Fugacity derivatives

For the Soave-Redlich Kwong equation, the fugacity derivatives are... 

Note that according to Eq, (El.10.5b), we need an array of fugacity derivatives with respect to mole fractions to calculate a single mole-number derivative. 

Note that we have dropped some of the subscripts on the fugacity derivatives. The above 2c equations provide the 2c composition variations. dZj/dT)p can be calculated from... 

An EOS can be used to calculate 3ZjfdT)p, idZj/ drij k)p as well as the fugacity derivatives of phase /The calculation of dZj/dT)p can be simplified... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

According to equation 184, all fluids having the same value of CO have identical values of Z when compared at the same T and P. This principle of corresponding states is presumed vaHd for all T and P and therefore provides generalized correlations for properties derived from Z, ie, for residual properties and fugacity coefficients, which depend on T and P through Z and its derivatives. 

When i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

Oxygen fugacity The relationship betweeen /sj and foj can be derived from the... 

The expression for the fugacity of a component j in a gas or liquid mixture, fj, based on the Trebble-Bishnoi EoS is available in the literature (Trebble and Bishnoi, 1988). This expression is given in Appendix 1. In addition the partial derivative, (dlnf/dx j>P, for a binary mixture is also provided. This expression is very useful in the parameter estimation methods that will be presented in this chapter. 

In real mixtures and solutions, the chemical potential ceases to be a linear function of the logarithm of the partial pressure or mole fraction. Consequently, a different approach is usually adopted. The simple form of the equations derived for ideal systems is retained for real systems, but a different quantity a, called the activity (or fugacity for real gases), is... 

The QWASI fugacity model contains expressions for the 15 processes detailed in Figure 2. For each process, a D term is calculated as the rate divided by the prevailing fugacity such that the rate becomes Df as described earlier. The D terms are then grouped and mass balance equations derived. 

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. 

The fugacity of the pure substance can be derived by means of equation 2.70,... 

Based on these estimates and literature values for the fugacity, Oldani and Bor obtained equation 14.31, from which they derived the reaction enthalpy and entropy at the mean temperature of the experimental temperature range (T = 272 K) Ar//2°72 = 58.6 2.6 kJ mol-1 and ATS 72 = 304 10 J K-1 mol-1. The uncertainty intervals are standard deviations multiplied by Student s t factor for 95% probability and 18 degrees of freedom (t = 2.101) [48]. 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

The dependence of fugacity on pressure can be derived by differentiating Equation (10.29) ... 

The partial derivative of the fugacity with respect to temperature 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. 

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