Fugacity capacities coefficients

Equilibrium. Equilibrium between compartments can be expressed either as partition coefficients K.. (i.e. concentration ratio at equilibrium) or in the fugacity models as fugacity capacities and Z. such that K.. is Z./Z., the relationships being depicted in Figur 1. Z is dellned as tfte ratio of concentration C (mol/m3) to fugacity f (Pa), definitions being given in Table I. 

An advantage of the fugacity capacity approach is that for N compartments N values of Z are defined while there may be N(N-l)/2 partition coefficients. Using Z values the partitioning properties between two phases are attributed independently to each phase. It is possible to assign (accidentally) three inconsistent partition coefficients between air, soil and water but the three Z values are inherently consistent. 

Figure 1. Relationships between fugacity capacities and partition coefficients. See Table 1 for symbol definitions.

The Level I calculation proceeds by deducing the fugacity capacities or Z values for each medium (see Table 1.5.3), following the procedures described by Mackay (2001). These working equations show the necessity of having data on molecular mass, water solubility, vapor pressure, and octanol-water partition coefficient. The fugacity f (Pa) common to all media is deduced as... 

The advantage of using fugacity to calculate the equilibrium distribution coefficients becomes apparent when one compares the fugacity capacities of a HOP for several different phases. For example, consider a region of the unsaturated zone just below the ground surface where naphthalene is distributed between air, water, pure phase octanol, and soil at equilibrium. The fugacity capacities for these phases are repeated below in Eqs. (46)-(49) ... 

The fugacity capacity for other phases is a function of both the chemical s partition coefficient between that phase and water and the chemical s Henry s law constant. For water, the fugacity capacity is... 

Fugacity modeling does not allow any new calculations to be made that cannot already be made with the partition coefficients described in the previous three sections. However, a comparison of the fugacity capacity of a chemical in different phases permits a direct assessment of which phase will have the highest chemical concentration at equilibrium. For further details, the reader is referred to Mackay and Paterson (1981) and Schwarzenbach et al. (1993). 

The ATaw and ATsw partition coefficients given above are a function of the properties of the chemical and the phase composition in two phases. In the fugacity approach, each phase is treated individually using a quantity termed the fugacity capacity or Z, value, which expresses the affinity of the chemical for that phase. The relationship between fugacity/ (Pa) and C (mol m ) is... 

Definitions of Fugacity Capacities and Partition Coefficients Used in the Soil Compartment Model ... 

In this expression, Area is the horizontal area of contact between two compartments, m Ca is the bulk contaminant concentration in the air compartment, mol m and Cs is the bulk contaminant concentration in the surface soil compartment, mol m . Table 8.3 provides definitions of the fugacity capacities Zair, Zs, Za, and Zap. Table 8.2 provides suggested values for Va, PC, pap, and rain. Chapter 6 gives more details on obtaining values for these deposition parameters. But the key parameters in this expression are the air-side mass-transfer coefficient (MTC) Ua and the soil-side MTC f/s (m d ) as given below ... 

The fugacity coefficient for C02(g) at this temperature and total pressure is 1.188 and the heat capacity change for the reaction is given by... 

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. 

The graphs are based on the Peng-Robinson equation of state (1) as improved by Stryjek and Vera (2, 3). The equations for thermodynamic properties using the Peng-Robinson equation of state are given in the appendix for volume, compressibility factor, fugacity coefficient, residual enthalpy, and residual entropy. Critical constants and ideal gas heat capacities for use in the equations are from the data compilations of DIPPR (8) and Yaws (28, 29, 30). 

Lee and Kesler (reference cited) found an accurate representation for compressibility of both gases and liquids by combining BWR-EOS with corresponding states law. They generated departure functions for enthalpy, entropy, fugacity coefficient and heat capacity. Tables are given in Reid et al. (1987), whereas illustrative graphs are presented in Perry (1997). The method is similar to that developed for compressibility. As an example, the enthalpy departure function may be calculated with the relation ... 

Route A requires an equation of state and sophisticated mixing rules for calculating the fugacity coefficient for both the vapor and the liquid phase. The advantage of using equations of state is that other information (e.g. molar heat capacities, densities, enthalpies, heats of vaporization), which is necessary for designing and optimizing a sustainable distillation process, is also obtained at the same time. 

Determine the equilibrium composition that is achieved at 300 bar and 700 K when the initial mole ratio of hydrogen to carbon monoxide is 2. You may use standard enthalpy and Gibbs free energy of formation data. For purposes of this problem you should not neglect the variation of the standard heat of reaction with temperature. You may assume ideal solution behavior but not ideal gas behavior. You may also use a generalized fugacity coefficient chart based on the principle of corresponding states as well as the heat capacity data listed below. 

W. M. Haynes and R. D. Goodwin, Thermophysical Properties of Normal Butane from 135 to 700 K at Pressures to 70 MPa, U.S. Dept, of Commerce, National Bureau of Standards Monograph 169, 1982, 192 pp. Tabulated data include densities, compressibility factors, internal energies, enthalpies, entropies, heat capacities, fugacities and more. Equations are given for calculating vapor pressures, liquid and vapor densities, ideal gas properties, second virial coefficients, heats of vaporization, liquid specific heats, enthalpies and entropies. 

Numerical calculations of phase equilibria require thermodynamic data or correlations of data. For pure components, the requisite data may include saturation pressures (or temperatures), heat capacities, latent heate, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determination of d/), and/or an expression for the molar excess Gibbs energy (fw determination of yt). We have discussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations of state and expressions for g, and the behavior of the fugacity coefficients and activity coefficients derived ftom them. 

NIST/ASME Steam Properties Database versiou 2.21 http //www.nist.gov/srd/nistlO.cfm (accessed November 10, 2010) (purchase required). Thermophysical properties include in the STEAM Database temperature, Helmholtz energy, thermodynamic derivatives, pressure, Gibbs energy, density, fugacity, thermal conductivity, volume, isothermal compressibility, viscosity, dielectric constant, enthalpy, volume expansivity, dielectric derivatives, internal energy, speed of sound, Debye-Hlickel slopes, entropy, Joule-Thomson coefficient, refractive index, heat capacity, surface tension. The STEAM database generates tables and plots of property values. Vapor-liquid-solid saturation calculations with either temperature or pressure specified are available. 

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