Margules equations binary solutions

In the A-B binary solution the activity coefficients are given by the three-suffix Margules equations ... 

Use the Margules formulation rx - exp( x2/2) for the activity coefficient of species 1 and relate the mole fraction to molarity through the expression n /Xi - 1000/14 derived in Section 2.10. (a) Specializing to the case of a binary solution, show that it is necessary to invoke an equation of the form (3.4.3) to fix the parameter /9, determine /9, and then write down an expression from which m may be found, (b) Let components 1 and 2 represent water and sucrose respectively determine /9 and m numerically. 

In both solid and gaseous solutions, virial equation-based Raoultian coefficients have often been proposed. For example, the Margules equations, often used in binary and sometimes in ternary solid solutions and which have a virial equation basis, were proposed originally for gaseous solutions. However, there is no satisfactory general model for Raoultian coefficients in multi-component solid solutions, and the tendency in modeling has been to treat these solutions as ideal (i.e., to use the mole fraction of a solid solution component as its activity see Equation (3.13)). 

Fig. 11.11. Raoultian activities of H2O and CO2 in the binary solution at 600°Cand 2 kb. Data from Bowers and Helgeson (1983). The curved lines are fit to the data with Margules equations, discussed in Chapter 15. The inset refers to a discussion of standard states in Chapter 12.

Next we can finally see how the activity coefficient relates to the Margules equations for this case. Recall from Chapter 9 that the partial molar quantity of one component in a binary solution can be obtained graphically from the tangent (as with the chemical potentials /j,b and /ta in the coexisting solutions of Figure 15.3a). From equation (9.6), the partial molar free energy or chemical potential of component A in a solution of A and B is given by... 

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... 

The rest of this chapter is in two parts. First, we consider how to calculate mole fractions in solid solutions assumed to have only long range order, and how to combine these mole fractions into Raoultian activities. Then we consider the determination of activity coefficients in (binary) solid solutions, and how regular solution and Margules equations are used to systematize these. 

Comparison of equations (26) and (ij) shows immediately that the functional properties of the model described by (l8) and those deriving from (l ), expanded to j=3, are identical. These equations comprise the two-parameter (asymmetric) Margules model for a binary solution. To complete the analogy it is necessary only to equate the coefficients in the expressions for chemical potential,... 

The derivatives of G p with respect to composition (X2) i or binary solutions conforming to the two-coefficient Margules model can be determined from the equation of state (2 ) ... 

At high pressures, both the vapor and liquid phases may be nonideal. Consider a binary mixture of a and b with vapor-phase mole fraction and T known. Develop a set of equations and a solution algorithm to determine the composition in the liquid phase and the system pressure. Use the van der Waals equation to quantify deviations from ideality in the vapor and the three-suffix Margules equation to model the nonideal liquid. Assume that critical properties, liquid volumes, and Antoine coefficients for each species are readily available and that the three-suffix Margules parameters have been determined. 

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. 

These equations can be solved for the coefficients c, d 0, c, and d 0 in terms of the infinite-dilution activity coefficients (which are binary-system properties), but the solution will not be unique. Equation 5.48d, connecting the ternary second-order Margules expansion coefficients to the binary infinite-dilution activity coefficients, shows that a constant (say, c0) can be added to any c and c[Pg.202]

The paper is organized as follows first, the thermodynamic relations for the solubility of poorly soluble solids in pure and multicomponent mixed solvents are written. Second, an equation for the activity coefficient of a solute at infinite dilution in a binary nonideal mixed solvent [23) is employed to derive an expression for its solubility in terms of the properties of the mixed solvent. Third, various expressions for the activity coefficients of the cosolvents, such as Margules and Wilson equations [19), are inserted into the above equation for the solubility. The obtained equations are used to correlate the HOP solubilities in binary aqueous mixed solvents and the results are compared with experiment. Finally, the case of an ideal multicomponent solvent is considered and used for ternary and higher mixed solvents. 

This model, which yields excellent results for polar and non-polar molecular liquids, is especially well suited for the study of liquid/ vapor equilibrium and the equilibrium between two liquids that are not completely miscible. Regardless of the number of components of the solution, the application of this model only requires the knowledge of two adjustment parameters per binary system, which can be deduced from the solution. The model is so widely applicable that it actually contains a number of previously classic models such as the models put forward by Van Laar, Wilson, Renon et al. (the NRTL - Non Random Two Liquids -model), Scatchard and Hildebrand, Flory and Huggins as special cases. In addition, it lends a physical meaning to the first three coefficients P, 5 and , in the Margules expansion (equation [2.1]). 

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... 

Equation (8.34) tells us how large the two-suffix Margules parameter A has to be for a binary mixture to be unstable and spontaneously separate into two components. The set of solutions to this equation is denoted the spinodal curve in Figure 8.12 which is represented by dashes. The compositions denoted by the spinodal curve are different from the set of equilibrium compositions indicated by the binodal curve. At compositions between these two curves, the liquid is metastable. While it is not at its lowest state of Gibbs energy, it will not necessarily spontaneously separate into two liquid phases. 

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