Applying The Margules Equations

The Margules equations such as those in Table 15.1 can be fitted by standard least-squares regression analysis to data for real solutions. For example, if data for the total free energy of a binary asymmetric solution is available over a range of compositions at different T and P, you could fit equation (15.41) for Greai (or the equation for in Table 15.1) and obtain Wq, and ITcj as regression parameters. The same could be done with the equations for excess enthalpy, entropy, and so on. This permits construction of phase diagrams and determination of thermodynamic properties based 

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Given in what follows are values of infinite-dilution activity coefficients and pure-spedes vapor pressures for binary systems at spedfied temperatures. For one of the systems, determine the Margules parameters, and then apply the Margules equation to a suffident number of VLE calculations to allow construction of a Pxy diagram for the given temperature. Base your calculations on the modified Raoult s-law expression, i.e., Eq. (11.74). 

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. 

Broul et al. (13) and Hala (14) developed a correlation scheme for systems containing two solvents and one salt, which they applied to several salt concentrations, not just to the saturation level as in the studies mentioned above. They utilized the binary VLE data for the three binaries (solvent 1-salt solvent 2-salt and solvent 1-solvent 2) along with the ternary data to correlate successfully the ternary results. They employed the Margules equation (15) with the addition of a term to account for the coulombic interactions. 

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. 

We liave presumed in applying tire Margules equations tliat the deviations of the experimental points for from tlie straight lines drawn to represent them result from... 

The Margules equations apply to many binary mixtures, including those that display positive deviations from ideality, mixtures that exhibit negative deviations from... 

There is an extensive hterature in both chemistry and geochemistry on these and other similar equations and their interrelations. Useful guides are Grover (1977) and Thompson (1967). They apply to any solutions, solid, liquid, or gaseous. Thompson points out that the Margules equations bear the same relationship to Henry s law that the virial equation ( 13.5) has to the ideal gas law, in that the first term in the virial series (in the form of 13.26) is the ideal gas law, and the Margules equation condenses to Henry s law, as shown above. 

Equation (18) was applied by Margules (1895) to calculate the heats of admixture of water and alcohol from the vapour-pressure data of Regnault the results agreed with the direct determinations of Winkelmann (1873). 

The Margules and van Laar equations apply only at constant temperature and pressure, as they were derived from equation 11.21, which also has this restriction. The effect of pressure upon y values and the constants and 2i is usually negligible, especially at pressures far removed from the critical. Correlation procedures for activity coefficients have been developed by Balzhiser et al.(ll Frendenslund et alSls>, Praunsitz et alS19>, Reid et al. 2 ) van Ness and Abbott(21) and Walas 22 and actual experimental data may be obtained from the PPDS system of the National Engineering Laboratory, UK1-23). When the liquid and vapour compositions are the same, that is xA = ya, point xg in... 

It is not permissible to express the constant of integration k in terms of the pressure jP, which the gas would exert if it could be obtained in the liquid state at the temperature of the experiment. Dolezalek Zeitschr. f. physikal.. Chemie 71, 206, 1910) falls into this error. Apart from the uncertainty of the extrapolation by which this pressure must be calculated when the temperature of the solution is above the critical temperature of the dissolved gas, it is also quite inadmissible to apply Margules equation to vapours which do not obey the gas laws. 

The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (5.1.5), and therefore they must obey the pure-component limit (5.4.12). In addition, as with Porter s equations, the parameters A and A2 are simply related to the activity coefficients at infinite dilution. In particular, when we apply the dilute-solution limit (5.4.13) to (5.6.12) and (5.6.13), we obtain... 

Departures from the above equations due to deviations from the law of the perfect gas mixture do not usually exceed a small percentage except at pressures above atmospheric or if there is association in the vapour phase, such as occurs in the case of formic and acetic acids. A procedure for applying the Duhem-Margules equation allowing for deviations from the gas law has been described by Scatchard and Raymond. ... 

Ganguly and Kennedy (l97lf). Margules equations for a "simple" ternary mixture (without specific ternary terms see Prigogine and Defay, 195 ) are applied to the quasi-temary system pyrope - grossiilar - (almandine + spessartine) and approximate values for the mixing coefficients (W terms) are derived from existing natural and experimental data. 

The parameter A is fit to experimental data for a given binary mixture. This parameter may change with temperature or pressure, but it is independent of the composition of the system. Moreover, as we will see shortly, we can apply the thermodynamic web to see how A varies with T and P. Equation (7.54) is called the two-suffix Margules equation it clearly satisfies condition 1 above. Example 7.8 shows that it also satisfies condition 2. 

We can use the two-suffix Margules equation to write the activity coefficients in terms of the Margules parameter, A. Applying Equations (7.55) and (7.56) gives ... 

To solve Equations (8.27) and (8.28), we need an activity coefficient model for g . We will illustrate the approach using the two-suffix Margules equation however, the same methodology can be applied to any model from Table 7.2. If we substitute Equation (7.55) for the activity coefficient of species a into Equation (8.27), we get ... 

To solve these equations, we need a model for g . We will illustrate the approach using the two-suffix Margules equation however, the same methodology can be applied to any model from Table 7.2. 

This applies to all steps such as (15.48), (15.49) and so on, which are used to calculate one Margules parameter from another. Finally, a general equation of state such as (15.50) could be derived to describe the mineral system, using the same assumptions and with the same warnings about accuracy. 

The values for the activity coefficients can be obtained from semi-empirical equations such van Laar. Margules. Wilson. UNIVAC and UNIQUAC (see chapter [ID. The constants, which appear in these equations can be found in literature as well [39J.. At the permeate side normally ideal behaviour is assiumed and the partial pressure is given by the product of pressure and mol fraction. The vapour pressure at the permeate side is minimum when a vacuum is applied in combination with liquid nitrogen temperatures (- 196°C or 77 K) which may occur in laboratory testing. In this case the driving force is determined completely by the vapour pressure of the feed liquid, which in turn can be strongly influenced by e temperature of the feed. 

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