Estimation of Fugacities

Estimation of fiigacity coefficients - and hence fugacities - can be accomplished with two approaches  

Estimation of fugacities with the virial EoS truncated after B uses Eq. 9.11.10, with second virial coefficient values obtained from the empiric correlations of Tsonopoulosor of Hayden and O Connell (Chapter 8). Use of the SRK EoS into Eq.9.11.6 leads to  

Results with this approach are presented in the next Example. 

Estimate the fiigacity coefficient of i-butane at 410K and the pressures shown in Table 9.E. 10 using the SRK EoS. Compare your results with the experimental values presented in Table 9.E.8. 

The calculation of fugacity coefficient using Eq.9.12.2requires the value of z obtained from the SRK EoS. To this purpose, it is more convenient to write the SRK EoS in the form  

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Example 1.16 Estimation of fugacity coefficients from virial equation Derive a relation to estimate the fugacity coefficients by the virial equation... 

The estimation of the two parameters requires not only conversion and head space composition data but also physical properties of the monomers, e.g. reactivity ratios, vapor pressure equation, liquid phase activity coefficients and vapor phase fugacity coefficients. 

Shikazono, N. (1985b) Gangue minerals from Neogene vein-type deposits in Japan and an estimate of their CO2 fugacity. Econ. Geol., 80, 754-768. 

Shikazono, N. and Takeuchi, K. (1984) Estimates of selenium and sulfur fugacities and formation temperatures for selenium-rich gold-silver vein-type deposits. Geochem. J. 18, 263-268. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

Application of Fugacity Models to the Estimation of Chemical Distribution and Persistence in the Environment... 

Mackay D, Paterson S, Joy M (1983) Application of fugacity models to the estimation of chemical distribution and persistence in the environment. In Swann Eschenroeder (eds) Fate of chemicals in the environment. American Chemical Society Symposium Series 225 175-196... 

An attractive feature of K<)A is that it can replace the liquid or supercooled liquid vapor pressure in a correlation. K,-ja is an experimentally measurable or accessible quantity, whereas the supercooled liquid vapor pressure must be estimated from the solid vapor pressure, the melting point and the entropy of fusion. The use of KOA thus avoids the potentially erroneous estimation of the fugacity ratio, i.e., the ratio of solid and liquid vapor pressures. This is especially important for solutes with high melting points and, thus, low fugacity ratios. 

As the evaluation of Equation (10.86) requires a great deal of data, and as adequate data are available for only a few mixtures of gases, it is useful to have approximate relationships that can be used to estimate the fugacity of components in a solution of gases. 

Estimation of the fugacity coefficients for methanol, carbon monoxide and hydrogen at 600 K and 300 atm using Newton s method [12]... 

In order to examine the roles of both the water column and the sediment bed in PCB bioaccumulation by the mussels, estimate the fugacities (ft) and chemical activities (a relative to the pure liquid chemical reference state) at 25°C for both PCB congeners in the water, in the sediment, and in the mussels. 

An initial guess for the pressure is assumed and the fugacity coefficient of each component in the liquid phase ( ) can be calculated. An initial guess is also assumed for the fugacity coefficient of each component in the vapour phase ( v), and consequently a first estimate of the vapour composition is evaluated. With this value of y, the fugacity coefficients in the vapour phase are recalculated using the equation of state and a second estimate for y,- is evaluated. This iterative procedure is continued until the difference between two successive values of the composition are below a predetermined error. At this point, the sum of y, is checked if the sum is different from unity a new value of the pressure is assumed for a new iteration. The iterative procedure ends when the y, differs from unity by less then a given value. 

Possible fate in the environment. An industrial chemical that has been released into the environment will exist in differing concentrations in the various environmental compartments. The concentrations of a substance in air, water, soil and other media following release can be modelled using the concept of fugacity.2 At its simplest, this involves only the use of standard physico-chemical data to estimate the partitioning between the various media. 

Fig. 3.24. The Duan et al. (1992b) model estimates of gas fugacity coefficients at 1 and 20 bars between 273 and 1473 K compared to extrapolations below 273 K generated by the FREZCHEM model for a methane and b carbon dioxide. Reprinted from Marion et al. (2006) with permission...

Table B.l lists all the chemical reactions and their temperature dependence. Table B.2 lists the Debye-Hiickel constants A,p and Av) as a function of temperature and pressure. Table B.3 lists the numerical arrays used for calculating unsymmetrical interactions (Equations 2.62 and 2.66). Table B.4 lists binary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.5 lists ternary Pitzer-equation parameters for cations and anions as a function of temperature. Table B.6 lists binary and ternary Pitzer-equation parameters for soluble gases as a function of temperature. Table B.7 lists equations used to estimate the molar volume of liquid water and water ice as a function of temperature at 1.01 bar pressure and their compressibilities. Table B.8 lists equations for the molar volume and the compressibilities of soluble ions and gases as a function of temperature. Table B.9 lists the molar volumes of solid phases. Table B.10 lists volumetric Pitzer-equation parameters for ion interactions as a function of temperature. Table B.ll lists pressure-dependent coefficients for volumetric Pitzer-equation parameters. Table B.12 lists parameters used to estimate gas fugacities using the Duan et al. (1992b) model.

Estimate the fugacity of liquid acetone at 110 C and 275 bar. At 110°C the vapor pressure acetone is 4.360 bar and the molar volume of saturated-liquid acetone is 73 cm3 mor1. 

The normal boiling point of n-butane is 0.5°C. Estimate the fugacity of liquid n-butane at temperature and 200 bar. 

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