Fugacity and activity coefficient

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. 

Since the fugacity and activity coefficients are mathematically complex functions of the compositions, finding corresponding compositions of the two phases at equilibrium when the equations are known requires solutions by trial. Suitable procedures for making flash calculations are presented in the next section, and in greater detail in some books on thermodynamics, for instance, the one by Walas (1985). In making such calculations, it is usual to start by assuming ideal behavior, that is,... 

After the ideal equilibrium compositions have been found, they are used to find improved values of the fugacity and activity coefficients. The process is continued to convergence. 

Since the equations for fugacity and activity coefficients are complex, solution of this kind of problem is feasible only by computer. Reference is made in Example 13.3 to such programs. There also are given the results of such a calculation which reveals the magnitude of deviations from ideality of a common organic system at moderate pressure. 

The Lee-Erbar-Edmister method is of the same type, but uses different expressions for the fugacity and activity coefficients. The vapor phase equation of state is a three-parameter expression, and binary interaction corrections are included. The liquid phase activity and fugacity coefficient expressions were derived to extend the method to lower temperatures and to improve accuracy. Binary interaction terms were included in the liquid activity coefficient equation. 

Unfortunately, K values are generally composition-dependent through the fugacity and activity coefficients. Only in ideal systems is the composition dependency removed ... 

It is interesting to note that the vapor and liquid compositions are usually different for ideal mixtures. We can see this from Eq. (6.6), since different pure component vapor pressures are rarely equal at the same temperature. This picture changes when nonideal mixtures are considered. As we see from Eq. (6.55, the vapor and liquid mole fractions can become equal when the fugacity and activity coefficients alter the pressure ratio enough to cause the K value to become unity. We then have an azeotrope. 

As the fugacities are not in themselves quantities which are easily established experimentally, it is necessary to relate them to easily determinable quantities—e.g., temperature, pressure, and composition. This is done by introducing the fugacity and activity coefficients and yi which are defined as follows,... 

Although the composition of an ideal solution can be predicted theoretically, few solutions are ideal, and fugacities and activity coefficients are seldom available for real systems. Hence, in general, too little is known of the direct relationships between solubilities and the specific properties of the solute and the solvent to permit prediction of solubility curves. The characteristics of each system must be determined experimentally. In many cases, it is not even possible to predict the effect of temperature on the solubility values of a given solute-solvent system. 

Since the equations for fugacity and activity coefficients are complex, solution of this kind of problem is feasible only by com-... 

Fugacity and activity coefficients are assumed to approach unity under the above conditions, and a standard free energy of adsorption AG r can be obtained... 

In summary, fugacity and activity coefficients were introduced in thermodynamics in order to explain deviations from ideal behavior and thus serving as correction factors for nonideal behavior. 

Thermodynamic Relationships for Fugacities and Activity Coefficients for the Liquid Phase... 

Here the component-2 fugacity and activity coefficients depend on the solute-free mole fractions. However, for a binary X ff = 1, Y2 = 1/ nd at low pressures the fugacity coefficients are unity, so (12.1.19) reduces to... 

Fugacity and activity are basically compositional terms. In ideal solutions they are not necessary pressure and various composition terms can be directly linked to the Gibbs energy. Real solutions have a variety of intermolecular forces, so that ideal solution models need correction factors. These corrections can be made either to the composition terms (fugacity and activity coefficients) or to the thermodynamic potentials (excess functions), and efforts to model these correction factors in mathematical terms have always been, and likely always will be, an important research field. 

The form taken by the equilibrium constant depends on the type of expression which is substituted in the above equation for the purpose of expressing the chemical potentials in terms of the composition this in its turn dex>ends on additional physical knowledge concerning whether or not the real system in question may be approximately represented by means of a model, such as the perfect gas or the ideal solution. If the system does not approximate to either of these models it is still possible, of course, to formulate an equilibrium constant in terms of fugacities or in terms of mole fractions and activity coefficients. However, this isapurely formal process the fugacities and activity coefficients are themselves defined in terms of the chemical potentials and therefore the knowledge contained in equation (10 1) is in no way increased, but is obtained in a more convenient form. 

The classical method of obtaining fugacity and activity coefficients of gases is by experimental measurement of the molar volume as a function of P and T (known as the P-V-T method) followed by application of equation (1) or (2). Unfortunately, the P-V-T method is difficult, expensive and time consuming when attempted at pressures above one Kbar and temperatures above 500 C. For those reasons very few measurements are available even for one-component (pure) substances. The P-T regions for which experimental P-V-T data are available for systems of geologic interest are listed in Table 1. 

In Equation 14.8 we include fugacity and activity coefficients in analogy to Equation 14.6 but in the lAST both are ignored. In lAST we assume that the gas phase is ideal (fugacity coefficients equal to unity) and that the activity coefficient of the gas adsorbate in the solid is also one. Thus, Equation 14.8 solved for the mole fraction of the gas in the solid (which is the property we wish to calculate) is ... 

Example 7.4 At 1 atm pressure, the ethanol-water azeotrope (discussed more in the next chapter) has the same composition of 10.57 mol% water, (and thus 89.43 mol% ethanol) in both vapor and liquid phases at a temperature of 78.15°C [5] (Figure 7.8). At this temperature the pure species vapor pressures are water 0.434 atm and ethanol 0.993 atm. Estimate the fugacity and activity coefficient for each species in each phase. 

This program performs bubble-point and dew-point calculations, with various fugacity and activity coefficient corrections. After the appropriate bubble-point or dew-point calculation type has been selected, the main window will be presented. 

Add a species to the mixture by using the Add button. Choose the desired fugacity and activity coefficient corrections at the bottom of the window. If the multicomponent Wilson model is used, the model parameters should entered by choosing Edit in the Wilson Model Parameters frame. Once everything is set, choose Solve Unknowns to perform the bubble-point or dew-point calculation. Choose More Information to see the values of the correction factors at equilibrium. 

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