Excess Gibbs energy Margules

To illustrate such a calculation, Balder (Bl) considered a simple case wherein he assumed that the (symmetric) excess Gibbs energy of the ternary system is given by a two-suffix Margules expansion ... 

The excess Gibbs energy for the system chiorofonn(l)/ethanol(2) at 55°C is well represented by the Margules equation, written ... 

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). 

Vapor/liquid equilibrium (VLE) block diagrams for, 382-386, 396,490 conditions for stability in, 452-454 correlation through excess Gibbs energy, 351-357, 377-381 by Margules equation, 351-357 by NRTL equation, 380 by Redlich/Kister expansion, 377 by the UNIFAC method, 379, 457, 678-683... 

Historically, this form of the excess Gibbs energy was suggested as the simplest function which obeys the requirements that g iX,S[ must be zero when either xA — 0 or X/j —> 0. This is known as Margules equation, see, e.g., Prausnitz et al (1986). From (L.6), one can obtain both equations (L.3) and (L.4). The latter was derived from theoretical arguments based on lattice model for mixtures (Guggenheim 1952). 

The one-constant Margules equation provides a satisfactory representation for activity coefficient behavior only for liquid mixtures containing constituents of similar size, shape, and chemical nature. For more complicated systems, particularly mixtures of dissimilar molecules, simple relations such as Eq. 9.5-1 or 9.5-5 are not valid. In particular, the excess Gibbs energy of a general mixture will not be a symmetric function of the mole fraction, and the activity coefficients of the two species in a mixture should not be expected to be mirror images. One possible generalization of Eq. 9.5-1 to such cases is to set... 

In this equation i denotes the species and has values of 1 and 2. These results are known as the two-constant Margules equations. In this case the excess Gibbs energy is not symmetric in the mole fractions and the two activity coefficients are not mirror images of each other as a function of concentration. 

Equations 11.2-11 through 11.2-14 are specific to the choice of the one-constant Margules equation to represent the excess Gibbs energy of the mixmre. The use of more realistic models for G will lead to other predictions for phase separation, such as the limit of stability at the upper consolute occurring away from X] = 0.5 (Problem... 

The temperature of a liquid mi.xture is reduced so that solids form. However, unlike the illustrations in Section 12.3, on solidification, a solid mixture (rather than pure solids) is formed. Also, the liquid phase is not ideal. Assuming that the nonideality of the liquid and solid mixtures can be described by the same one-constant Margules excess Gibbs energy expression, derive the equations for the compositions of the coexisting liquid and solid phases as a function of the freezing point of the mixlure and the pure-component propenies. 

The following tables provide values for parameters in models for the excess Gibbs energy of selected binary liquid mixtures. Table E.l contains values for the Porter equation ( 5.6.2), Table E.2 for the Margules equation ( 5.6.3), and Table E.3 for Wilson s equation ( 5.6.5). 

Table E.2 Selected binary liquid mixtures in which the excess Gibbs energy approximately obeys the Margules equation (5.6.11) ...

The Margules equation models the excess Gibbs free energy by a two-parameter Redlich-Kister polynomial. The excess Gibbs energy and the activity coefficients are given by the following equations ... 

Still with the aim of having mathematical expressions for the representation of the solution, Redlich and Kister offered a representation that provides an expansion of the excess Gibbs energy, a pure-substance reference in the same state of segregation as the solution (reference (I)), the equivalent of ihe Margules expansion for the activity coefficients. For a two-component solution, the pol5momial expansion up to order m is written ... 

In chemical systems of interest, we usually have more than two components. In this section we will briefly explore the extension of the activity coefficient models above to multicomponent systems. We begin with an extension of the two-suffix Margules equation to a ternary system. The excess Gibbs energy is written as follows ... 

A binary vapor mixture contains 48% ethanol a) in water (b) at 70°C. Determine the pressure at which this vapor develops the first drop of liquid. What is the fiquid composition The excess Gibbs energy can be described by the three-suffix Margules equation with parameters ... 

A binary mixture exhibits vapor-liquid-liquid equilibrium at 300 K. The excess Gibbs energy is described by the two-suffix Margules equation with A = 6235 [j/mol]. Determine the composition of the three phases and the total pressure. The saturation pressures are given by Pf = 100 [kPa] andP = 50 [kPa]. 

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 following Margules equation is one form in which we can approximately represent nonideal behavior, allowing us to estimate the excess Gibbs free energy as a function of composition ... 

This last result, with only one adjustable parameter, is too simple to be useful but does show that, to a first approximation, the Margules model is symmetric in mole fraction. This is evident because the activity coefficients are mirror images of each other, and the excess Gibbs free energy is symmetric around Xj = 0.5. The higher-order terms in eqn. (2.4.1) lead to more realistic, unsymmetric behavior. 

The authors propose a simple method to distinguish between ideal and non-ideal systems. This consists in correlating excess Gibbs free energy with a simple one-suffix Margules equation of the form /RT=Ax X2. Theoretically A=0 for non-ideal mixtures, but practically this limit may be set to A=0.6. Note that ideal systems may be... 

Comments The particular expressions fitted to the excess Gibbs free energy and the resulting activity coefficients are known as the Margules model. This and other models for the activity coefficient are discussed in Section 12.6. 

Brown and Skinner (197 ) Symmetric Margules functions are used to represent the excess Gibbs free energy of non-ideal mixtures in this iterative approach to the rapid calculation of the equilibrium compositions of coexisting phases in multicomponent systems (c > 15). 

Margules activity coefficient model A simple thermodynamic model used to describe the excess Gibbs free energy of a liquid mixture. It uses activity coefficients that are a measure of the deviation from ideality of solubility of a compound in a liquid. See raoult s law. In the case of a binary mixture, the excess Gibbs free energy is expressed as a power series of the mole fraction in which the constants are regressed with experimental data. The activity coefficients are found by differentiation of the equation. Unlike other... 

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