Vapor-Phase Nonideality

Eor nonideal vapor-phase behavior, the fugacity coefficient for component i in the mixture must be determined ... 

K VALUE FOR IDEAL LIQUID PHASE, NONIDEAL VAPOR PHASE 3.7... 

For nonideal vapor phase, corrections for nonideaUty should be included. ... 

Thus at all pressures (low enough that nonideal vapor phase corrections can be ignored) we have Th2 =Ti2 = 0.03462 ym = 0.93075... 

It is, of course, possible to derive equations analogous to Eqs. 10.2-17 and 10,2-18 for a nonideal vapor phase. 

NONIDEAL LIQUID MIXTURES 3.5 K VALUE FOR IDEAL LIQUID PHASE, NONIDEAL VAPOR PHASE 3.7 K VALUE FOR IDEAL VAPOR PHASE, NONIDEAL LIQUID PHASE 3.8 THERMODYNAMIC CONSISTENCY OF EXPERIMENTAL VAPOR-LIQUID EQUILIBRIUM DATA 3.8 ESTIMATING INFINITE-DILUTION ACTIVITY COEFFICIENTS 3.10... 

We proceed now with a discussion of the more general case nonideal liquid and nonideal vapor phases. Notice that the type of calculation involved depends on the information given, as described in Table 13.1. 

The method proposed in this monograph has a firm thermodynamic basis. For vapo/-liquid equilibria, the method may be used at low or moderate pressures commonly encountered in separation operations since vapor-phase nonidealities are taken into account. For liquid-liquid equilibria the effect of pressure is usually not important unless the pressure is very large or unless conditions are near the vapor-liquid critical region. 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Conclusion Effect of Independent Variables on Vapor-Phase Nonideality... 

As discussed in Chapter 3, at moderate pressures, vapor-phase nonideality is usually small in comparison to liquid-phase nonideality. However, when associating carboxylic acids are present, vapor-phase nonideality may dominate. These acids dimerize appreciably in the vapor phase even at low pressures fugacity coefficients are well removed from unity. To illustrate. Figures 8 and 9 show observed and calculated vapor-liquid equilibria for two systems containing an associating component. 

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. 

At low pressures, it is often permissible to neglect nonidealities of the vapor phase. If these nonidealities are not negligible, they can have the effect of introducing a nonrandom trend into the plotted residuals similar to that introduced by systematic error. Experience here has shown that application of vapor-phase corrections for nonidealities gives a better representation of the data by the model, oven when these corrections... 

Cases 3 and 4 show strong vapor-phase nonidealities as well. 

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

With flashes carried out along the appropriate thermodynamic paths, the formalism of Eqs. (6-139) through (6-143) applies to all homogeneous equihbrium compressible flows, including, for example, flashing flow, ideal gas flow, and nonideal gas flow. Equation (6-118), for example, is a special case of Eq. (6-141) where the quahty x = and the vapor phase is a perfect gas. 

In systems that exhibit ideal liquid-phase behavior, the activity coefficients, Yi, are equal to unity and Eq. (13-124) simplifies to Raoult s law. For nonideal hquid-phase behavior, a system is said to show negative deviations from Raoult s law if Y < 1, and conversely, positive deviations from Raoult s law if Y > 1- In sufficiently nonide systems, the deviations may be so large the temperature-composition phase diagrams exhibit extrema, as own in each of the three parts of Fig. 13-57. At such maxima or minima, the equihbrium vapor and liqmd compositions are identical. Thus,... 

Pure-component vapor pressures can be used for predicting solu-bihties for systems in which RaoiilFs law is valid. For such systems Pa = Pa a, where p° is the pure-component vapor pressure of the solute andp is its partial pressure. Extreme care should be exercised when attempting to use pure-component vapor pressures to predict gas-absorption behavior. Both liquid-phase and vapor-phase nonidealities can cause significant deviations from the behavior predicted from pure-component vapor pressures in combination with Raoult s law. Vapor-pressure data are available in Sec. 3 for a variety of materials. 

Vapor-phase fugacity coefficients are needed not only in high-pressure phase equilibria, but are also of interest in high-pressure chemical equilibria (D6, K7, S4). The equilibrium yield of a chemical reaction can sometimes be strongly influenced by vapor-phase nonideality, especially if reactants and products have small concentrations due to the presence in excess of a suitably chosen nonreactive gaseous solvent (S4). 

Example 6.5 Repeat the calculations from Example 6.4 taking into account vapor-phase nonideality. Fugacity coefficients can be calculated from the Peng-Robinson Equation of State (see Poling, Prausnitz and O Connell6 and Chapter 4). 

This is a simple and important result. It equates VPIE to the isotopic difference of standard state free energies on phase change, plus a small correction for vapor phase nonideality, here approximated through the second virial coefficient. Therefore Equation 5.8 is limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at even higher pressures the virial expansion must be abandoned for a more accurate equation of state. 

Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

In equation 21 the vapor phase is considered to be ideal, and all nonideality effects are attributed to the liquid-phase activity coefficient, y. For an ideal solution (7t = 1), equation 21 becomes Raoult s law for the partial pressure,, exerted by the liquid mixture ... 

Since pf is a function of temperature, the dewpoint and bubble-point temperatures for an ideal vapor or liquid mixture can be determined as a function of the total pressure tr from Eq. (9) or (10), respectively. An analogous procedure can be used for real mixtures, but the nonidealities of the liquid and vapor phases must be accounted for. 

The values of yx and y2 calculated in parts a) and b) differ by less than 1%. Therefore, the effects of vapor-phase nonidealities is here minimal. 

The number of equations, M5C + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal if-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required. 

Like gases, solutions can also be thought of as ideal. Raoult s law only works for ideal solutions. Ideal solutions are described as those solutions that follow Raoult s law. Solutions that deviate from Raoult s law are nonideal. What makes a solution deviate from ideal behavior The main reason is intermolecular attractions between solute and solvent. When the attraction between solute and solvent is very strong, the particles attract each other a great deal. This makes it more difficult for solute particles to enter the vapor phase. As a result, fewer particles will enter that state and the vapor pressure will be lower than expected. Remember, Raoult s law operates on the assumption that the reason for a decrease in the number of particles leaving the solution is that fewer can be on the surface in order to leave. If, in addition to this, the solute particles are also holding more tightly to the solvent particles, then fewer will leave the surface than expected. The most ideal solutions are those where the solvent and solute are chemically similar. 

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