Margules correlation

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. 

The Margules correlation gives the two activities for a binary mixture (in plate i) about a reference temperature. To, as ... 

It is possible to extend the Margules correlation to multicomponent mixtures although the expressions become more complicated, they depend still on the same, binary interaction parameters. See Perry (1963), Wohl (1946, 1953), Sevems (1955), Chien and Null (1972). For ternary distillation, the activity coefficient at the reference temperature is given for component 1 by ... 

Although the development of the equations has allowed for the activity coefficients, to be functions of temperature, we have seen that the Margules correlation is intended to be valid over a range of temperatures. Hence, if the Margules correlation is used, we may set the temperature derivatives of the activity coefficients to zero ... 

Mixtures of diethyl ether (C4HioO)(l) and ethanol(2) are to be separated into essentially pure components by a distillation column operating at low pressure. Estimate the bounds on the relative volatility ax2- Assume the liquid mixtures obey the Margules correlation (5.6.11) with Ax = 0.1665 + 233.74/T(K) and A2 = 0.5908 -I-197.55 / T(K). Pure-component vapor pressures are in Appendix D. 

Outlined below are the steps required for of a X T.E calciilation 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 binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty 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 the 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 binaiy. 

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. 

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... 

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. 

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. 

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. 

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... 

This equation concisely stores the infonnationof the data set. Indeed, the Margules eqnations for In Y and In Y2 allow constmctionof a correlation of Uie original P-xi-yi data set. Eqnatioii (12.1)is rearrangedand written for species I and 2 of a binary system as ... 

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. 

Equation (12.8), writtenfor derived property values, i.e., those given by a correlation, such as tire Margules equations, is subtracted from tins equation to yield ... 

The data are well correlated by the tliree-parameter Margules equation [an extension of Eq. (12.9)] ... 

All of the necessary experimental data [Vf, H2,i, 7 2,3, and E (Margules parameter)] were taken from the original publications (indicated as footnotes to Table 1) or calculated using the data from Gmehling s vapor-liquid equilibrium data compilation. Figure 1 and Table 1 show that the present eq 25 is in much better agreement with experiment than Krichevsky s eq 1 and equations A2-3—5 from Appendix 2, which involve the Margules expression for the activity coefficient. The new eq 25 provides predictions that are comparable to those of an empirical correlation for aqueous mixtures of solvents, which involves three adjustable parameters. 

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. 

Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978 67 (1978)] and excess enthalpy data [Morris et al.,/, Chem. Eng. Data 20 403-T05 (1975)] are available. The VLE data are well correlated by the Margules equations. As noted in connection with Eq. (4-270), parameters Ai and A i relate directly to infinite dilution values of the activity coefficients. Thus, we have from the VLE data at 323.15 K ... 

Correlations Representing VLE Data—Margules, van Laar, Wilson, and UNIQUAC... 

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