The Margules Equation

The Margules formulation for G x has been especially useful in dealing with systems in which phase separation (positive deviation from ideality) occurs or which exhibit extrema in the variation of the activity coefficients with composition. The remainder of this paper will be devoted to a brief discussion of the properties of the Margules equations. 

Hardy (1953, equation 5) also presents an expression for the excess free energy of a binary mixture  

The expression for the total free energy of binary mixing corresponding to equations (5) and (18) is  

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, 

These same results were obtained implicitly by Carlson and Colburn 

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The relationship between viscosity, angular velocity, and torque for a Newtonian fluid in a concentric cylinder viscometer is given by the Margules equation (eq. 26) (21,146), where M is the torque on the inner cylinder, h the length of the inner cylinder, Q the relative angular velocity of the cylinder in radians per second, T the radius of the inner cylinder wall, the radius of the outer cylinder wall, and an instmment constant. 

Therefore, the viscosity can be determined from the torque and angular velocity. However, the viscosity is usually calculated from the shear rate and shear stress, which can be obtained from the Margules equation. The shear rate is given by equation 27, where r is any given radius. 

The Margules equations (Eqs. [4-244], [4-245], and [4-246]) are well suited to this system, and the parameters for this equation are given as... 

We can rearrange this in terms of the viscosity to give the Margules equation ... 

Clearly the Riener-Riwlin equation reduces to the Margules equation when the Bingham yield value is zero, but there is an important consequence in that it is assumed that all the material is flowing, i.e. the shear stress at the wall of the outer cylinder must be... 

If only B and C are 0, the resulting equation is frequently called the Margules equation [9]. 

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. 

Because of the limitations of the Margules equation—especially in predicting multicomponent VLE data—the Wilson, NRTL, and LEMF (16) equations are employed in this study. The experimental data on the systems presented in Table I were used in this work. These are the only systems for which both binary and ternary data could be found in the literature. As a matter of fact, uncertainties do exist about the accuracy of the two HgC systems. The maximum boiling... 

Figure 13.28. Vapor-liquid equilibria of some azeotropic and partially miscible liquids, (a) Effect of pressure on vapor-liquid equilibria of a typical homogeneous azeotropic mixture, acetone and water, (b) Uncommon behavior of the partially miscible system of methylethylketone and water whose two-phase boundary does not extend byond the y = x line, (c) x-y diagram of a partially miscible system represented by the Margules equation with the given parameters and vapor pressures Pj = 3, = 1 atm the broken line is not physically significant but is...

Although the correlations provided by the Margules equations for the two sets of VLE data presented here are satisfactory, they are not perfect. The two possible reasons are, first, that the Margules equations are not precisely suited to the data set second, that the data themselves are systematically in error such that they do not conform to the requirements of the Gibbs/ Duhem equation. 

U4 The following is a set of VLE data for the system acetone(l)/chloroform(2) at 50°C [H. R and W. Schroder, Z. Phys. Chem. (Frankfurt), II 41, 1957]. Assuming the validity of Eq. (11. find parameter values for the Margules equation that provide a suitable correlation of these and prepare a Pxy diagram that compares the experimental points with curves determined from correlation. 

The Redlich/Kister expansion, the Margules equations, and the van equations are all special cases of a very general treatment based on ratio functions, i.e., on equations for Ge given by ratios of polynomials. These... 

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

An alternative is to process just the P-x, data this is possible because the P-Xj -y, data set includes more information than necessary. The procedure requires a computer, but in principle is simple enough. Assuming that the Margules equation is appropriate to the data, one merely searches for values of the parameters A,2 and A2) that yield pressures by Eq. (11.75) that are as close as possible to the measured values. The... 

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). 

The Margules equations [Eq. (11.72)] provide an example of this functionality. 

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. 

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