Activity coefficient-models Margules

Given an activity coefficient model, we can use Eq. (6.22) to determine the azeotropic composition. For example, if we use a two-suffix Margules model, where... 

Conversely, given the composition of the coexisting liquid phases, Eq. (7.7) can also be used to fit an activity coefficient model. For example, using the two-suffix Margules model for a binary mixture. 

Since we have the concentrations of the coexisting equilibrium liquid phase we can determine two binary parameters. Also, since we are interested in two different temperatures (LLE at 20°C and VLE at 73.4° C) we want an activity coefficient model with some built in temperature dependence (otherwise, we will get LLE with the same compositions at all temperatures.) Consequently, I will use the two constant Margules equation... 

Determine the parameter values in the two- constant Margules equation that best fit the P ver-sus X] data of Problem 10.2-46, and then use this activity coefficient model to estimate the vapor-phase compositions. 

Margules activity coefficient model A simple thermodynamic model used to describe the excess Gibbs free energy of a liquid mixture. It uses activity coefficients that are a measure of the deviation from ideality of solubility of a compound in a liquid. See raoult s law. In the case of a binary mixture, the excess Gibbs free energy is expressed as a power series of the mole fraction in which the constants are regressed with experimental data. The activity coefficients are found by differentiation of the equation. Unlike other... 

For liquids and solids, determine the activity coefficients for binary and multicomponent mixtures through activity coefficient models, including the two-suffix Margules equation, the three-suffix Margules equation, the van Laar equation, and the Wilson equation. Identify when the symmetric activity coefficient model is appropriate and when you need to use an asymmetric model. 

In chemical systems of interest, we usually have more than two components. In this section we will briefly explore the extension of the activity coefficient models above to multicomponent systems. We begin with an extension of the two-suffix Margules equation to a ternary system. The excess Gibbs energy is written as follows ... 

Since the value for A based on species a is close to that from species b, the system is reasonably symmetric and we can use the two-suffix Margules equation. Alternatively, we could use an asymmetric activity coefficient model the van Laar equation is commonly used for azeotropes. 

In Examples 8.9 and 8.10, we explore different ways to fit model parameters of the two-suffix and three-suffix Margules equations, respectively, using experimental VLE data. We will use an entire data set to find the best value for the two-suffix Margules parameter A or the three-suffix Margules parameters A and B. When a model equation can be written in a linear form, a least-squares linear regression can be employed to determine the model parameters. Example 8.11 uses this method on the same data as used in Examples 8.9 and 8.10. This latter method is restricted to simpler activity coefficient models that can be written in linear form. 

To solve Equations (8.27) and (8.28), we need an activity coefficient model for g . We will illustrate the approach using the two-suffix Margules equation however, the same methodology can be applied to any model from Table 7.2. If we substitute Equation (7.55) for the activity coefficient of species a into Equation (8.27), we get ... 

VLE data can be used to obtain best-fit values of activity coefficient model parameters. A general approach appficable to any model is through the use of objective functions. An objective function takes account of the entire measured data set. It is written in terms of the difference between the calculated value of a given property and the experimental value of the same property. The model parameters are determined when the objective function is minimized. Alternatively, equations such as the two-suffix and three-suffix Margules equations can be rewritten to find model parameters through averages and linear regression. 

Aij Three-suffix Margules activity coefficient model parameters (one form)... 

A,B Three-suffix Margules or van Laar activity coefficient model parameters... 

Adopting a subregular Margules model for the NaAlSi30g-KAlSi308 (Ab-Or) binary mixture and assuming that the activity coefficient of the albite component is not affected by the presence of limited amounts of the third component in the mixture (i.e., CaAljSijOg), equation 5.260 may be transformed into... 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

Related Calculations. The constants for the binary Margules and Van Laar models for predicting activity coefficients (see Related Calculations under Example 3.4) are simply the natural logarithms of the infinite-dilution activity coefficients A t2 = I n y(XJ and /12,1 = I n y2XJ. 

The activity coefficients of a solute in a mixed solvent could be also calculated by employing various well-known phase equilibria models, such as the Wilson, NRTL, Margules, etc., which using information for binary subsystems could predict the activity coefficients in ternary mixtures (Fan and Jafvert, 1997 Domanska, 1990). 

To model a mixture that phase separates into two coexisting liquid phase, we need to add non-ideal terms (activity coefficients) to the ideal solution model. As an example of this, we examine the stability of the two-suffix Margules model, which has a molar Gibbs free energy of... 

This last result, with only one adjustable parameter, is too simple to be useful but does show that, to a first approximation, the Margules model is symmetric in mole fraction. This is evident because the activity coefficients are mirror images of each other, and the excess Gibbs free energy is symmetric around Xj = 0.5. The higher-order terms in eqn. (2.4.1) lead to more realistic, unsymmetric behavior. 

The Margules expansion model has been tested on some ionic systems over very wide ranges of composition, but over limited ranges of temperature and pressure (33,34). In this study, the model is applied over a wider range of temperature and pressure, from 25-350 C and from 1 bar or saturation pressure to 1 kb. NaCl and KCl are major solute components in natural fluids and there are abundant experimental data from which their fit parameters can be evaluated. Models based on the ion-interaction ajiproach are available for NaCl(aq) and KCl(aq) (8,9), but these are accurate only to about 6 molal. Solubilities of NaCl and KCl in water, however, reach 12 and 20 m, respectively, at 350 C, and ionic strengths of NaCl-KCl-H20 solutions reach more than 30 m at this temperature (35). The objective of this study is to describe the thermodynamic properties, particularly the osmotic and activity coefficients, of NaCl(aq) and KCl(aq) to their respective saturation concentrations in binary salt-H20 mixtures and in ternary NaCl-KCl-H20 systems, and to apply the Margules expansion model to solubility calculations to 350 C. 

Figure 4. Solubilities of halite (NaCl) in water to 350°C. The curve represents values calculated using the Margules expansion model for activity coefficients (infinite dilution reference state), and standard state Gibbs energies for NaCl(aq) derived from the equations of Pitzer et al. to 300°C, and of Tanger and Helgeson above 300 C.

J. Sci.. in press). Indeed, in the case of a solid-solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, ao, is usually sufficient (8). Equations 5 and 6 then become identical to those of the "regular" solid-solution model of Hildebrand (9). For the case where both ao and ai parameters are needed, equations 5 and 6 become equivalent to those of the "subregular" solid-solution model of Thompson and Waldbaum (10). a model much used in high-temperature work. Equations 5 and 6 can also be shown equivalent to Margules activity coefficient series (11). 

Equations 11.2-2 can be used with experimental phase equilibrium data to calculate the activity coefficient of a species in one phase from its known value in the second phase or, with Eqs. 11.2-3 and experimental activity coefficient data or appropriate solution models, to compute the compositions of both coexisting liquid phases (see Illustration 11.2-2). For example, using the one-constant Margules equation to represent the activity coefficients, we obtain from Eq. 11.2-2 the following relationship between the phase compositions ... 

The first term describes the contribution of the ideal mixing itself, while the second one describes the excess energy due to interactions The variation can exhibit a particular shape, as Illustrated by a numerical example in the Figure 6.18, where the activity coefficients have been calculated by Margules model with A,2=2 and A2,=1.5. It may be observed that in the immiscibility region a-b the value, resulting by... 

Figure 5.7 Activity coefficients for each component in three binary liquid mixtures, all at 60°C. Top acetone-chloroform. Middle acetone-methanol. Bottom methanol-chloroform. Note the scale change from one ordinate to the next. These y,- are based on the Lewis-Randall standard state and were comjnited using the Margules model, with parameters from Table E.2. Note in the top panel that y, < 1, while in the middle and bottom panels y, > 1. After a similar figure in Prausnitz et al. [2] and based on original data in Severns et al. [9].

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