Fugacity calculated from

The dependence of solubility on % fill in 0.5M (OH) solution is shown in Fig. 5.(10) The solubility in pure water is an order of magnitude smaller under similar conditions. In pure water, activity coefficient (actually fugacity calculated from appropriate compressibility) estimates enable one to get reasonably accurate values for the equilibrium constant. This treatment suggests that the solubizing reaction in pure water is ... 

The permselectivity for membrane separations can also be calculated by substituting fugacities calculated from an equation of state, here using the Beattie-Bridgeman equation, into Equation (3) for the partial pressure values (4). The values of the permselectivities in Table IV are relatively constant at a fixed feed composition in agreement with the approximately linear behavior noted in Figures 9-11. 

Initial values. Before a trial is begun, stage temperatures, 7ys, and total flow rates, V/s and L/s, have to be given initial values. The stage component rates, v-fs and l a, do not have to be estimated since these can be calculated from the component balances. The component balances are dependent on the Zf-values and for the first.component balances, composition-independent Jf-values must be used. A composition-independent /C-value can be found from the pure component fugacities calculated from an equation of state ... 

The analyses here differ from those of Gardiner (1996), Kotas (1995) and Moran and Shapiro (1993) because of the use of the fugacity calculations from the JANAF tables (Chase etal., 1998), and, more importantly, because the contents of the isothermal enclosure of the fuel cell are at concentrations determined by the equilibrium constant (high vacuum of reactants, high concentration of products). The introduction of a Faradaic reformer is new. 

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. 

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 constants in Equation (5) are not the same as those in Equation (4). Using this saturation pressure, the pure-liquid reference fugacity at zero pressure is then calculated from the equation... 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

The vapor pressures (fugacities) shown in Figure 6.14 were reported by J. J. Fritz and C. R. Fuget, Vapor Pressure of Aqueous Hydrogen Chloride Solutions", Chem. Eng. Data Ser., 1, 10-12 (1956). The vapor pressures are too small to measure directly. The values reported were calculated from the results of emf measurements made on an electrochemical cell. In Chapter 9, we will describe this and other cells in detail. 

The fugacity coefficients in Equation (7.29) can be calculated from pressure-volume-temperature data for the mixture or from generahzed correlations. It is frequently possible to assume ideal gas behavior so that = 1 for each component. Then Equation (7.29) becomes... 

A number of problems formulated with data from the literature are given next as exercises. In addition, to the objective function given by Equation 15.11 the reader who is familiar with thermodynamic computations may explore the use of implicit objective functions based on fugacity calculations. 

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). 

In the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

Figures 1.7.15 to 1.7.18 show the mass distributions obtained in Level I calculations and the removal distribution from Level II fugacity calculation of pentachlorophenol (PCP) at two different environmental pHs for the generic...

Fugacity coefficients are empirical quantities and are calculable from correlations such as equations of state. They differ appreciably from unity at high pressures or near the critical state. 

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. 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

This behavior of a two-component mixture is illustrated in Figure 15.2, which shows the actual fugacity, the values calculated from Henry s law, and the values calculated from Raoult s law, as a function of mole fraction. 

The fugacity coefficient can be calculated from an equation of state by one of the following expressions [2] ... 

The standard state fugacicity at pressure P can be calculated from the standard state fugacity at pressure P°... 

Fig. 7. Mixed gas sorption isotherms for PMMA-C02> C2H4 in terms of fugacities f, at fC02 = 1.50 + 0.05 atm, T = 35 °C. Experimental data ( ) in comparison with lines calculated from pure...

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