Margules equations

Different forms of this equation have been proposed (Wohl, 1946). One of these is the three-suffix or third-degree form  

It can be proved that these equations are solutions to the Gibbs-Duhem equation by taking the differentials d In Yi and d In Yt and substituting them in Equation 1.30 written for a binary. It should also be recalled that -1- = 1 and that dX = -dX  

The binary constants A12 and A21 are usually determined by regressing binary isothermal VLE data. If these data are accurate in the vicinities around Xj = 0 and Xi = 1, the binary constants may be determined from the limiting conditions  

The constants may, in fact, be calculated from a single equilibrium data point (vapor-liquid or liquid-liquid if intended for liquid-liquid equilibrium calculations) although the results would be heavily weighted to that point, probably resulting in less accuracy for the remaining range of compositions. 

The Margules equation is generalized in its four-suffix form to multi-component mixtures as follows (Wohl, 1946)  

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SUFFIX MARGULES EQUATION UNIOUAC EQUATION NRTL EGU AT ION WILSON EQUATION... 

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... 

Equation (6.77) is known as the Duhem-Margules equation. It can also be written as... 

When deviations from ideal solution behavior occur, the changes in the deviations with mole fraction for the two components are not independent, and the Duhem-Margules equation can be used to obtain this relationship. The allowed combinations"1 are shown in Figure 6.10 in which p /p and P2//>2 are... 

Figure 6.10 Representative deviations from ideal solution behavior allowed by the Duhem-Margules equation. The dotted lines are the ideal solution predictions. The dashed lines giveP2IP2 (lower left to upper right), and p jp (upper left to lower right).

From the Duhem-Margules equation [equation (6.77)], we know that... 

The Wilson equation is superior to the familiar Van-Laar and Margules equations (see Volume 2, Chapter 11) for systems that are severely non-ideal but, like other three suffix equations, it cannot be used to represent systems that form two phases in the concentration range of interest. 

Margules equation, 21 733-734 Mariculture, 3 182 Maridomycin complex, 15 287 Marignac process, 24 319 Marine antifoulant, environmentally safe, 12 811-812... 

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... 

When the Krichevsky-Kasarnowsky equation fails it may be because of either changing activity coefficient of the solute gas with composition, changing partial molal volume of the gas with pressure, or both. The Krichevsky-Ilinskaya equation takes into account the variation in the activity coefficient of the solute gas with mole fraction by means of a two-suffix Margules equation. 

These equations become identical to the three-suffix Margules equations when = 21, and the functional form of these two types of equations is not greatly different unless the constants i2 and differ by more than about 50 per cent. 

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... 

Fan, C., and C. T. Jafvert, Margules equations applied to PAH solubilities in alcohol-water mixtures , Environ. Sci. Technol., 31, 3516-3522 (1997). 

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...

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. 

These equations are known as three suffix Margules equations. 

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

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