Activity coefficient from excess Gibbs energy

Equation 9.9-3 does not give values tor the mixture parameters a and b separately, but only for their sum. A second equation comes from requiring that the excess Gibbs energy predicted from an equation of state at liquidlike densities be equivalent to that from excess Gibbs energy or activity coefficient models discussed in Secs. 9.5 and 9.6. Since, from an equation of state, as F —> oo, V b and Ymix mix. so that liquid densities are obtained, the second equation that is used is... 

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... 

The activity coefficient y, can be calculated from a model for the molar excess Gibbs energy gE... 

The excess Gibbs energy GE calculated from a liquid phase activity coefficient model, and the excess Gibbs energy GE calculated from the equation-of-state are equal at infinite pressure ... 

Figs. 5.5 and 5.6 show the deviation in the activity coefficients predicted by COSMOSPACE and the BGY model from those obtained directly from the MC simulations using an addition to our MC code, which allows us to evaluate the activity coefficients of the components. We see from these results that COSMOSPACE is in much better agreement with the MC simulations than the BGY model. We have not calculated the activity coefficients for the AD model since it is not a model for the excess Gibbs energy. 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

Similarly, the activity coefficient equations (which can be derived from the excess Gibbs energy expression) have the general form ... 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

Activity coefficients yk have traditionally been calculated from correla equations for GE/RT by application of Eq. (11.62). The excess Gibbs energy a function of Tt P, and composition, but for liquids at low to moderate, pressi it is a very weak function of P. Under these conditions, its pressure dependen and therefore the pressure dependence of the activity coefficients are usual neglected. This is consistent with our earlier omission of the Poynting factor fr... 

Analytical representation of the excess Gibbs energy of a system impll knowledge of the standard-state fugacities ft and of the frv. -xt relationshi Since an equation expressing /, as a function of x, cannot recognize a solubili limit, it implies an extrapolation of the /i-vs.-X[ curve from the solubility I to X) = 1, at which point /, = This provides a fictitious or hypothetical va for the fugadty of pure species 1 that serves to establish a Lewis/ Randall 1 for this species, as shown by Fig. 12.21. ft is also the basis for calculation of activity coefficient of species 1 ... 

The curves in Figs. 13.18 and 13.19 provide an excellent correlation c VLE data. They result from BUBL P calculations carried out as indicated in 12.12. The excess Gibbs energy and activity coefficients are here express functions of liquid-phase composition by the 4-parameter modified, Ma... 

Activity Coefficient at Infinite Dilution. A procedure similar to that employed by Wilson will be used here to obtain an expression for the excess Gibbs energy. Wilson started from the Flory and Huggins expression" 2 for the excess free energy of athermal solutions, but expressed the volume fractions in terms of local molar fractions. We selected Wilson s approach from a number of approaches, because it provided a better description of phase equilibria and because the interactions that count the most are the local one, but started from the more... 

From the excess Gibbs energy of mixing, we can calculate the activity coefficients of the components, however, the mole fractions must be calculated from the amounts of substances... 

The surface tension of the system KF-KBF4 decreases with the increasing content of KBF4, obviously due to the covalent character of the bonds in the BFJ complex anions, which are surface active and concentrate on the melt surface. Similar values as well as the shape of the surface adsorption curve were found when it was calculated from the polynomial coefficients and from the excess Gibbs energy of mixing in the liquid phase. Even both the calculated interaction parameters B are relatively close. 

The matrix [G] is symmetric. Its elements may be obtained from activity coefficient models in much the same way that the matrix [T] is obtained. Expressions for the G y for some models of the excess Gibbs energy are given in Appendix D. 

There are several ways of reporting experimental data on the deviations from ideal behavior. The most common ones are either the activity coefficients of each component, or the total excess Gibbs energy of the system. If the vapor above the liquid -mixture can be assumed to be an ideal gas, then it is also convenient to plot Pa/Pa as a function of xA where PA and PA are the partial pressure and the vapor pressure of A, respectively. 

The activity coefficients are typically computed from a model for the excess Gibbs energy g, as described in the thermodynamics chapter (Chapter 4). The most popular are the Wilson, NRTL, and Uniquac models, described in detail in many places [15, 36 0]. They contain two or three adjustable (and possibly temperature-dependent) parameters per binary. One cannot predict which model will be best for a given system however, the Wilson equation is incapable of describing LEE. 

Equations 16c and 16d relate the volumetric parameters Wjj and Uj x 1 pressure dependence of the parameters Wjj and U, j for the excess Gibbs energy and osmotic/activity coefficients. In this study values of Wj and NaCl(aq) were evaluated at temperatures from... 

Suppose that we have availshle an expression for g , the molar excess Gibbs energy, which holds for the entire composition ranga. We can Red activity coefficients, as discusred in Section 1.4, from the relationship... 

There is a very important point in this illustration that is easily overlooked. To obtain the activity coefficient from the expression for the excess Gibbs energy, we used the definition... 

All activity coefficients derived from an excess Gibbs energy expression that satisfies the boundary conditions of being zero at. V = 0 and 1 will satisfy the Gibbs-Duhem equation. Can you prove this 0... 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

Excess Gibbs energy and activity coefficients are linked. From the equations (6.30) and (6.33) the following fundamental relation is obtained ... 

Hence, the total excess Gibbs energy ean be easily determined from experimental values of activity coefficients, as it will be shown in the next section. Because In /, is a partial property, the following relation may be written ... 

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