Vapor phase 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. 

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... 

PHIS calculates vapor-phase fugacity coefficients, PHI, for each component in a mixture of N components (N 5. 20) at specified temperature, pressure, and vapor composition. 

PHIS CALCULATES VAPOR PHASE FUGACITY COEFFICIENTS PHI, FOR ALL N... 

CALCULATE VAPOR PHASE FUGACITY COEFFICIENTS FOR ACTUAL COMPOSITION OF... 

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). 

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. 

Thus, equilibrium is achieved when the escaping tendency from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, fj, can be defined by the expression ... 

If an activity coefficient model is to be used at high pressure (Equation 4.27), then the vapor-phase fugacity coefficient can be predicted from Equation 4.47. However,... 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

Equation (4.5) is probably the form that first comes to mind when Henry s law is mentioned. It is applicable up to two atmospheres of pressure (P) and liquid mole fractions (x,) of up to 1%. Higher values of pressure or liquid mole fractions require corrections using vapor phase fugacity coefficients and liquid activity coefficients or fugacity coefficients. 

Although values for vapor-phase fugacity coefficient are easily calculated (Secs. 11.6... 

Solubilitiesattemperaturesand pressures above the critical values of the solvent liave important applications for supercritical separation processes. Examples are extraction of caffeine from coffee beans and separation of asplraltenes from heavy petroleum fractions. For a typical solid/vapor equilibrium (SVE) problem, tire solid/vapor saturation pressure P is very small, and the saturated vapor is for practical purposes an ideal gas. Hence 0 for pure solute vapor at this pressure is close to unity. Moreover, exceptfor very low values of the system pressure P, the solid solubility yj is small, and can be approximated by j, the vapor-phase fugacity coefficient of the solute at infinite dilution. Finally, since is very small, the pressure difference P — in the Poyntingfactor is nearly equal to P at any pressure where tins factor... 

Fugacity coefficient of the solute in the vapor phase Fugacity coefficient of the solute at the saturation pressure Activity coefficient of the solute in the solution Polarizability of the solute or its chemical class... 

The /<-values are calculated by Equation 1.25, with the vapor phase fugacity coefficients calculated from the equation of state and the liquid phase fugacity coefficients for an ideal solution calculated as... 

The following alternative routes are possible for calculating the vapor phase fugacity coefficients ... 

The vapor phase fugacity coefficient, may be calculated, as before, from Equation 1.23, using, for instance, an equation of state. The pure component fugacity in the liquid state is equal to its fugacity in the vapor at equilibrium with the liquid ... 

The vapor phase fugacity coefficient is arbitrarily calculated first followed by the liquid phase fugacity coefficient. 

The goal of a subctitical VLE calculation is to quantitatively predict or correlnte the various kinds of behavior illustrated by Fig. 1.5-1 or by its iso baric or multicomponent coimlerparta. The basis for the calculation is phase-equilibrium formulation of Section 1.2-5, whare liquid-phase fogaciries are elimienled in favor of liquid-phase activity coefficients, aed vapor-pbase fugaciries in favor of vapor-phase fugacity coefficients. Raoult s Law standard states are chosen (tor all components in the liquid phase hence, (fTf- = f > and Eq. (1.2-63) becomes... 

The Chao-Seader method12 is an example of the use of multiple equations of state for the calculation of K values. The Redlich-Kwong equation of state is used to compute the vapor-phase fugacity coefficient the Hildebrand equation for the calculation of the liquid-phase activity coefficient y/% and an extension of Pitzer s modified form of the principle of corresponding states for the calculation of the liquid-phase fugacity coefficient 4> ... 

P. L. Chuch and J. M. Prausnitz Vapor-Phase Fugacity Coefficients in Nonpolar and Quantum-Gas Mixtures. Ind. Eng. Chem. Fundam., 6 192 (1967). 

The vapor phase fugacity coefficient was calculated exactly as described in Chap. 3, in Prausnitz et al., and Eqs. (10) through (23) were used directly. 

Note that the enhancement factor E has contributions from both the Poynting factor and the vapor-phase fugacity coefficient, both of which are important at high pressure, and that —> 1 as 7 —> 7 . 

In a significant departure from conventional practice, Chueh and Prausnitz (11,12) proposed that the critical constraints on the RK equation be relaxed, and that parameters b and c be treated as empirical constants, determined separately for the liquid phase and for the vapor phase of a given substance. The conventional RK expression for (T) was retained the application was to vapor-liquid equilibrium calculations, in which the vapor-phase version of the equation was used for computation of vapor-phase fugacity coefficients, but in which the liquid-phase version was used only for Poynting corrections. Thus, they proposed that... 

Inner loop. The search begins by computing the liquid phase and vapor phase fugacity coefficients. The equations of state used in these problems are explicit in pressure, so the (ps are determined from (4.4.23), which involves an integration over the volume. This requires us to compute the molar volumes and from the equation of state. If the equation is cubic in v, then it should be solved analytically using Cardan s method (Appendix C). However, if the equation is fifth order or higher, then it will have to be solved by trial for v. With the qis known, we compute the K-factors from (11.1.2) and hence get calculated values for the vapor mole fractions xf). Typi-... 

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