Empirical activity coefficient models

In order to perform quantitative thermodynamic calculations using the Gibbs free energy for a nonideal solution (see Eqs. (6.9)—(6.11)), we need explicit expressions for the activity coefficients. A few empirical expressions that are typically employed are  

The parameters A, B, and C in the above models are, in general, functions of temperature and pressure, but are independent of composition. These parameters are typically obtained by fitting experimental data. Given the parameters of these activity coefficient models, we can predict all the thermodynamic properties of the system. 

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At present, there is no exact high-density boundary condition for mixture equations of state. However, there is the ob.servation that, at liquid densities, the empirical activity coefficient models (such as those of van Laar, Wilson, NRTL, UNIQUAC, etc.) discussed earlier provide a good representation of the excess or nonideal part of the free energy of mixing. Therefore, another boundary condition that could be imposed is as follows ... 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

Kontogeorgis, G.M. et ah. Improved models for the prediction of activity coefficients in nearly athermal mixtures. Part I. Empirical modifications of free-volume models. Fluid Phase Equilibria, 92, 35, 1994. Coutinho, J.A.P., Andersen, S.I., and Stenby, E.H., Evaluation of activity coefficient models in prediction of alkane SEE, Fluid Phase Equilibria, 103, 23, 1995. 

With increasing electrolyte concentration, the short-range interactions become more and more dominating. Therefore, in activity coefficient models the Debye-Hiickel term, which describes the long-range interactions, has to be extended by a term describing the short-range interactions. A well-known empirical extension of the Debye-Hiickel theory is the Bromley equation [5] ... 

In actual experiments, as indicated above, ionization quotients Q are usually measured in a solution at finite ionic strength made up by the addition of supporting electrolytes such as NaCl, KCl, or NaCFsSOs. Therefore, activity coefficient models are needed to extrapolate the Q values to infinite dilution for such equilibria. All of these models are based on some version of the Debye-Hiickel equation, which determines the initial slope the logio0 versus ionic strength dependence, with additional empirical ionic strength terms which are typically derived from those used in the Pitzer ion interaction model (Pitzer, 1991). An example of this empirical approach is given in Equation (3.29). 

The clay ion-exchange model assumes that the interactions of the various cations in any one clay type can be generalized and that the amount of exchange will be determined by the empirically determined cation-exchange capacity (CEC) of the clays in the injection zone. The aqueous-phase activity coefficients of the cations can be determined from a distribution-of-species code. The clay-phase activity coefficients are derived by assuming that the clay phase behaves as a regular solution and by applying conventional solution theory to the experimental equilibrium data in the literature.1 2 3... 

In the B-dot model, as currently applied (Wolery, 1992b), the activity coefficients of electrically neutral, nonpolar species [B(OH)3, C>2(aq), SiC>2(aq), CH aq), and H2(aq) are calculated from ionic strength using an empirical relationship,... 

As the basis for predicting ionic activity coefficients we chose to adopt an. empirical modification of Bromley s ( 5) extension of the Debye-Huckel model. The mean activity coefficient of a pure salt in water is given by... 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

We present results describing the solid-liquid and the vapor-liquid equilibria in the NaCl-HCl-HpO system. In the first part, purely empirical relations are used to describe the activity coefficients and the second part includes use of a semi-empirical model (Z) to describe the compositional dependence of the activity coefficients. 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

It is very satisfying and useful that the COSMO-RS model—in contrast to empirical group contribution models—is able to access the gas phase in addition to the liquid state. This allows for the prediction of vapor pressures and solvation free energies. Also, the large amount of accurate, temperature-dependent vapor pressure data can be used for the parameterization of COSMO-RS. On the other hand, the fundamental difference between the liquid state and gas phase limits the accuracy of vapor pressure prediction, while accurate, pure compound vapor pressure data are available for most chemical compounds. Therefore, it is preferable to use experimental vapor pressures in combination with calculated activity coefficients for vapor-liquid equilibria predictions in most practical applications. 

Although COSMO-RS generally provides good predictions of chemical potentials and activity coefficients of molecules in liquids, its accuracy in many cases is not sufficient for the simulation of chemical processes and plants, because even small deviations can have large effects on the behavior of a complex process. Therefore, the chemical engineer typically prefers to use empirical thermodynamic models, such as the UNIQUAC and NRTL, for the description of liquid-phase activity coefficients with... 

One of the concerns regarding the use of COSMO-RS thermodynamics directly in simulations is the considerably larger computation time that is required for the evaluation of the activity coefficients compared to simpler empiricEd models with... 

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... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

All current activity coefficient estimation models are by necessity semi-empirical in nature, because too little is known about solution theory for outright estimation. Chemical modeling is not readily available and is not far enough developed to make this type of calculation. The constants required by these models must be estimated using either experimental data (e.g. an infinite dilution activity coefficient or a molar volume) or group contributions derived from experimental data (e.g. interaction constants, molecular volumes and surface areas). 

Although Procedure C is a good predictive method, it should not be used as a substitute to reducing good experimental data to obtain activity coefficients. In general, higher accuracy can be obtained from empirical models when these models are used with binary interaction parameters obtained from experimental data. 

There are several limitations which lead to the discrepancies in Tables IV-X. First of all, no model will be better than the assumptions upon which it is based. The models compiled in this survey are based on the ion association approach whose general reliability rests on several non-thermodynamic assumptions. For example, the use of activity coefficients to describe the non-ideal behavior of aqueous electrolytes reflects our uncertain knowledge of ionic interactions and as a consequence we must approximate activity coefficients with semi-empirical equations. In addition, the assumption of ion association may be a naive representation of the true interactions of "ions" in aqueous solutions. If a consistent and comprehensive theory of electrolyte solutions were available along with a consistent set of thermodynamic data then our aqueous models should be in excellent agreement for most systems. Until such a theory is provided we should expect the type of results shown in Tables IV-X. No degree of computational or numerical sophistication can improve upon the basic chemical model which is utilized. 

Pitzer (1973) re-examined the statistical mechanics of aqueous electrolytes in water and derived a different but semi-empirical method for activity coefficients, commonly termed the Pitzer specific-ion-interaction model. He fitted a slightly different function for behavior at low concentrations and used a virial coefficient formulation for high concentrations. The results have proved extremely fruitful for modeling activity coefficients over a very large range of molality. The general equation is... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

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