Margules equations applications

This equation is extremely important (see Section 5.12 for some applications). It is known as the Gibbs-Duhem equation, and such equations as the Duhem-Margules equation may be derived from it. Since no limitation has been put on the type of system considered in the derivation, this equation must be satisfied for every phase in a heterogenous system. We recognize that the convenient independent variables for this equation are the intensive variables the temperature, the pressure, and the chemical potentials. 

An increasing amount of attention has been given recently to the problem of manipulating the Duhem-Margules equation in the most convenient way for any particular application. 

Sc. Liquid and Vapor Compositions.—Some general rules concerning the relative compositions of liquid and vapor in equilibrium, which are applicable to systems of all types, may be derived from the Duhem-Margules equation, using the form of (35.1). Since the increase in the mole fraction of one component of a binary mixture must be equal to the decrease for the other component, dNi is equal to — c/N2, as seen in 34b hence equation (35.1) may be written as... 

Problem Assuming the simplified Margules equation to be applicable to the fused LiBr-AgBr system, use the result obtained in the last problem in 38e to derive general expressions for the activity coefficient s of LiBr (ai/xi) and AgBr (027x2) as functions of the respective mole fractions, at 500 C. 

In this chapter we will describe some of the non-ideal properties of solids, and the set of Margules equations commonly used for solid solutions. However, although solid solutions have been the main area of application of the Margules equations in the Earth Sciences to date, they work equally well for non-ionic liquid and gaseous solutions. Aqueous electrolyte solutions are sufficiently different that we will give them a later chapter of their own (Chapter 17). 

A direct application of the power series is the development of Margules equations. If B=C=0 then one gets a two-suffix equation, because is a second degree function ... 

In this section we consider the equilibrium between a condensed phase and vapor. We derive the Duhem-Margules equation and investigate its application to the determination of vapor fugacities. 

For unsymmetrical systems beyond the capabilities of the van Laar equations, the Scatchard-Hamer equations, although less convenient, are better. If Va = Vb, they reduce to the Margules equations, while if Aab/Aba = Va/Vb, they reduce to those of van Laar. Carlson and Col-bum (5) consequently suggest that the ratio of Va/Vb may be taken as a guide as to which of the equations are applicable. For systems that cannot be handled by any of these, the more complex equations suggested by Wohl (35) may be tried. 

The upper critical solution point for furfural-water lies at 122.7 C., 51 wt. per cent furfural. Calculate the van Laar and Margules constants from this datum, and compare with the values of activity coefficient at x = 0 and 1.0 obtained from vapor-liquid data, Chemical Engineers Handbook. Explain the results in terms of the applicability of the van Laar and Margules equations to this system. 

Values of parameters for the Margules, van Laar, Wilson, NRTL, and UNIQUAC equations are given for many binary pairs by Gmehling et al.t in a summary collection of the world s published VLE data for low to moderate pressures. These values are based on reduction of data through application of Eq. (11.74). On the other hand, data reduction for determination of parameters in the UNIFAC method (App. D) is carried out with Eq. (12.1). 

Outlined below are the steps required for of a VLE calculation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binary. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicability for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in me equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binary. 

Although it is one of the oldest activity coefficient equations, the Margules correlation is still commonly used for a wide range of applications. Its accuracy diminishes, however, as the molecules of a binary are more and more dissimilar in size or chemical structure. 

Other activity coefficient equations have been proposed since the earlier equations of Margules and van Laar. It is outside the scope of this book to present the detailed development of these equations, which may be found in the indicated references. The objective here is to present the basis for each of the more commonly used equations, their features, and applicability. 

The first relations have already been proposed by Margules (95MAR1) more than one eentury ago, and later on they have appeared many others, e.g. the equations by Van Laar, Wohl, Seatehard-Hammer, Carlson-Colbum etc. some of them were modified in mareh of time. At present, several newer ones beeame popular because of their easy application to the deseription of multieomponent systems and for the phase behavior prediction methods based on knowledge of binary data. 

Each equation used, whether for ((i°, f , or y, has its particular advantages and disadvantages, and limitations on its range of applicability these and other factors influence the selection of the equations, or particular combination of equations used. For example, in many Kellogg design applications the four-suffix Margules form of the Wohl equation for activity coefficients is... 

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]). 

Equations 12.10.5 through 12.10.12 are based on the use of the two-suffix (one-parameter) Margules expression for the excess Gibbs free energy (Eq. 12.10.1). It is a very simple model which, even with a temperature dependent A, is applicable to a very limited number of systems where is a symmetric function of a i, mainly hydrocarbons of similar size. 

---