Fugacity coefficients

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. 

The fugacity fT of a component i in the vapor phase is related to its mole fraction y in the vapor phase and the total pressure P by the fugacity coefficient ... 

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure... 

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) ... 

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. 

Figures 3 and 4 show fugacity coefficients for two binary systems calculated with Equation (10b). Although the pressure is not large, deviations from ideality and from the Lewis rule are not negligible.

The virial equation is appropriate for describing deviations from ideality in those systems where moderate attractive forces yield fugacity coefficients not far removed from unity. The systems shown in Figures 2, 3, and 4 are of this type. However, in systems containing carboxylic acids, there prevails an entirely different physical situation since two acid molecules tend to form a pair of stable hydrogen bonds, large negative... 

Figure 3-3. Fugacity coefficients for the system acetone-cyclohexanone. ...

As shown elsewhere (Nothnagel et al., 1973), the fugacity coefficient of component i is given by... 

To use Equation (13), it is first necessary to calculate the true fugacity coefficient (ft. This calculation is achieved by utilizing the Lewis fugacity rule... 

Figure 3-6. Fugacity coefficients for saturated mixtures containing two carboxylic acids formic acid (1) and acetic...

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... 

Figure 3-7. Fugacity coefficients for a saturated mixture of propionic acid (1) and raethylisobutylketone (2). Calculations based on chemical method show large variations from ideal behavior.

A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... 

Details for calculating fugacity coefficients are given in Appendix A. 

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. 

Figure 13 presents results for a binary where one of the components is a supercritical, noncondensable component. Vapor-phase fugacity coefficients were calculated with the virial... 

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. 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

The Lewis fugacity rule is used for calculating the fugacity coefficients of the true species, and (2) the second virial co-... 

If the data are correlated assuming an ideal vapor, the reference fugacity is just the vapor pressure, P , the Poynting correction is neglected, and fugacity coefficient is assumed to be unity. Equation (2) then becomes... 

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. 

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE SECOND VIRIAL COEFFICIENTS. 

CALCULATE FUGACITY COEFFICIENTS FOR NQN-ASSOCI ATING COMPONENTS ... 

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