Liquid Phase Fugacities

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. 

In Chapter 2 we discuss briefly the thermodynamic functions whereby the abstract fugacities are related to the measurable, real quantities temperature, pressure, and composition. This formulation is then given more completely in Chapters 3 and 4, which present detailed material on vapor-phase and liquid-phase fugacities, respectively. 

Equation (1) is of little practical use unless the fuga-cities can be related to the experimentally accessible quantities X, y, T, and P, where x stands for the composition (expressed in mole fraction) of the liquid phase, y for the composition (also expressed in mole fraction) of the vapor phase, T for the absolute temperature, and P for the total pressure, assumed to be the same for both phases. The desired relationship between fugacities and experimentally accessible quantities is facilitated by two auxiliary functions which are given the symbols (f... 

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

We can now consider the most convenient form for writing the liquid-phase fugacity of component i. First we consider a condensable component and write... 

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 ... 

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. 

Equation (23) holds only when, for every component i, the same standard-state fugacity is used in both liquid phases. 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

At the user s option, one of two methods can be used to calculate the liquid-phase reference fugacity (1) An empirical... 

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. ... 

In the calculation of vapor phase partial fugacities the use of an equation of state is always justified. In regard to the liquid phase fugacities, there is a choice between two paths ... 

The partial fugacity of component i in the liquid phase is expressed as a function of the total fugacity of this same component in the pure liquid state, according to the following relation ... 

Calculation of the total fugacity of the pure component in the liquid phase. ... 

The fugacity in the liquid phase is determined by methods we have seen previously. 

The properties of the solids most commonly encountered are tabulated. An important problem arises for petroleum fractions because data for the freezing point and enthalpy of fusion are very scarce. The MEK (methyl ethyl ketone) process utilizes the solvent s property that increases the partial fugacity of the paraffins in the liquid phase and thus favors their crystallization. The calculations for crystallization are sensitive and it is usually necessary to revert to experimental measurement. 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

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... 

Fig. I. Effect of pressure on the melting temperature of argon, (a) Pure argon, (b) argon-helium, (c) argon-hydrogen.Qualitative difference between curves (b) and (c) is due to the effect of composition on the liquid-phase fugacity of argon (M6).

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... 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

When two liquid phases exist, the compositions of the two phases a and / are governed by the equality of fugacities for each component ... 

Case I. At sufficiently low pressures, the solubility curve does not intersect the coexistence curve. In this case, the gas solubility is too low for liquid-liquid immiscibility, since the coexistence curve describes only liquid-phase behavior. Stated in another way, the points on the coexistence curve are not allowed because the fugacity f2L on this curve exceeds the prescribed vapor-phase value f2v. The ternary phase diagram therefore consists of only the solubility curve, as shown in Fig. 28a where V stands for vapor phase. 

In Chapter 5, we considered systems in which composition becomes a variable, and defined and described the chemical potential. We showed that the chemical potential provides the condition for spontaneity or equilibrium. It is the potential that drives the flow of mass in a chemical process, A useful quantity related to the chemical potential is the fugacity. It can also be thought of as a measure of the flow of mass in a chemical process, and can be used to determine the point of equilibrium. It is often known as the escaping tendency since it can be used to describe the ease with which mass flows from one phase to another, particularly the flow from a solid or liquid phase to a gas phase. 

Binary (vapor + liquid) equilibria studies involve the determination of / as a function of composition. the mole fraction in the liquid phase. Of special interest is the dependence of/ on composition in the limit of infinite dilution. In the examples which follow, equilibrium vapor pressures, p,. are measured and described. These vapor pressures can be corrected to vapor fugacities using the techniques described in the previous section. As stated earlier, at the low pressures involved in most experiments, the difference between p, and / is very small, and we will ignore it unless a specific application requires a differentiation between the two. 

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