Liquid fugacity coefficient

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Three constants must be known for each component in order to determine its liquid-fugacity coefficient These are the critical temperature, critical pressure, and the acentric factor co. Values of o presented by Chao and Seader12 are listed in Table 14-10. 

Coefficients for use in the Grayson-Streed correlation of the liquid fugacity coefficient ... 

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only Tc, Pc, and o> were required for the calculation. The proposed equation successfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. 

The applicability of the proposed equation was tested in terms of its predicted values of the compressibility factors, liquid fugacity coefficients, and isothermal enthalpy departures of pure compounds. 

At low pressures the vapor will be ideal, so q) =1 then, on using (5.5.8) for the liquid fugacity coefficient, neglecting the Poynting factor at low pressures, and taking the pure-i vapor pressure as the standard-state pressure, (12.1.5) reduces to... 

The Chao-Seader correlation and its many modifications, and the Chueh-Prausnitz correlation are examples of this approach. In the Chao-Seader correlation f and ir are combined as a pure liquid fugacity coefficient, v, so that Eq. (5) has three distinct parts —each requiring a unique equation, as just described. 

Figure 2. Comparison of Chao-Seader and Grayson-Streed liquid fugacity coefficient correction terms...

Coefficient in Wohl Eq.(13) y = Constant used in BWR-11 Eq. (Table II) y = Activity coefficient of a component in liquid 6 = Solubility parameter v(0) = Simple fluid liquid fugaclty coefficient v(l) = Liquid fugacity coefficient correction term TT = System pressure... 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

Chapter 3 discusses calculation of fugacity coefficient < ). Chapter 4 discusses calculation of adjusted activity coefficient Y fugacity of the pure liquid f9 [Equation (24)], and Henry s constant H. 

It is important to be consistent in the use of fugacity coefficients. When reducing experimental data to obtain activity coefficients, a particular method for calculating fugacity coefficients must be adopted. That same method must be employed when activity-coefficient correlations are used to generate vapor-liquid equilibria. 

Two additional illustrations are given in Figures 6 and 7 which show fugacity coefficients for two binary systems along the vapor-liquid saturation curve at a total pressure of 1 atm. These results are based on the chemical theory of vapor-phase imperfection and on experimental vapor-liquid equilibrium data for the binary systems. In the system formic acid (1) - acetic acid (2), <() (for y = 1) is lower than formic acid at 100.5°C has a stronger tendency to dimerize than does acetic acid at 118.2°C. Since strong dimerization occurs between all three possible pairs, (fij and not... 

P the other terms provide corrections which at low or moderate pressure are close to unity. To use Equation (2), we require vapor-pressure data and liquid-density data as a function of temperature. We also require fugacity coefficients, as discussed in Chapter 3. 

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. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

The ratio fj/Pyj is written as (/>, and called the partial fugacity coefficient in the liquid phase, this coefficient is written as , and in the gas phase 0. ... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

This fugacity coefficient applies equally to saturated vapor and to saturated liquid at given temperature T Equation (4-278) can therefore equally well be written... 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

The binary interaction parameters are evaluated from liqiiid-phase correlations for binaiy systems. The most satisfactoiy procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, Rearrangement of the logarithmic form yields... 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

The estimation of the two parameters requires not only conversion and head space composition data but also physical properties of the monomers, e.g. reactivity ratios, vapor pressure equation, liquid phase activity coefficients and vapor phase fugacity coefficients. 

Phase Equilibria Models Two approaches are available for modeling the fugacity of a solute in a SCF solution. The compressed gas approach includes a fugacity coefficient which goes to unity for an ideal gas. The expanded liquid approach is given as... 

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