Activity coefficient method for

Equation 19 shows that the activity coefficient method for high-pressure fluid-phase equilibria requires information for the partial molar volume, Vi1-, which is very difficult to obtain. 

As in the case of ideal solutions, the equilibrium constant involves a ratio of factors for the equilibrium concentration variables xj, raised to the appropriate power. This ratio is now preceded by two factors, enclosed in curly brackets, that attend to the nonideality of the participants in the chemical reaction. Departures from ideality of the unmixed components (first factor) are discussed immediately below under normal conditions, the corresponding activity coefficients do not differ greatly from unity. For the intermixed components, one must look up in appropriate tabulations values of the various activity coefficients Methods for their experimental determination are also introduced below. 

For the monomers in the polymerization under consideration the fugacity coefficients were estimated by Redlich-Kwong equation of state and were found to be close to unity. The activity coefficients (8) for the monomers were estimated by Scatchard-Hildebrand s method (5) for the most volatile monomer there was a temperature dependence but none for the other monomer. These were later confirmed by applying the UNIFAC method (6). The saturation vapor pressures were calculated by Antoine coefficients (5). 

Unless liquid phase activity coefficients have been used, it is best to use the same equation of state for excess enthalpy that was selected for the vapour-liquid equilibria. If liquid-phase activity coefficients have been specified, then a correlation appropriate for the activity coefficient method should be used. 

Solubility modelling with activity coefficient methods is an under-utilized tool in the pharmaceutical sector. Within the last few years there have been several new developments that have increased the capabilities of these techniques. The NRTL-SAC model is a flexible new addition to the predictive armory and new software that facilitates local fitting of UNIFAC groups for Pharmaceutical molecules offers an interesting alternative. Quantum chemistry approaches like COSMO-RS [25] and COSMO-SAC [26] may allow realistic ab-initio calculations to be performed, although computational requirements are still restrictive in many corporate environments. Solubility modelling has an important role to play in the efficient development and fundamental understanding of pharmaceutical crystallization processes. The application of these methods to industrially relevant problems, and the development of new... 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

Although there is no straightforward and convenient method for evaluating activity coefficients for individual ions, the Debye-Hiickel relationship permits an evaluation of the mean activity coefficient (y+), for ions at low concentrations (usually <0.01 moll-1) ... 

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

The local composition model (LCM) is an excess Gibbs energy model for electrolyte systems from which activity coefficients can be derived. Chen and co-workers (17-19) presented the original LCM activity coefficient equations for binary and multicomponent systems. The LCM equations were subsequently modified (1, 2) and used in the ASPEN process simulator (Aspen Technology Inc.) as a means of handling chemical processes with electrolytes. The LCM activity coefficient equations are explicit functions, and require computational methods. Due to length and complexity, only the salient features of the LCM equations will be reviewed in this paper. The Aspen Plus Electrolyte Manual (1) and Taylor (21) present the final form of the LCM binary and multicomponent equations used in this work. 

A second example is provided by a semiempirical correlation for multi-component activity coefficients in aqueous electrolyte solutions shown in Fig. 2. This correlation, developed by Fritz Meissner at MIT [3], presents a method for scale-up activity-coefficient data for single-salt solutions, which are plentiful, are used to predict activity coefficients for multisalt solutions for which experimental data are rare. The scale-up is guided by an extended Debye-Hilckel theory, but essentially it is based on enlightened empiricism. Meissner s method provides useful estimates of thermodynamic properties needed for process design of multieffect evaporators to produce salts from multicomponent brines. It will be many years before sophisticated statistical mechanical techniques can perform a similar scale-up calculation. Until then, correlations such as Meissner s will be required in a conventional industry that produces vast amounts of inexpensive commodity chemicals. 

For the analytical equations, there are two methods to compute the vapour-liquid equilibrium for systems. The equation of state method (also known as the direct or phi-phi method) uses an equation of state to describe both the liquid and vapour phase properties, whereas the activity coefficient method (also known as the gamma-phi approach) describes the liquid phase via an activity coefficient model and the vapour phase via an equation of state. Recently, there have also been modified equation of state methods that have an activity coefficient model built into the mixing mles. These methods can be both correlative and predictive. The predictive methods rely on the use of group contribution methods for the activity coefficient models such as UNIFAC and ASOG. Recently, there have also been attempts to develop group contribution methods for the equation of state method, e.g. PRSK. " For a detailed history on the development of equations of state and their applications, as well as activity coefficient models, refer to Wei and Sadus, Sandler and Walas. ... 

In which yi is the activity coefficient of component i in the solution, yf is the combinatorial part and ytR is the residual part. Up to this point, all of the group contribution and activity coefficient methods (i.e. NRTL-SAC) have been the same, but the methods in which the activities have been calculated are different. In the UNIFAC model, the combinatorial part for component i is found from the following equation [8] ... 

This mixing rule has been successful in several ways. First, when combined with any cubic EOS that gives the correct vapor pressure and an appropriate activity coefficient model for the term, it has been shown to lead to very good correlations of vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria, indeed generally comparable to those obtained when the same activity coefficient models are used directly in the y-

mixing rule extends the range of application of equations of state to mixtures that previously could be correlated only with activity coefficient models. 

The goal of predictive phase equilibrium models is to provide reliable and accurate predictions of the phase behavior of mixtures in the absence of experimental data. For low and moderate pressures, this has been accomplished to a considerable extent by using the group contribution activity coefficient methods, such as the UNIFAC or ASOG models, for the activity coefficient term in eqn. (2.3.8). The combination of such group contribution methods with equations of state is very attractive because it makes the EOS approach completely predictive and the group contribution method... 

Furthermore, for most vapor mixtures allow pressure, 0,- is very close to unity (there are exceptions to this assumption for example, associating gases such as hydrogen fluoride or acetic acid), and that leads to the equilibrium relation we used in this monograph to calculate the vapor-liquid phase equilibrium by the direct use of activity coefficient methods ... 

Click and then the box on the left (in the Data Browser option). The part of the process represented by that box is then described. You can also click on the box and then the FI key to obtain the same information. The Help menu provides detailed help in all areas. For example, the Help/Physical Property Methods/Choosing a Property Method gives advice about which thermodynamic model is recommended for different applications. The Guidelines for Choosing a Property Method and Guidelines for Choosing an Activity Coefficient Method are sub-menus that outline decision trees to guide your choice. 

Equations of state offer a number of advantages over activity coefficient models for example, they can be applied to both low and high pressures, for properties other than phase equilibria, and the density is not required as an input parameter. However, often they are more difficult to develop for complex fluids and mixtures than are activity coefficient models. Very many equations of state have been proposed for polymers Section 16.7 discusses the reason. Recent reviews have been presented. " " We will not attempt to cover all the various approaches, but essentially discuss in detail only two of them, which seem promising for polymer solutions and blends the cubic equations of state and the SAFT (Statistical Associating Fluid Theory) method. 

Bofh EOS and activity-coefficient methods require binary interaction parameters. In process simulation software, the necessary parameters may already be built into a data bank. Sometimes, parameters for the system of interest may be found in the literature. If not, however, the parameters must be fitted to mixture data. 

The individual-ion activity coefficients for the free ions were based on the Macinnis (18) convention, which defines the activity of Cl to be equal to the mean activity coefficient of KCl in a KCl solution of equivalent ionic strength. From this starting point, individual-ion activity coefficients for the free ions of other elements were derived from single-salt solutions. The method of Millero and Schreiber (14) was used to calculate the individual-ion, activity-coefficient parameters (Equation 5) from the parameters given by Pitzer (19). However, several different sets of salts could be used to derive the individual-ion activity coefficient for a free ion. For example, the individual-ion activity coefficient for OH could be calculated using mean activity-coefficient data for KOH and KCl, or from CsOH, CsCl, and KCl, and so forth. 

Equilibrium constants for the formation of nitrate complexes at hydrogen ion concentrations of 2 and 4 M and metal ion concentrations of 5 X 10 M or less, were determined using ion exchange techniques (353, 465) (Table XVIII). Activity coefficient data for aqueous zirconium and hafnium species are scarce, although there is one report (319) of activity coefficients for metal nitrate solutions as determined by the isopiestic method. 

We could write all non-ideal properties of solutions in terms of activity coefficients. For electrolyte (salt) solutions we usually do use activity coefficients, but for solid and gaseous solutions Earth scientists have traditionally used other approaches based on excess properties. This is not to say the other approaches are fundamentally different or better. They are simply different ways of representing the same physical properties. However, we have just seen that using excess functions instead of activity coefficients can simplify notation, and this is always an advantage. The next section discusses the method most commonly used to express the activity coefficients and excess functions of solid solutions. 

As an alternative, group contribution methods can be applied to predict the required activity coefficients. Methods like UNIFAC or modified UNIFAC are based on the functional groups of the components and group interaction parameters, which are fitted to a large number of experimental data. The calculated activity coefficients therefore have a high accuracy (Gmehling et ah, 2002) and can be used for reactive systems. Especially for very fast reactions, group contribution methods are often recommended. 

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