Gibbs energy excess function

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... 

Figure 10.2-6 Excess Gibbs energy, excess enthalpy, and excess entropy as a function of mole fraction for the benzene-2,2,4-trimethyl pentane system.

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

Following the idea of van Laar, Chueh expresses the excess Gibbs energy per unit effective volume as a quadratic function of the effective volume fractions. For a binary mixture, using the unsymmetric convention of normalization, the excess Gibbs energy gE is found from6... 

As in the nonelectrolyte case, the problem of representing the thermodynamic properties of electrolyte solutions is best regarded as that of finding a suitable expression for the non-ideal part of the chemical potential, or the excess Gibbs energy, as a function of composition, temperature, dielectric constant and any other relevant variables. 

One of the main models which is available in CALPHAD calculation programmes is that based on Pitzer (1973, 1975), Pitzer and Mayorga (1973) and Pitzer and Kim (1974). The model is based on the development of an explicit function relating the ion interaction coefficient to the ionic strength and the addition of a third virial coefficient to Eq. (5.83). For the case of an electrolyte MX the excess Gibbs energy is given by... 

The excess term should allow the total Gibbs energy to be fitted to match that of Eq. (6.3) while at the same time incorporating a return to the inclusion of f 6) and f i) in the lattice stabilities. With the increased potential for calculating metastable Debye temperatures and electronic specific heats from first principles (Haglund et al. 1993), a further step forward would be to also replace Eq. (6.5) by some function of Eq. (6.8). 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

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. 

The thermodynamic characteristics of solutions are often expressed by means of excess functions. These are the amounts by which the free energy, entropy, enthalpy, etc. exceed those of a hypothetical ideal solution of the same composition (Denbigh, 1981). The excess free energy is closely related to the activity coefficients. The total free enthalpy (Gibbs free energy) of a system is ... 

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 ... 

For a binary system the excess Gibbs energy of the liquid phase is given by an equation of the form GE/RT = Bx,Xi, where B is a function of temperature only. Making the usual assumptions for low-pressure VLB, show that... 

We note with respect to this equation that all terms have the units of m moreover, in contrast to Eq. (10.2), the enthalpy rather than the entropy app on the right-hand side. Equation (13.12) is a general relation expressing as a function of all of its canonical variables, T, P, and the mole numb reduces to Eq. (6.29) for the special case of 1 mole of a constant-compo phase. Equations (6.30) and (6.31) follow from either equation, and equ for the other thermodynamic properties then come from appropriate def equations. Knowledge of G/RT as a function of its canonical variables evaluation of all other thermodynamic properties, and therefore implicitly tains complete property information. However, we cannot directly exploit characteristic, and in practice we deal with related properties, the residual excess Gibbs energies. 

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... 

The calculation of yi9 the activity coefficient, establishes y as a function of composition, as well as temperature and pressure. This is done by relating yi to the excess Gibbs energy GE,—i.e., by the equation... 

The problem of expressing the excess Gibbs energy as a function of composition has been researched extensively, and many methods of varying accuracy and usefulness have been proposed. An extensive discussion of these methods is given by Hala et al. (2), who show that many common expressions—e.g., those of van Laar and Margules—are deduced from the general expression of Wohl (3). 

The excess Gibbs energy of a particular ternary liquid mixture is represented by tlte empirical expression, witli parameters An, A13, and A23 functions of T and P only ... 

Excess properties find application in the treatment of liquid solutions. Of primary importance for engineering calculations is the excess Gibbs energy G, because its canonical variables are T, P, and composition, the variables usually specified or sought in design calculations. Knowing as a function of T, P, and composition, one can in principle compute from it all other excess properties. 

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