Fugacity coefficient, liquid phase

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. 

The activity coefficient y relates the liquid-phase fugacity... 

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... 

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. 

As discussed in Chapter 2, for noncondensable components, the unsymmetric convention is used to normalize activity coefficients. For a noncondensable component i in a multicomponent mixture, we write the fugacity in the liquid phase... 

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 computer subroutines for calculation of vapor-phase and liquid-phase fugacity (activity) coefficients, reference fugac-ities, and molar enthalpies, as well as vapor-liquid and liquid-liquid equilibrium ratios, are described and listed in this Appendix. These are source routines written in American National Standard FORTRAN (FORTRAN IV), ANSI X3.9-1978, and, as such, should be compatible with most computer systems with FORTRAN IV compilers. Approximate storage requirements and CDC 6400 execution times for these subroutines are given in Appendix J. 

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. ... 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), 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... 

For any component i in a liquid phase, the fugacity of f is most conveniently related to the mole fraction xt by use of the activity coefficient, y(, according to... 

For liquid mixtures at low pressures, it is not important to specify with care the pressure of the standard state because at low pressures the thermodynamic properties of liquids, pure or mixed, are not sensitive to the pressure. However, at high pressures, liquid-phase properties are strong functions of pressure, and we cannot be careless about the pressure dependence of either the activity coefficient or the standard-state fugacity. 

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

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... 

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 ... 

The liquid-phase fugacity coefficient

[Pg.60]

Before an equation of state can be applied to calculate vapor-liquid equilibrium, the fugacity coefficient < />, for each phase needs to be determined. The relationship between the fugacity coefficient and the volumetric properties can be written as ... 

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. 

Furthermore, 5 vapor-liquid phase equilibria are involved, e.g. for molecular NH3, C02, H2S and S02 and for water. Applying the concept of Henry s constant Hi for the solution of a gas i in pure water and fugacity-coefficient for describing the influence of inter-molecular forces in the vapor phase, the resulting equations are ... 

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

The first method, which is the more flexible, is to use an activity coefficient model, which is common at moderate or low pressures where the liquid phase is incompressible. At high pressures or when any component is close to or above the critical point (above which the liquid and gas phases become indistinguishable), one can use an equation of state that takes into account the effect of pressure. Two phases, denoted a and P, are in equilibrium when the fugacity / (for an ideal gas the fungacity is equal to the pressure) is the same for each component i in both phases ... 

Effective use of this general equation requires explicit introduction of the compositions of the phases. This is done either through the activity coefficient, y, or the fugacity coefficient, ( ) A Two procedures are in common use. By the gamma—phi approach, activity coefficients for the liquid phase enter by equation 202 and fugacity coefficients for the vapor phase by equation 164 equation 220 then becomes equation 221 ... 

The second common procedure for VLE calculations is the equation-of-state approach. Here, fugacity coefficients replace the fugacities for both liquid and vapor phases, and equation 220 becomes equation 226 ... 

Use of equation 247 for actual calculations requires explicit introduction of composition variables. As in phase-equilibrium calculations, this is normally done for gas phases through the fugacity coefficient and for liquid phases through the activity coefficient. Thus, either... 

An initial guess for the pressure is assumed and the fugacity coefficient of each component in the liquid phase ( ) can be calculated. An initial guess is also assumed for the fugacity coefficient of each component in the vapour phase ( v), and consequently a first estimate of the vapour composition is evaluated. With this value of y, the fugacity coefficients in the vapour phase are recalculated using the equation of state and a second estimate for y,- is evaluated. This iterative procedure is continued until the difference between two successive values of the composition are below a predetermined error. At this point, the sum of y, is checked if the sum is different from unity a new value of the pressure is assumed for a new iteration. The iterative procedure ends when the y, differs from unity by less then a given value. 

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

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