Fugacities in Mixtures

As described in Section 2.4.3, the behavior of a real fluid can also be expressed by means of the fugacity, which is applicable to mixtures as well The fugadty of a species in a mixture is defined in the same way as the fugacity of a pure component 

The fugacity coefficient of a species in a mixture is also defined analogously to the definition of a pure component fugacity coefficient, where the partial pressure replaces the total pressure  

The fugacity coefficient in a mixture may be represented as a function of the variables temperature, pressure, or volume, corresponding to the equations for pure components in Section 2.4.3  

The equations for mixtures additionally contain the mole numbers, that is, the composition as a variable. They are valid for both vapor and liquid. With the help of these equations, the fugacity coefficient can be determined from experimental PvT Xi(yi) data or it can be calculated using an equation of state.  

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The data base contains provisions for a simple augmentation by up to eight additional compounds or substitution of other compounds for those included. Binary interaction parameters necessary for calculation of fugacities in liquid mixtures are presently available for 180 pairs. 

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. ... 

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... 

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. 

Fugacity in Liquid Mixtures Raoult s Law and Henry s Law Each component in a liquid mixture has an equilibrium vapor pressure, and hence, a vapor fugacity. These fugacities are functions of the composition and the nature of the components, with the total vapor fugacity equal to the sum of the fugacities of the components, That is,... 

Equation (6.38) defines fugacity in a mixture through the relationship... 

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. 

In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, fA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal ... 

In 1977 De Santis et al. (J5) as well as Heidemann et al. ( ) calculated the gas-phase fugacities in the systems HjO-air and H2O-N2-CO2 by equation of state in these calculations the liquid phase was not included. One of the authors (7J showed in 1978 that aqueous systems with some inert gases and alkanes as well as H2S and C02 could be represented by an equation of state if the molecular weight of water was artificially increased. An extension of this method applied to alcohols was found to be only partially successful. Gmehling et al. (8) treated polar fluids such as alcohols, ketones and water as monomer-dimer mixtures using Donohue s equation of state (9) various systems including water-methanol and water-ethanol were succussfully represented. 

If we consider, for example, compound i in a liquid mixture, e.g., in organic or in aqueous solution (subscript t see Fig. 3.9pure liquid compound by [note that for convenience, we have chosen the pure liquid compound (superscript ) as our reference state] ... 

The fugacity in Equation 2-39 is that of the component in the equilibrium mixture. However, fugacity of only the pure component is usually known. It is also necessary to know something about how the fugacity depends on the composition in order to relate the two, therefore, assumptions about the behavior of the reaction mixture must be made. The most common assumption is that the mixture behaves as an ideal solution. In this case, it is possible to relate the fugacity, f, at equilibrium to the fugacity of the pure component, f, at the same pressure and temperature by... 

A system consisting of a liquid mixture and vapor is in equilibrium if, for any component i, the fugacities in the vapor and liquid phases, fiY and are equal. 

Sum of squares of residuals Feedback-reset time Fugacity coefficient of pure component Entrainment or occlusion ratio Fugacity coefficient in mixture... 

Activity is the ratio of the fugacity of a component in a system relative to the standard-state fugacity. In a drying system, it is the ratio of the vapor pressure of a solvent (e.g., water) in a mixture to the pure solvent vapor pressure at the same temperature. Boiling occurs when the vapor pressure of a component in a liquid exceeds the ambient total pressure. 

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