Phase calculation from fugacity

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... 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

Example 6.5 Repeat the calculations from Example 6.4 taking into account vapor-phase nonideality. Fugacity coefficients can be calculated from the Peng-Robinson Equation of State (see Poling, Prausnitz and O Connell6 and Chapter 4). 

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. 

Table I lists experimental results, comprising derived values of the fugacity of benzene at known total molarity in the aqueous phase, [B], and known molarity of 1-hexadecylpyridinium chloride [CPC] or sodium dodecylsulfate [SDS]. Fugacities have been calculated from total pressures by subtracting the vapor pressure of the aqueous solution in the absence of benzene from the measured total pressure and correcting for the small extent of nonideality of the vapor phase (15, 22). Results are given for temperatures varying from 25 to 45°C for the CPC systems and 15 to 45°C for the SDS systems.

If it is assumed that the specific volume, u , of the phase is very nearly constant over a large pressure increase, then when the pressure difference in the exponential is small enough, the exponential will be nearly one. Thus, at moderate pressures the fugacity of a condensed phase is nearly equal to the vapor pressure. This approximation was used in Eq. (11) to write Raoult s Law. When the pressure is low enough that the fugacity coefficient is nearly 1.00, Eq. (13) reduces to Raoult s Law extended with the activity coefficient included or Eq. (11). The fugacity coefficient can be calculated from equations of state, and the activity coefficient can be found from various correlations as discussed earlier. 

The fugacity of the solute in fluid phase (02) can be calculated from (Prausnitz et al. 1999) ... 

The /<-values are calculated by Equation 1.25, with the vapor phase fugacity coefficients calculated from the equation of state and the liquid phase fugacity coefficients for an ideal solution calculated as... 

The fugacities in the gas mixtures are obtained from the gas-phase data using an equation of state, one can calculate the fugacity coefficient (p from the experimental and p and then form the product to give... 

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... 

The liquid-phase fugacity coefficient = /f/P may be calculated from a generalized correlation in terms of reduced temperature and pressure such as those of Lydersen et al.42 and Curl and Pitzer.15 Chao and Seader used a modified form of the Curl and Pitzer correlation. The correlation was modified by use of experimental data such that appropriate values of could be computed for the case where a component does not exist as a liquid and for the case of low temperatures. The following expression was proposed for the calculation of the fugacity coefficient for any component / in the liquid phase... 

The molar volume V of each equilibrium phase is calculated by solving Equation 5. At a given temperature, pressure, and composition of the mixture, three values for V will be obtained, the largest of which is the vapor molar volume and the smallest the liquid molar volume. Although the calculated liquid molar volume, VL, does not accurately reproduce experimental data on molar volume, the fugacity coefficients, < iL, calculated from the inaccurate VL, are in good agreement with those derived from experimental K-data. 

Thus for this and any similar reaction, the equilibrium constant can always be made equal to the gas fugacity at equilibrium with the (pure) minerals in the reaction. The physical situation is illustrated in Figure 1.3.1a. The equilibrium constants at various temperatures have been calculated from the data in Robie et al. (1978) (referred to here asRHF) and are shown in Tables 13.1 and 13.4 and plotted in Figure 13.2. We should perhaps emphasize that although K is completely independent of the compositions of the minerals or of the gas phase, it only equals the water fugacity in equilibrium with brucite and periclase when those two minerals are pure. 

Equations (1.3-14) and (1.3-15) thus give the prediction from transition-state theory for the rate of a reaction in terms appropriate for an SCF. The rate is seen to depend on (i) the pressure, the temperature and some universal constants (ii) the equilibrium constant for the activated-complex formation in an ideal gas and (iii) a ratio of fugacity coefficients, which express the effect of the supercritical medium. Equation (1.3-15) can therefore be used to calcu-late the rate coefficient, if Kp is known from the gas-phase reaction or calculated from statistical mechanics, and the ratio (0a 0b/0cO estimated from an equation of state. Such calculations are rare an early example is the modeling of the dimerization of pure chlorotrifluoroethene = 105.8 °C) to 1,2-dichlor-ohexafluorocyclobutane (Scheme 1.3-2) and comparison with experimental results at 120 °C, 135 °C and 150 °C and at pressures up to 100 bar [15]. 

Vapor-liquid equilibria are calculated from the equation through the application of the thermodynamic conditions of equilibrium, namely, that the pressure be the same in both phases and that the fugacity of each component be the same in both phases. These conditions are determined indirectly, since an equivalent condition of equilibrium is that the Helmholtz free energy must be a minimum at constant total volume and number of moles of each component. This latter condition can be more directly applied to numerical computations on the computer. The equations for the Helmholtz free energy and the fugacity of a component are as follows ... 

Solution The starting point is the equilibrium criterion in eq. fio.2oI. which relates the fugacity coefficients of a component to the mole fraction of the component in each phase. However, one coefficient must be known for the other to be calculated from this equation. In the absence of additional information, we must make some suitable... 

Experimental studies were carried out to derive correlations for mass-transfer coefficients, reaction kinetics, liquid holdup and pressure drop for the new catalytic packing MULTIPAK (see [9,10]). Suitable correlations for ROMBOPAK 6M were taken from [70] and [92], The vapor-liquid equilibrium is calculated using the modification of the Wilson method [9]. For the vapor phase, the dimerization of acetic acid is taken into account using the chemical theory to correct vapor-phase fugacity coefficients [93]. Binary diffusion coefficients for the vapor phase and for the liquid phase are estimated via the method purposed by Fuller et al. and Tyn and Calus, respectively (see [94]). Physical properties like densities, viscosities and thermal conductivities are calculated from the methods given in [94]. 

From the historical point of view and also from the number of applications in the literature, the common method is to use activity coefficients for the liquid phase, i.e., the polymer solution, and a separate equation-of-state for the solvent vapor phase, in many cases the truncated virial equation of state as for the data reduction of experimental measurements explained above. To this group of theories and models also free-volume models and lattice-fluid models will be added in this paper because they are usually applied within this approach. The approach where fugacity coefficients are calculated from one equation of state for both phases was applied to polymer solutions more recently, but it is the more promising method if one has to extrapolate over larger temperature and pressure ranges. 

AG = 0, and the equilibrium pressure as ouqiut of the simulation. Then, a second set of runs (G-NPT) was performed with the same feed mixtures and the pressure fixed at the values obtained fiom the G-NVT runs. In this case < Fb) is an output of the simulation. Finally, a third set of runs (GCMC) was carried out to check the validity of our simulation technique. These runs consisted of standard multicomponent GCMC simulations with mixture fugacities calculated from the virial equation of state using the pressures and gas-phase compositions obtained in the G-NVT runs. 

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