Fugacity coefficient derivatives

The lower bound of the integral is defined at P = 0, where <]), = 1, resulting in the following expression for the fugacity coefficient, derived in a manner similar to Equation 1.21 ... 

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). 

Equations for fugacity coefficients are derived from equations of state. Table 13.1 has them for the popular Soave equation of state. At pressures below 5-6 atm, the ratio of fugacity coefficients in Eq. (13.8) often is near unity. Then the VER may be written... 

The fugacity coefficient cj> can be calculated from a valid equation of state the activity coefficient y can be derived from an applicable GE expression. The activity a, is the product of y( and y. 

It is believed that ASPEN provides a state-of-the-art capability for thermodynamic properties of conventional components. A number of equation-of-state (EOS) models are supplied to handle virtually any mixture over a wide range of temperatures and pressures. The equation-of-state models are programmed to give any subset of the properties of molar density, residual enthalpy, residual free energy, and the fugacity coefficient vector (and temperature derivatives) for a liquid or vapor mixture. The EOS models (named in tribute to the authors of such work) made available in ASPEN are the following ... 

In addition to these ordinary thermodynamic properties, the temperature and composition derivatives of the enthalpy and the fugacity coefficients are required in some calculations. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

The Lee-Erbar-Edmister method is of the same type, but uses different expressions for the fugacity and activity coefficients. The vapor phase equation of state is a three-parameter expression, and binary interaction corrections are included. The liquid phase activity and fugacity coefficient expressions were derived to extend the method to lower temperatures and to improve accuracy. Binary interaction terms were included in the liquid activity coefficient equation. 

These equations are restatements of Eqs. (6.37) and (6.38) wherein the restriction of the derivatives to constant composition is shown explicitly. They lead to Eqs. (6.40), (6.41), (6.42), and (11.20), which allow calculation of residual properties and fugacity coefficients from PVT data and equations of state. It is through the residual properties that this kind of experimental information enters into the practical application of thermodynamics. 

Calculate the compressibility factor for the mixture. In a manner similar to that used in the previous problem, an expression for the fugacity coefficient in vapor mixtures can be derived from any equation of state applicable to such mixtures. If the Redlich-Kwong equation of state is used, the expression is... 

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

Write out expressions for K, Kc, and Kp for homogeneous gas phase reactions in terms of fugacities or fugacity coefficients, based on derivations analogous to those of Sections 2.9 and 2.11. 

The difference approximation of this expression is somewhat more involved since we have to include the derivative of the activity (or fugacity) coefficient in the approximation. If, as often is assumed to be the case (not always with justification), the resistance to mass transfer in the liquid phase is negligible, then the MS equations for the liquid phase can safely be replaced by... 

This equation is analogous to Eq. 5 of Ch. 1 for the solubility of a solid in a SCF. In this equation, the subscript 2 refers to the liquid component. The superscript s refers to saturation conditions at temperature T. Pj refers to the saturation vapor pressure of the liquid at temperature T. The variable uf is the molar volume of the liquid, ( )2 is the fugacity coefficient at saturation pressure and is the fiigacity coefficient in the high pressure gas mbrture. For a detailed derivation of this equation, see Prausnitz. " As is stated in the derivation, it is the escaping tendency of the liquid into the supercritical fluid phase, as described by the fugacity coefficient, ( >2, which is responsible for the enhanced solubility of liquids in compressed gases. 

Recently, a method [5] for the prediction of the solubility of a solute in a SC fluid in the presence of an entrainer has been proposed. The method, based on the Kirkwood-Buff (KB) formalism, was however developed for cases in which the entrainer was in dilute amounts. The present paper is focused on the solubility of a solid in a non-dilute mixture of a SC fluid and an entrainer. The theoretical treatment, which is more complex than for the dilute case, is also based on the KB formalism. In this paper the following aspects will be addressed (1) general equations for the solubility in binary and ternary mixtures will be written for the cases involving a small amount of solute (2) the KB formalism will be used to obtain expressions for the derivatives of the fugacity coefficients in a ternary mixture with respect to mole fractions (3) these expressions will be employed to derive an equation for the solubility of a solute in a SC fluid containing an entrainer at any concentration (4) a predictive method for this solubility will be proposed in terms of the solubilities of the solute in the SC fluid and in the entrainer (5) the derived equation will be compared with experimental results from literature regarding the solubility of a solute in a mixture of two SC fluids. 

For the cases in which only the solute concentration is small, the derivation of an expression for the fugacity coefficient 02 (see Eq. (2)) is still critical for the prediction of the solubility x . Let us consider those compositions of the ternary mixture which are located on the line between the points (xf" = 0,... 

Equation 14 allows one to calculate the fugacity coefficient of a solute at infinite dilution in the binary mixture of two SC fluids, in terms of the fugacity coefficients of the solute at infinite dilution for each of the SC fluids. This expression will be used in the next section to derive an expression for the solubility of a solid in a gaseous mixture of two SC fluids. 

In eqs 2 and 3, V is the molar volume, z is the compressibility factor, n is the total number of moles in the system, and ni is the number of moles of component i. Equations 2 and 3 show that the fugacity coefficient their derivatives with respect to the number of moles of solute are known. While near the critical point the fluctuations are important and an EOS involving them should be used, we neglect for the time being their effect. 

From Equation 11, the equation to evaluate fugacity coefficient 02 is derived as... 

If the vapor phase is ideal, then the fugacity coefficients are equal to one, and we see that Eq. (11.25) becomes the modified Raoult s law that we have derived previously. 

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