Virial methods

The virial methods differ conceptually from other techniques in that they take little or no explicit account of the distribution of species in solution. In their simplest form, the equations recognize only free ions, as though each salt has fully dissociated in solution. The molality m/ of the Na+ ion, then, is taken to be the analytical concentration of sodium. All of the calcium in solution is represented by Ca++, the chlorine by Cl-, the sulfate by SO4-, and so on. In many chemical systems, however, it is desirable to include some complex species in the virial formulation. Species that protonate and deprotonate with pH, such as those in the series COg -HCOJ-C02(aq) and A1+++-A10H++-A1(0H), typically need to be included, and incorporating strong ion pairs such as CaSO aq) may improve the model s accuracy at high temperatures. Weare (1987, pp. 148-153) discusses the criteria for selecting complex species to include in a virial formulation. 

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. 

The virial methods work by assuming that the solution s excess free energy Gex (i.e., the free energy in excess of that in an ideal solution) can be described by a 

and k are subscripts representing the various species in solution and /dh is a function of ionic strength similar in form to the Debye-Hiickel equation. The terms Xy and Hijk are second and third virial coefficients, which are intended to account for short-range interactions among ions the second virial coefficients vary with ionic strength, whereas the third virial coefficients do not. 

Equation 8.10 is notable in that it ascribes specific energetic effects to the interactions of the aqueous species taken in pairs (the first summation) and triplets (second summation). The equation s general form is not ad hoc but suggested by statistical mechanics (Anderson and Crerar, 1993, pp. 446 -51). The values of the virial coefficients, however, are largely empirical, being deduced from chemical potentials determined from solutions of just one or two salts. 

and k are subscripts representing the various species in solution and 

An expression for the ion activity coefficients y, follows from differentiating Eqn. 7.10 with respect to The result in general form is 

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Virial methods, the second type, employ coefficients that account for interactions among the individual components (rather than species) in solution. The virial methods are less general, rather complicated to apply, require considerable amounts of data, and allow little insight into the distribution of species in solution. They can, however, reliably predict mineral solubilities even in concentrated brines. 

Considering the rather large amount of data required to implement virial methods even at 25 °C (e.g., Tables 8.4—8.7), it is tempting to dismiss the methods as... 

It is interesting to compare the Debye-Hiickel and virial methods, since each has its own advantages and limitations. The Debye-Hiickel equations are simple to apply and readily extensible to include new species in solution, since they require few coefficients specific to either species or solution. The method can be applied as well over the range of temperatures most important to an aqueous geochemist. There is an extensive literature on ion association reactions, so there are few limits to the complexity of the solutions that can be modeled. 

The virial methods, on the other hand, provide remarkably accurate results over a broad range of solution concentrations and with a variety of dominant solutes. The methods, however, are limited in breadth. Notably lacking at present are data for redox reactions and for components such as aluminum and silica with low solubilities. Data for extending the methods to apply beyond room temperature (e.g., Mpller, 1988 Greenberg and Mpller, 1989), furthermore, are limited currently to relatively simple chemical systems. 

Unlike the Debye-Hiickel equations, the virial methods provide little or no information about the distribution of species in solution. Geochemists like to identify the dominant species in solution in order to write the reactions that control a system s behavior. In the virial methods, this information is hidden within the complexities of the virial equations and coefficients. Many geochemists, therefore, find the virial methods to be less satisfying than methods that predict the species distribution. The information given by Debye-Hiickel methods about species distributions in concentrated solutions, however, is not necessarily reliable and should be used with caution. 

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. 

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... 

Another necessary condition for equilibration is that the average pressure calculated during the simulation be close to the set pressure of the NPT simulation. Since in this MC simulation of polymer chains the bond lengths and angles are infinitely stiff, the determination of the stress tensor from interatomic forces is non-trivial. There are in fact many techniques for computing the stress tensor these are reviewed elsewhere [30]. Here we used both the so-called Molecular Virial method (see Appendix A of [37] and also [30]) and an inter-chain force-based method [30] to calculate the stress tensor and the pressure. We found the calculated pressure to be in excellent agreement with the set pressure for both methods within 1% for the Molecular Virial method and within 10% for the inter-chain force-based method. 

In using the reduced virial method of predicting compressibility factors or other thermodynamic properties, the pure component virials are found by interpolation among the values of Table I or Fig. 1 which represent (13) at different Q values. The necessary values of 6 and cp are read from Fig. 2. Interaction virials are also obtained from Table I but the virials and temperature are reduced as in (18). The correction term is read from Fig. 3. In order to interpolate between values in Table I it is necessary to define a Qi 2value for an unlike pair virial. This is done by calculating 12... 

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