain mathematical relations. These relations comprise, at every tray, heat and material balances, vapor-liquid equilibrium relations, and mol fraction constraints. In a later section, these equations will be stated in detail. For now, it can be said that for a separation of C components in a column of n trays, there still remain a number, C + 6, of variables besides those involved in the dted equations. These must be fixed in order to define the separation problem completely. Several different combinations of these C + 6 variables may be feasible, but the ones commonly fixed in column operation are the following ... 

TT Then salt is added to a volatile solvent mixture, there is a salt effect—a change in the vapor-liquid equilibrium relation. This salt effect occurs because salt forms a preferential solvate with a particular component of the solvent mixture, causing a drop in partial pressure of the particular component which forms the preferential solvate. Results of the studies conducted based on this idea are reported by the author in References 1 and 2. In the past studies, the vapor-liquid equilbrium relation of the system for which formation of preferential solvate had been expected was observed, preferential solvation number was calculated based on the actually observed values, and further, salt effect was predicted based on the preferential solvate number. The author has... 

When a preferential solvate is formed across salt and a particular component in a solvent mixture, the preferentially solvated component is assumed to be nonvolatile. Hence, the essential concentration of the preferentially solvated component in the solvent mixture is reduced as much as the solvated component. The vapor-liquid equilibrium relation obtained under the addition of a salt may well be considered to be the same as the vapor-liquid equilibrium without the salt for liquid-phase composition from which the solvents forming solvates are excluded. Based on this idea, the essential concentration at the time when salt forms a preferential solvate with the primary component is given by Equation 1. Then we can obtain the preferential solvation number from the observed values of the salt effect. As the concentration of solvent is decreased by the number of solvated molecules, the actual solvent composition participating in the vapor-liquid equilibrium is changed. Assuming that a salt forms the solvate with the first component, the actual composition Xia is given by... 

Therefore, the solvation number can be calculated by determining x1sl from the measured values using the vapor-liquid equilibrium relation obtained without adding a salt. When a salt forms the solvation with the second component, the following three equations can be derived in a similar manner. 

We can eliminate eioh and > h,o in this equation in favor of XetoH and Xh o hy the vapor/liquid equilibrium relation ... 

For a process system that involves liquid and gas streams in equilibrium and vapor-liquid equilibrium relations for all distributed components, draw and label the flowchart, carry out the degree-of-freedom analysis, and perform the required calculations. 

For a system consisting of C components, the phase rule indicates that, in the two-phase region, there are F=C-2 + 2 = C degrees of freedom. That is, it takes C independent variables to define the thermodynamic state of the system. The independent variables may be selected from a total of 2C intensive variables (i.e., variables that do not relate to the size of the system) that characterize the system the temperature, pressure, C - 1 vapor-component mole fractions, and C - 1 liquid-component mole fractions. The number of degrees of freedom is the number of intensive variables minus the number of equations that relate them to each other. These are the C vapor-liquid equilibrium relations, Yj = K,X, i=l,. .., C. The equilibrium distribution coefficients, AT, are themselves functions of the temperature, pressure, and vapor and liquid compositions. The number of degrees of freedom is, thus, 2C - C = C, which is the same as that determined by the phase rule. 

Mathematical solution of absorbers, like other multistage separation processes, involves setting up material and energy balance equations and vapor-liquid equilibrium relations that describe the entire column. The resulting set of simultaneous equations is then solved by some suitable technique. 

The column section can be solved by simultaneous solution of the component mass balance and energy balance equations and the vapor-liquid equilibrium relations. Additional equations include the temperature, pressure, and composition dependence of the equilibrium coefficients and enthalpies. The equations for stage j are as follows ... 

The equations describing total column operation include vapor-liquid equilibrium relations. Equation 5.12 component balances in the rectifying and stripping sections, Equations 5.13 and 5.14 feed stage component balance. Equation 5.15 feed stage energy balance and overall material balance, expressed as Equations 5.16 and 5.17 and overall column component balance. Equation 5.18 ... 

Section 3.3.3). The product rates and compositions in such a column are determined by the energy and material balances and the vapor-liquid equilibrium relations. Under these circumstances the operator has no control over the column performance. 

The SO2 vapor-liquid equilibrium relation at the column temperature and pressure is expressed as Y = 40X. The column cross-sectional area is 1.5 m and the packing height is 3.5 m. It is required to calculate the following ... 

M. Hirata and H. Komatsu Vapor-Liquid Equilibrium Relations Accompanied with Chemical Reaction, Kagaku Kogaku (abridged edition), 5(1) 143 (1967). 

A common method of plotting the equilibrium data is shown in Fig. 11.1-2, where Sec. 11.1 Vapor-Liquid Equilibrium Relations 641... 

Equation (6.3.13) has been plotted in Figure 6.7 for different values of rj and r2 and in Eq. (6.3.16) in Figure 6.8 for different values of rj. The relation between Fj and/, as shown in these plots, is quite similar to the vapor-liquid equilibrium relations. The vapor pressure of an ideal hquid behaves in the same way as the eopolymer monomer mixture eomposition in Figure 6.8. Accordingly, copolymerizations with rjr2 equal to 1 are sometimes termed ideal. 

---