Virial activity models

In the virial methods, therefore, the activity coefficients account implicitly for the reduction in the free ion s activity due to the formation of whatever ion pairs and complex species are not included in the formulation. As such, they describe not only the factors traditionally accounted for by activity coefficient models, such as the effects of electrostatic interaction and ion hydration, but also the distribution of species in solution. There is no provision in the method for separating the traditional part of the coefficients from the portion attributable to speciation. For this reason, the coefficients differ (even in the absence of error) in meaning and value from activity coefficients given by other methods. It might be more accurate and less confusing to refer to the virial methods as activity models rather than as activity coefficient models. 

We could, of course, attempt more sophisticated simulations of scale formation. Since the fluid mixture is quite concentrated early in the mixing, we might use a virial model to calculate activity coefficients (see Chapter 8). The Harvie-Mpller-Weare (1984) activity model is limited to 25 °C and does not consider barium or... 

Considering the rather large amount of data required to implement virial methods even at 25°C (e.g., Tables 7.4-7.7), it is tempting to dismiss the methods as no more than statistical fits to experimental data. In fact, however, virial methods take chemical potentials measured from simple solutions containing just one or two salts to provide an activity model capable of accurately predicting species activities in complex fluids. Eugster et al. (1980), for example, used the virial method of Harvie and Weare (1980) to accurately trace the evaporation of seawater almost to the point of desiccation. Using any other activity model, such a calculation could not even be contemplated. Other... 

The NRTB activity coefficient model (Renon and Prausnitz ) was used for the VBBB for the ternary system. The Hayden-O Connell second virial coefficient model, with association parameters, was used to account for the dimerization of acetic acid in the vapor phase. The Aspen Plus built-in association parameters were employed to compute ffigacity coef-hcients. The extended Antoine equation is used to calculate the vapor pressure of each component in the system. The Aspen Plus built-in parameters were again used in the simulation. 

To account for nonideal vapor-hquid equilibrium and possible vqxjr-liquid-liquid equilibrium (VLLE) for these quaternary systems, the NRTL model or UNIQUAC model is used for activity coefficients. Table 7.2 provides the model parameters for these five quaternary systems where the EtAc, IPAc, and AmAc systems are described by the NRTL model and MeAc and BuAc systems are represented using the UNIQUAC model. Because of the near atmosphaic pressure, the only vapor phase nonideality considered is the dimerization of acetic acid as described by the Hayden-O CoimeU second virial coefficient model. The Aspen Plus built-in association parameters are used to compute fiigacity coefficients. 

To account for nonideal vapor-liquid equilibrium and possible VLLE for this quaternary system, the NRTL model is used to calculate the activity coefficients. Model parameters are taken from Chapter 7. Vapor-phase nonideality caused by the dimerization of acetic acid is also taken into consideration using the Hayden-O Connell second virial coefficient model. Aspen Plus built-in parameters values are used. 

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 a series of papers, Harvie and Weare (1980), Harvie el al. (1980), and Eugster et al (1980) attacked this problem by presenting a virial method for computing activity coefficients in complex solutions (see Chapter 8) and applying it to construct a reaction model of seawater evaporation. Their calculations provided the first quantitative description of this process that accounted for all of the abundant components in seawater. 

One method takes into account the individual characteristics of the ionic media by using a medium-dependent expression for the activity coefficients of the species involved in the equilibrium reactions. The medium dependence is described by virial or ion interaction coefficients as used in the Pitzer equations and in the specific ion interaction model. 

It can be shown that the virial type of activity coefficient equations and the ionic pairing model are equivalent, provided that the ionic pairing is weak. In these cases, it is in general difficult to distinguish between complex formation and activity coefficient variations unless independent experimental evidence for complex formation is available, e.g., from spectroscopic data, as is the case for the weak uranium(VI) chloride complexes. It should be noted that the ion interaction coefficients evaluated and tabulated by Cia-vatta [10] were obtained from experimental mean activity coefficient data without taking into account complex formation. However, it is known that many of the metal ions listed by Ciavatta form weak complexes with chloride and nitrate ions. This fact is reflected by ion interaction coefficients that are smaller than those for the noncomplexing perchlorate ion (see Table 6.3). This review takes chloride and nitrate complex formation into account when these ions are part of the ionic medium and uses the value of the ion interaction coefficient (m +,cio4) for (M +,ci ) (m +,noj)- Io... 

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. 

Due to lack of space, the few results presented here are primarily intended to demonstrate the validity of the proposed method. The pore space of the adsorbent is assumed to consist of slit-shaped pores of width 15 A, with parameters chosen to model activated carbon. The porosity values are fixed at q = 0.45 and qp = 0.6. The feed stream is atemary gas mixture of H2/CH4/C2H6. The vtqx>r-phase fugacities were computed from the virial equation to second order, using coefficients taken from Reid et al ... 

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... 

Empirical models for the induced trace have also been obtained from (nonspectroscopic) measurements of the second virial dielectric coefficient of the Clausius-Mosotti and Lorentz-Lorenz expansions [30]. Excellent surveys with numerous references to the historical as well as the modern dielectric research activities were given by Buckingham [27], Kielich [89], and Sutter [143] in 1972 see also a recent review with a somewhat more spectroscopic emphasis [11]. 

Evaluation of d is usually by Eq. (4-243), based on the two-term virial equation of state. The activity coefficient y is ultimately based on Eq. (4-251) applied to an expression for G /RT, as described in the section Models for the Excess Gibbs Energy. ... 

DH-type, low ionic-strength term. Because the DH-type term lacks an ion size parameter, the Pitzer model is also less accurate than the extended DH equation in dilute solutions. However, a.ssuming the necessary interaction parameters (virial coefficients) have been measured in concentrated salt solutions, the model can accurately model ion activity coefficients and thus mineral solubilities in the most concentrated of brines. 

The WS mixing rule satisfies the low-density boundary condition that the second virial coefficient be quadratic in composition and the high-density condition that excess free energy be produced like that of currently used activity coefficient models, whereas the mixing rule itself is independent of density. This model provides a correct alternative to the earlier ad hoc density-dependent mixing rules (Copeman and... 

More recently, Ustinov and coworkers [72, 73] developed a thermodynamic approach based on an equation of state to model the gas adsorption equilibrium over a wide range of pressure. Their model is based on the Bender equation of state, which is a virial-like equation with temperature dependent parameters based on the Benedict-Webb-Rubin equation of state [74]. They employed the model [75, 76] to describe supercritical gas adsorption on activated carbon (Norit Rl) at high temperature, and extended this treatment to subcritical fluid adsorption taking into account the phase transition in elements of the adsorption volume. They argued that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of He expansion at room temperature can lead to erroneous results due to the adsorption of He in narrow micropores of activated carbon. 

Pitzer s formulation offers a satisfactory and desirable way to model strong electrolyte activity coefficients in concentrated and complex mixtures. When sufficient experimental data are available, one can make calculations which are considerably more accurate than those presented in this paper. Attaining high accuracy requires not only experimentally-based parameters but also that one employ third virial coefficients and additional mixing terms and include explicit temperature dependencies for the various parameters. 

In both solid and gaseous solutions, virial equation-based Raoultian coefficients have often been proposed. For example, the Margules equations, often used in binary and sometimes in ternary solid solutions and which have a virial equation basis, were proposed originally for gaseous solutions. However, there is no satisfactory general model for Raoultian coefficients in multi-component solid solutions, and the tendency in modeling has been to treat these solutions as ideal (i.e., to use the mole fraction of a solid solution component as its activity see Equation (3.13)). 

Ion Interaction. Ion-interaction theory has been the single most noteworthy modification to the computational scheme of chemical models over the past decade this option uses a virial coefficient expansion of the Debye-Huckel equation to compute activities of species in high ionic strength solutions. This phenomenological approach was initially presented by Pitzer ( ) followed by numerous papers with co-workers, and was developed primarily for laboratory systems it was first applied to natural systems by Harvie, Weare and co-workers (45-47). Several contributors to the symposium discussed the ion interaction approach, which is available in at least three of the more commonly used codes SOLMNEQ.88, PHRQPITZ, and EQ 3/6 (Figure 1). 

The ion-interaction model is a theoretically based approach that uses empirical data to account for complexing and ion pair formation by describing this change in free ion activity with a series of experimentally defined virial coefficients. Several philosophical difficulties have resulted from the introduction of this approach the lack of extensive experimental database for trace constituents or redox couples, incompatibility with the classical ion pairing model, the constant effort required to retrofit solubility data as the number of components in the model expand using the same historical fitting procedures, and the incompatibility of comparing thermodynamic solubility products obtained from model fits as opposed to solubility products obtained by other methods. 

For solution compositions in which the virial coefficients are well defined, the mineral phase boundaries, ionic activity, and the activity of water, can be modeled remarkably well. Numerous applications are already benefiting from the existing database, such as the ability to predict the solubility of minerals in brine environments. Despite the advancement in the description of high ionic strength solutions, the incompatibility of the ion-pairing model and the semi-empirical... 

In Pitzer s model the Gibbs excess free energy of a mixed electrolyte solution and the derived properties, osmotic and mean activity coefficients, are represented by a virial expansion of terms in concentration. A number of summaries of the model are available (i,4, ). The equations for the osmotic coefficient (( )), and activity coefficients (y) of cation (M), anion (X) and neutral species (N) are given below ... 

---