Fugacity mixture coefficient

It should be noted that, while the expressions for the derivatives of the fugacity (activity) coefficients in ternary mixtures with respect to mole fractions (Eqs. (25) and (26)) employed to derive Eq. (42) are rigorous, Eq. (42) for the solubility also implies that lim o 2 i3 = 0. The latter approximation is probably not entirely accurate at very high pressures when the intermolecular interactions between the SC fluids become comparable to those in liquids. Indeed, the predicted solubilities of phenanthrene in the mixture of SC CO2 and SC ethane (Eig. 3) deviate somewhat from the experimental data [26] at high pressures. This deviation might have been caused also by the experimental solubilities of phenanthrene in SC ethane [26] employed in our calculations, which are quite different from other experimental data [27]. 

Note that with this modification to the van der Waals one-fluid model, the fugac-ity coefficient expression for species i given in eqn. (3.3.9) will change because an additional compositional dependence has been introduced to the a term of the EOS. For the PR EOS, with the van der Waals one-fluid model, a more general form of the fugacity coefficient expression of species i in a mixture is... 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

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... 

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

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 ... 

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. 

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). ... 

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. 

Eor nonideal vapor-phase behavior, the fugacity coefficient for component i in the mixture must be determined ... 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

When i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

The fugacityy) of pure compressed liqiiid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor 9i = by an integrated form of Eq. (4-161), written for pure species i and evalu-atea at temperature T and the corresponding vapor pressure P = Equation (4-276) written for pure species i becomes... 

We have a considerable body of knowledge to help us to say something about the third coefficient, the variation of fugacity with composition. Many empirical and semiempirical expressions (e.g., Margules, Van Laar, Scat-chard-Hildebrand) have been investigated toward that end. Most of our experience in this regard is limited to liquid mixture at low pressures, where... 

The fugacity of a component i in a gas mixture is related to the total pressure P and to its mole fraction yt through the fugacity coefficient [Pg.144]

The fugacity coefficient is a function of pressure, temperature, and gas composition. It has the useful property that for a mixture of ideal gases (Pi = 1 for all i. The fugacity coefficient is related to the volumetric properties of the gas mixture by either of the exact relations (B3, P5, R6) ... 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

To obtain the fugacity coefficient of a component / in the mixture, we must differentiate In q>M with respect to composition. For example, for a binary solution having composition yl we have the rigorous relation ... 

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. 

In Section I, we indicated that significant progress in understanding high-pressure thermodynamics of mixtures requires a quantitative description of the variation of fugacity with pressure as given by Eq. (3). To obtain the effect of pressure on activity coefficient we substitute as follows ... 

In Section HI, we discussed the relation between fugacities and activity coefficients in liquid mixtures, and we emphasized that we have a fundamental choice regarding the way we wish to relate the fugacity of a component to the pressure and composition. This choice follows from the freedom we have in choosing a convention for the normalization of activity coefficients. 

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... 

For more comprehensive calculations of the fugacity coefficients in mixtures, see J. M. Prausnitz, R. N. Lichtenthaler, and E. G. de Azevedo. Modular Thermodynamics of Fluid Phase Equilibria, Prentice Hall. Englewood Cliffs. N.J., 19S6. Chapter 5. 

The fugacity coefficients in Equation (7.29) can be calculated from pressure-volume-temperature data for the mixture or from generahzed correlations. It is frequently possible to assume ideal gas behavior so that = 1 for each component. Then Equation (7.29) becomes... 

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