Models Margules

Nonpolar -1- nonpolar compounds All of the models (Margules, van Laar, Wilson. UNIQUAC, and NRTL) will give good correlations of data for these mixtures. 

Nonpolar + strongly polar compounds Although all of the models (Margules, van Laar. Wilson, UNIQUAC. and NRTL) can be used for mixtures that are not too nonideal, the UNIQUAC model appears to give the best correlation for somewhat nonideal systems, wheras the Wilson model may be better for mixtures that are more nonideal (but not so nonideal as to result in liquid-liquid immiscibility see Sec 11.2). 

Both models (Margules, van Laar) are hardly used for process simulation today. 

Saxena (1973). A general account of the mixing properties of crystalline solutions, with detailed discussions of the van Laar, regular (one coefficient, symmetric), "sub-regular" (two-coefficient, asymmetric) and quasi-chemical models. Margules and other equations for i presented for binary... 

There are many simple two-parameter equations for Hquid mixture constituents, including the Wilson (25), Margules (2,3,18), van Laar (3,26), nonrandom two-Hquid (NRTI.v) (27), and universal quasichemical (UNIQUAC) (28) equations. In the case of the NRTL model, one of the three adjustable parameters has been found to be relatively constant within some homologous series, so NRTL is essentially a two-parameter equation. The third parameter is usually treated as a constant which is set according to the type of chemical system (27). A third parameter for Wilson s equation has also been suggested for use with partially miscible systems (29,30,31). These equations all require experimental data to fit the adjustable constants. Simple equations of this type have the additional attraction of being useful for hand calculations. 

Margules, and Scatchard-Hildebrand) are particular mathematical solutions to Eq. (48) these models do not satisfy Eqs. (45) and (46), except in the limiting case where the right-hand sides of these equations vanish. This limiting case provides a good approximation for mixtures at low pressures but introduces serious error for mixtures at high pressures, especially near critical conditions. 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

The third virial coefficients for molecule-molecule interactions can be taken as zero for aqueous systems containing molecular solutes at low concentration. The remaining term for the molecule-molecule interaction contribution is equivalent to the unsymmetric two-suffix Margules model. 

The earliest equations for Gibbs excess energy, like Margules and van Laar, were largely empirical. More recent equations and NRTL and UNIQUAC are based on a semiempirical physical model, called the two-liquid theory, based on local composition. The molecules do not mix in a random way, but because of different bonding effects, the molecules prefer a certain surrounding. This results in a composition at the molecular level, the local composition, which differs from the macroscopic composition. 

If the mixture is subregular, definition of the limits of spinodal decomposition is more complex. For a subregular Margules model (figure 3. IOC and D), we have... 

In pyroxenes, exsolutive processes proceed either by nucleation and growth or by spinodal decomposition (see sections 3.11, 3.12, and 3.13). Figure 5.30B shows the spinodal field calculated by Saxena (1983) for Cag sMgo sSiOj (diop-side) and MgSi03 (chnoenstatite) in a binary mixture, by application of the subregular Margules model of Lindsley et al. (1981) ... 

Table 5.60 Volumetric [Wy, J/(bar X mole)] enthalpic Wjj kJ/mole), and entropic [Ws J/(mole X K] terms of subregular Margules model for (Na,K)Al2Si3A10io(OH)2 binary mixture, according to various authors.

Newton et al. (1980) calorimetrically measured the enthalpy of the NaAl-Si30g-CaAl2Si208 mixture at P = 970 K and P = 1 bar and found a positive excess enthalpy reproduced by a subregular Margules model ... 

Saxena and Ribbe (1972) have shown that the excess Gibbs free energy of mixing of the mixture, based on the data of Orville (1972), may be reproduced by a subregular Margules model ... 

The subsolidus properties of the KAlSi30g-CaAl2Si208 system are essentially those of a mechanical mixture (cf section 7.1). According to Ghiorso (1984), the miscibility gap is not influenced by pressure and may be described by a strongly asymmetric Margules model ... 

Adopting a subregular Margules model for the NaAlSi30g-KAlSi308 (Ab-Or) binary mixture and assuming that the activity coefficient of the albite component is not affected by the presence of limited amounts of the third component in the mixture (i.e., CaAljSijOg), equation 5.260 may be transformed into... 

An accurate representation of the phase equilibrium behavior is required to design or simulate any separation process. Equilibrium data for salt-free systems are usually correlated by one of a number of possible equations, such as those of Wilson, Van Laar, Margules, Redlich-Kister, etc. These correlations can then be used in the appropriate process model. It has become common to utilize parameters from such correlations to obtain insight into the fundamentals underlying the behavior of solutions and to predict the behavior of other solutions. This has been particularly true of the Wilson equation, which is shown below for a binary system. 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

Related Calculations. The constants for the binary Margules and Van Laar models for predicting activity coefficients (see Related Calculations under Example 3.4) are simply the natural logarithms of the infinite-dilution activity coefficients A t2 = I n y(XJ and /12,1 = I n y2XJ. 

Try to improve upon the Margules model by including a cubic term in the expansion of Eqs. (3.14.9a,b), and note the resulting complications. 

These are the Margules equations, and tliey represent a commonly used empirical model of solutionbeliavior. For the limiting conditions of infinite dilution, they become ... 

The activity coefficients of a solute in a mixed solvent could be also calculated by employing various well-known phase equilibria models, such as the Wilson, NRTL, Margules, etc., which using information for binary subsystems could predict the activity coefficients in ternary mixtures (Fan and Jafvert, 1997 Domanska, 1990). 

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