Activity Coefficients from Fugacity Coefficients

A particular method for emphasizing the deviation from ideality is to introduce the so-called activity coefficient or fugacity coefficient defined by yi = ft/Pi, so as to rewrite Eq. (3.1.4a) in the form... 

In 5.3 we showed how excess properties, which are difference measures for deviations from ideal-solution behavior, can be obtained from residual properties, which are difference measures for deviations from ideal-gas behavior. In this section we establish a similar set of equations that relate activity coefficients to fugacity coefficients. As a result, the equations given here, together with those in 5.3, establish a complete connection between the description of mixtures based on models for PvTx equations of state and the description based on models for and y. ... 

The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials,... 

The dependence of solubility on % fill in 0.5M (OH) solution is shown in Fig. 5.(10) The solubility in pure water is an order of magnitude smaller under similar conditions. In pure water, activity coefficient (actually fugacity calculated from appropriate compressibility) estimates enable one to get reasonably accurate values for the equilibrium constant. This treatment suggests that the solubizing reaction in pure water is ... 

In terms of activity and the fugacity coefficients, the vapor-liquid equilibrium from Eq. (1.189) becomes... 

The data necessary for the evaluation of the activity coefficients (or fugacities) of the individual gases in a mixture ( 30e) are not usually available, and an alternative treatment for allowing for departure from ideal behavior is frequently adopted. 

Experimental VLE data, namely equilibrium temperature, pressure, and vapor and liquid compositions, can be used to calculate the activity coefficients from Equation 1.29. The X-values are calculated from the composition data, /f = T/X . The fugacities and fugacity coefficients, /T, are calculated from the compositions, temperature, and pressure, using, for instance, an equation of state and liquid density data as described earlier. The activity coefficients are then calculated by rearranging Equation 1.29 ... 

Besides being expressed in terms of activity coefficients, the fugacities of a liquid solution can also be calculated from equations of state in the form of a fugacity coefficient ( ) . The equality of fugacities of two liquid phases at equilibrium becomes expressed by... 

In addition to the excess properties, which are difference measures for deviations from ideal-solution behavior, we also find it convenient to have ratio measures. In particular, for phase equilibrium calculations, it proves useful to have ratios that measure how the fugacity of a real mixture deviates from that of an ideal solution. Such ratios are called activity coefficients. The activity coefficients can be viewed as special kinds of a more general quantity, called the activity so we first introduce the activity ( 5.4.1) and then discuss the activity coefficient ( 5.4.2). 

We would extract an appropriate expression for the activity coefficients from FFF 2-5. Usually we use the same FFF for the same component in both phases, but this is not necessary, and we usually choose the same standard-state fugacity for the same component in both phases, then (12.1.24) reduces to a simple ratio of activity coefficients. 

Deviation from ideality of real liquid solutions manifests itself by departure of the various characteristics such as partial pressure, fugacity, activity, and activity coefficient, from the simple linear relationships previously shown. Of these, perhaps the most convenient to study is the activity coefficient 7. Thus, for a binary solution of A and 5,... 

As can be seen from Eq. (14), the solubility of a solid in an SCE depends not only on solid-state parameters, such as sublimation pressure and molar volume V , but additionally on the fiigacity coefficient (]), . The fugacity coefficient is the supercritical analogue to the activity coefficient (5). The fugacity coefficient varies not only with the type of fluid but with temperature and pressure (53). Therefore, solubility of solids can be significantly influenced by changing the density of SCFs on alteration of temperature and/or pressure. The fugacity coefficient is the key variable that explains the different solubility of solids in SCFs compared with ordinary liquids. 

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

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. 

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... 

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... 

The ratio f/f° is called activity, a. Note This is not the activity coefficient. The activity is an indication of how active a substance is relative to its standard state (not necessarily zero pressure), f°. The standard state is the reference condition, which may be anything however, most references are to constant temperature, with composition and pressure varying as required. Fugacity becomes a corrected pressure, representing a specific component s deviation from ideal. The fugacity coefficient is ... 

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... 

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. 

With the standard state expressed in this manner, the activity of the gas becomes the fugacity expressed in bars. We will usually follow this convention as we work with activities of gases. An added convenience comes from being able to relate fugacity to pressure through the fugacity coefficient [Pg.284]

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

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,... 

In the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

The program produces in its output dataset a block of results that shows the concentration, activity coefficient, and activity calculated for each aqueous species (Table 6.4), the saturation state of each mineral that can be formed from the basis, the fugacity of each such gas, and the system s bulk composition. The extent of the system is 1 kg of solvent water and the solutes dissolved in it the solution mass is 1.0364 kg. 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

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