Gamma-phi approach

Effective use of this general equation requires expHcit introduction of the compositions of the phases. This is done either through the activity coefficient, y, or the fugacity coefficient, ( ) Two procedures are in common use. By the gamma—phi approach, activity coefficients for the Hquid phase enter by equation 202 and fugacity coefficients for the vapor phase by equation 164 equation 220 then becomes equation 221 ... 

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 identifying superscripts I and v are omitted here with the understanding that Ji a.nd f are liquid-phase properties, whereas 9 is a vapor-phase property. Applications of Eq. (4-277) represent what is known as the gamma/phi approach to XT E calculations. 

Solute/Solvent Systems The gamma/phi approach to X T.E calculations presumes knowledge of the vapor pressure of each species at the temperature of interest. For certain binary systems species I, designated the solute, is either unstable at the system temperature or is supercritical (T > L). Its vapor pressure cannot be measured, and its fugacity as a pure liquid at the system temperature/i cannot be calculated by Eq. (4-281). 

Equation-of-State Approach Although the gamma/phi approach to X- E is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactoiy for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for is included only for the local-composition equations, and it is at best only approximate. 

A generally apphcable alternative to the gamma/phi approach results when both the hquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase ... 

This information allows prediction of X T.E at 323.15 K and at the higher temperatures, 372.8, 397.7, and 422.6 K, for which measured X T.E values are given by Wilsak, et al. (Fluid Phase Equilibria, 28, pp. 13-37 [1986]). Values of In yX and hence of the Margules parameters at the higher temperatures are given by Eq. (4-325) with Cf = 0. The pure-species vapor pressures in all cases are the measured values reported with the data sets. Res lilts of these calculations are displayed in Table 4-1, where the parentheses enclose values from the gamma/ phi approach as reported in the papers cited. 

For the analytical equations, there are two methods to compute the vapour-liquid equilibrium for systems. The equation of state method (also known as the direct or phi-phi method) uses an equation of state to describe both the liquid and vapour phase properties, whereas the activity coefficient method (also known as the gamma-phi approach) describes the liquid phase via an activity coefficient model and the vapour phase via an equation of state. Recently, there have also been modified equation of state methods that have an activity coefficient model built into the mixing mles. These methods can be both correlative and predictive. The predictive methods rely on the use of group contribution methods for the activity coefficient models such as UNIFAC and ASOG. Recently, there have also been attempts to develop group contribution methods for the equation of state method, e.g. PRSK. " For a detailed history on the development of equations of state and their applications, as well as activity coefficient models, refer to Wei and Sadus, Sandler and Walas. ... 

Equation-of-State Approach Although the gamma/phi approach... 

Results of calculations with the Margules equations are displayed as the primary entries at each temperature in Table 4-7. The values in parentheses are from the gamma/phi approach as reported in the papers cited. 

Gas solubility usually refers to the liquid-phase mole fraction x, that occurs in a VLB situation in which the system temperature T is above the critical temperature of the pure solute (T > Tcil but below the critical temperature of at least one other component (T < Tg ). An example is CO2 dissolved in a carbonated beverage. These situations can be described using either a phi-phi or a gamma-phi approach the gamma-phi method is the traditional approach. 

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