Activity Coefficient Models for Electrolyte Solutions

For the development of activity coefficient models for electrolyte solutions, the theory of Debye and Huckel is usually the starting point. It can be regarded as an exact equation to describe the behavior of an electrolyte system at infinite dilution. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

This chapter traces the history of activity coefficient models for aqueous solutions of single strong electrolytes. Of great importance is the Debye-Huckel equation, which has been the cornerstone of more recent models. In addition to Debye-Huckel, the following, more recent methods are outlined ... 

We consider Pitzer s model for the calculation of activity coefficients in aqueous electrolyte solutions (Pitzer, 1991). It is the most widely used thermodynamic model for electrolyte solutions. 

The model of Debye and Hiickel enabled the construction of a relatively simple equation for the determination of activities coefficients in diluted electrolyte solutions. In first approximation their equation, which is called Debye-Huckel equation, looks as follows ... 

Due to the need to model the equilibria of solutions containing multiple weak electrolytes, such as the H2O - NHs CO2 system, it became necessary to go beyond the Setschenow equation for activity coefficient calculations. For such solutions to be modeled well, the ion-molecule interactions must affect not only the molecular activity coefficients, but also the ionic activity coefficients and water activities. An early attempt by Edwards. Newman and Prausnitz (P5) used the Guggenheim equation for activity coefficients and assumed the water activity to be unity. This application was felt to be good for low weak electrolyte concentrations at temperatures no higher than 80° C. 

The regularities and estimation methods of the activity coefficients of concentrated electrolyte solutions have been investigated for many years, but up to now these problems are not completely solved yet. In this field, Pitzer s ion-ion interaction model [109], as a semi-empirical model, has been most widely used. In Pitzer s method, the thermodynamic function (such as logarithm value of activity coefficient, log y ) is expanded into a series, and the coefficients of the terms of this series, and P and C, are used to calculate the activity coefficients of solutions of different concentrations. The coefficients P ° P and C have to be calculated from the experimental data of activity coefficients of concentrated... 

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

In applying this equation to multi-solute systems, the ionic concentrations are of sufficient magnitude that molecule-ion and ion-ion interactions must be considered. Edwards et al. (6) used a method proposed by Bromley (J7) for the estimation of the B parameters. The model was found to be useful for the calculation of multi-solute equilibria in the NH3+H5S+H2O and NH3+CO2+H2O systems. However, because of the assumptions regarding the activity of the water and the use of only two-body interaction parameters, the model is suitable only up to molecular concentrations of about 2 molal. As well the temperature was restricted to the range 0° to 100 oc because of the equations used for the Henry1s constants and the dissociation constants. In a later study, Edwards et al. (8) extended the correlation to higher concentrations (up to 10 - 20 molal) and higher temperatures (0° to 170 °C). In this work the activity coefficients of the electrolytes were calculated from an expression due to Pitzer (9) ... 

Two activity coefficient models have been developed for vapor-liquid equilibrium of electrolyte systems. The first model is an extension of the Pitzer equation and is applicable to aqueous electrolyte systems containing any number of molecular and ionic solutes. The validity of the model has been shown by data correlation studies on three aqueous electrolyte systems of industrial interest. The second model is based on the local composition concept and is designed to be applicable to all kinds of electrolyte systems. Preliminary data correlation results on many binary and ternary electrolyte systems suggest the validity of the local composition model. 

Nitric acid is a strong electrolyte. Therefore, the solubilities of nitrogen oxides in water given in Ref. 191 and based on Henry s law are utilized and further corrected by using the method of van Krevelen and Hofhjzer (77) for electrolyte solutions. The chemical equilibrium is calculated in terms of liquid-phase activities. The local composition model of Engels (192), based on the UNIQUAC model, is used for the calculation of vapor pressures and activity coefficients of water and nitric acid. Multicomponent diffusion coefficients in the liquid phase are corrected for the nonideality, as suggested in Ref. 57. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Second, we need an equation-of-state for electrolyte solutions. Equations-of-state are needed for modeling high-pressure applications with electrolyte solutions. Significant advances are being made in this area. Given that the electrolyte NRTL model has been widely applied for low-pressure applications, we are hopeful that, some day, there will be an equation-of-state for electrolytes that is compatible with the electrolyte NRTL activity coefficient model. 

In addition to the short-range interactions between species in all solutions, long-range electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theory or the mean spherical approximation (MSA). These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. 

Models are often developed to explain certain kinds of data, ignoring other kinds that also might be pertinent. The initial development of Pitzer s equations (33.34) for activity coefficients in concentrated solutions was focused on explaining measurements of vapor pressure equilibrium and of electromotive force (emf). The data could be explained by assuming that the electrolytes examined were, at least in a formal sense, fully dissociated. Later work using these equations to explain solubility data required the formal adoption of a few ion pair species (30). Even so, no speciation/activity coefficient model based on Pitzer s equations is presently consistent with the picture of much more extensive ion-pairing based on other sources, such as Smith and Martell s (35) compilation of association constants. This compilation is a collective attempt to explain other kinds of data, such as electrical conductance, spectrophotometry, and acoustic absorption. 

Application of the restricted primitive model leads to the limiting Debye-Hiickel expression for the mean molal activity coefficient of an electrolyte that pertains to very dilute solutions ... 

The coefficients J R) and J2 R) depend on the cutoff distance R and thus include the influence of the short-range forces on the transport phenomenon for the activity coefficient of the chemical model, see Electrolyte Solutions,... 

The binary sodium chloride-water system has been the object of many studies. As a result there is a wealth of published data for a wide range of temperatures. This data includes solubility, density, vapor pressure lowering and heat of solution measurements. Because of this availability of data and the straightforward strong electrolyte behavior of the system, sodium chloride has almost always been included as an example when illustrating activity coefficient modeling techniques. For this application, Meissner s method of activity coefficient calculation will be used. 

Setchenow coefficients for hydrocarbons and for volatile solutes In sea water the osmotic coefficient and density of sea water as a function of temperature and salinity. Thermodynamic solubility products of minerals In brines the activity coefficient of carbon dioxide In sea water speclatlon calculations on copper, zinc, cadmium, and lead In sea water excess Gibbs energies of mixing of electrolyte solutions at 25 C and pairwise and triplet Interaction terms for electrolyte solutions in terms of various models. 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

A more quantitative prediction of activity coefficients can be done for the simplest cases [18]. However, for most electrolytes, beyond salt concentrations of 0.1 M, predictions are a tedious task and often still impossible, although numerous attempts have been made over the past decades [19-21]. This is true all the more when more than one salt is involved, as it is usually the case for practical applications. Ternary salt systems or even multicomponent systems with several salts, other solutes, and solvents are still out of the scope of present theory, at least, when true predictions without adjusted parameters are required. Only data fittings are possible with plausible models and with a certain number of adjustable parameters that do not always have a real physical sense [1, 5, 22-27]. It is also very difficult to calculate the activity coefficients of an electrolyte in the presence of other electrolytes and solutes. Even the definition is difficult, because electrolyte usually dissociate, so that extrathermodynamical ion activity coefficients must be defined. The problem is even more complex when salts are only partially dissociated or when complex equilibriums of gases, solutes, and salts are involved, for example, in the case of CO2 with acids and bases [28, 29]. 

The above approach is empirical. Thermodynamic models for describing solution behavior can also be employed to determine gas solubilities, and these models are amenable to the estimation of gas solubilities in multicomponent systems from sets of single salt data. The thermodynamic approach employed is known as the Pitzer species interaction model, and it is used to determine the activity coefficient of the gas from a summation of interaction terms with anions, cations, and neutral species [3, 10, 11]. These interaction parameters are determined empirically from solubility data in a range of electrolyte solutions and have been tabulated for a wide range of salts, permitting the solubility of oxygen to be determined in mixed electrolyte solutions over a wide range of temperature and concentrations. 

---