Vapor fugacity coefficient

The vapor fugacity coefficient, It accounts for the effect of vapor nonideality on vapor fugacity, It is usually estimated from an equation of state and is based on system temperature, pressure, and vapor mole fraction. 

When t is zero, the relationship expresses essentially ideal behavior. Moving t to unity slowly introduces the nonideality expressed by liquid activity and the vapor fugacity coefficient. Taylor and others report solving some very difficult problems using this approach. 

P = total system pressure d>, = vapor fugacity coefficient ... 

Case 1 T < Tci and P > Pt Since the vapor phase does not exist, the fugacity of the vapor must be determined by extrapolation. Extrapolations may be carried out either isothermally or isobarically. For example, if the pressure is held fixed, then the values of the vapor fugacity coefficient may be evaluated at... 

For all components, the vapor-fugacity coefficients were computed by use of the Redlich-Kwong equation of state as originally proposed by Chao and Seader. The fugacity coefficient 0f = /f /P) for any component i in the liquid... 

The fugacity coefficient v" of a pure species at temperature T and pressure P can be determined directly from an equation of state by means of (4-51). If PP , v° is the fugacity coefficient of the liquid. Saturation pressure corresponds to the condition vi = vy. Integration of (4-51) with the R-K equation of state gives... 

To determine pure vapor fugacity coefficients (4-51) is rewritten as... 

If a subcode is missing, the basic method code for that property will be substituted. In the liquid activity coefficient examples, if the subcode for the liquid volume method is missing, the basic method code specified for liquid volume calculations will be used. If the basic method codes are missing, default values are assigned. The default method code for a property is the simplest model requiring the least input data, e.g., ideal-gas model for vapor fugacity coefficients. 

The system in the second example is plagued with all of the usual problems and, in addition, it contains acetic acid. The person who developed a model for this system decided that it was necessary to account for the vapor phase association of acetic acid. Thus, a user equation of state subroutine was written, wherein the association was rigorously treated then appropriate correction factors were determined for the vapor fugacity coefficient and vapor enthalpy pf the apparent species. The flexibility in the computing system made this possible. 

The separation of the vapor and liquid fugacities and the activity coefficients in the fundamental equilibrium relationship allow great flexibility, and a multitude of choices, in the selection of the thermodynamic relationships or empirical equations for estimation of each of these quantities. For the vapor fugacity coefficient any of the equations of state mentioned earlier or some other, such as the virial equation, may be used. In the latter case, the virial coefficients may be determined experimentally or estimated using three- or four-parameter generalized correlations. 

There are many types of EOS with a wide range of complexity. The Redlich-Kwong (RK) EOS is a popular EOS that relies only on critical temperatures and critical pressures of all components to compute equilibrium properties for both liquid and vapor phases. However, the RK EOS does not represent liquid phases accurately and is not widely used, except as a method to compute vapor fugacity coefficients in activity-coefficient approaches. On the other hand, the Benedict-Webb-Rubin-Starling (BWRS) EOS [6] has up to sixteen constants specific for a given component This EOS is quite complex and is generally not used to predict properties of mixture with more than few components. 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

The fugacity fT of a component i in the vapor phase is related to its mole fraction y in the vapor phase and the total pressure P by the fugacity coefficient ... 

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... 

It is important to be consistent in the use of fugacity coefficients. When reducing experimental data to obtain activity coefficients, a particular method for calculating fugacity coefficients must be adopted. That same method must be employed when activity-coefficient correlations are used to generate vapor-liquid equilibria. 

Two additional illustrations are given in Figures 6 and 7 which show fugacity coefficients for two binary systems along the vapor-liquid saturation curve at a total pressure of 1 atm. These results are based on the chemical theory of vapor-phase imperfection and on experimental vapor-liquid equilibrium data for the binary systems. In the system formic acid (1) - acetic acid (2), <() (for y = 1) is lower than formic acid at 100.5°C has a stronger tendency to dimerize than does acetic acid at 118.2°C. Since strong dimerization occurs between all three possible pairs, (fij and not... 

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 ... 

P the other terms provide corrections which at low or moderate pressure are close to unity. To use Equation (2), we require vapor-pressure data and liquid-density data as a function of temperature. We also require fugacity coefficients, as discussed in Chapter 3. 

As discussed in Chapter 3, at moderate pressures, vapor-phase nonideality is usually small in comparison to liquid-phase nonideality. However, when associating carboxylic acids are present, vapor-phase nonideality may dominate. These acids dimerize appreciably in the vapor phase even at low pressures fugacity coefficients are well removed from unity. To illustrate. Figures 8 and 9 show observed and calculated vapor-liquid equilibria for two systems containing an associating component. 

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. 

Figure 13 presents results for a binary where one of the components is a supercritical, noncondensable component. Vapor-phase fugacity coefficients were calculated with the virial... 

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. 

If the data are correlated assuming an ideal vapor, the reference fugacity is just the vapor pressure, P , the Poynting correction is neglected, and fugacity coefficient is assumed to be unity. Equation (2) then becomes... 

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. 

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. 

PHIS CALCULATES VAPOR PHASE FUGACITY COEFFICIENTS PHI, FOR ALL N... 

CALCULATE VAPOR PHASE FUGACITY COEFFICIENTS FOR ACTUAL COMPOSITION OF... 

The fugacity coefficient of component i at saturation is obtained after the calculation of the vapor fugacity at saturation, by the relation ... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

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