Virial coefficient equations for

Debye-Huckel effects are significant in the dilute range and are not considered, and (2) the usual composition scale for the solute standard state is molality rather than mole fraction. Both of these problems have been overcome, and the more complex relationships are being presented elsewhere (17). However, for most purposes, the virial coefficient equations for electrolytes are more convenient and have been widely used. Hence our primary presentation will be in those terms. 

A very severe test of these virial-coefficient equations for the sea-water-related Na-K-Mg-Ca-Cl-S0,-H 0 system has been made by Harvie and Weare (37) who calculated tne solubility relationships for most of the solids which can arise from this complex system. There are 13 invariant points with four solids present in the system Na-K-Mg-Cl-SO - O and the predicted solution compositions in all 13 cases agree with the experimental values of Braitsch (38) substantially within the estimated error of measurement. In particular, Harvie and Weare found that fourth virial coefficients were not required even in the most concentrated solutions. They did make a few small adjustments in third virial coefficients which had not previously been measured accurately, but otherwise they used the previously published parameters. 

There are also many less severe tests (11) of predictions for mixed electrolytes which illustrate the accuracy to be expected in various cases. Thus it is well-established that the virial coefficient equations for electrolytes yield reliable predictions of... 

The importance of the virial-coefficient equations is especially great for mixed electrolytes. Of the needed virial coefficients for a complex mixture such as sea water, most are determined by the pure electrolyte measurements and all the others of any significance are determined from data on simple mixtures such as NaCl-KCl, NaCl-MgC, NaCl-Na.SO, etc., which have been measured. The effect of the terms obtained from mixtures is very small in any case and these terms can be ignored for all but the most abundant species. 

Parameters for mixed electrolytes with the virial coefficient equations (at 25°C)... 

On the basis of these relationships, using expressions for the chemical potentials of the components at the level of approximation of the second virial coefficients (equations 3.17 and 3.18), the spinodal curve (equation 3.25) can be expressed mathematically in the following form (Edmond and Ogston, 1968) ... 

Equation (1.3) is by no means the only power law found in the excluded volume limit. For the second virial coefficient Akf, for instance, we find... 

Dymond and Smith [11] give an excellent compilation of virial coefficients of gases and mixtures. Cholinski et al. [12] provide second virial coefficient data for individual organic compounds and binary systems. The latter book also discusses various correlational methods for calculating second virial coefficients. Mason and Spurling [13] have written an informative monograph on the virial equation of state. 

Both qf these equations are known as virial expansions, and the parameters B C, D etc., and B, C, D, etc., are called virial coefficients. Parameters B and B are second virial coefficients C and C are third virial coefficients etc. For a given gas the virial coefficients are functions of temperature only. 

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... 

Mairy otlrer equations of state Irave been proposed for gases, but the virial equations are tire oirly ones having a finrr basis in theory. Tire methods of statistical mechanics allow derivation of tire virial equations and provide physical sigirificance to the virial coefficients. Thus, for the expansionin 1/ F, the term B/V arises on account of interactions between pairs of molecules (Sec. 7d.2) the C / term, on account of tlrree-body interactions etc. Since two-... 

The partition coefficients predicted by the theory for the four globular protein, lysozyme, chymotrypsin, albumin and catalase were determined by inserting into Equation 31 the second virial coefficients calculated for each... 

Low pressure for a mixture that contains an associating component Use the vinal equation of state, retaining only the second virial coefficient, and search for experimental pure component and cross virial coefficient data for all components in the mixture. 

The results themselves have a subtlety associated with their interpretation owing to the presence of the volume-ratio parameter and, optionally, the initial density parameter. The Burnett equations have more flexibility to fit Burnett data than only a density series to PVT data. The statistical uncertainties reflect the quality of the experimental data relative to the particular model used to describe the experiment. The estimation of accuracy for Burnett results is necessarily somewhat subjective since the effect of systematic errors on parameter values is not explicit in nonlinear equations, such as the Burnett equations. Accuracy, however, can be estimated from a study of the effects of systematic errors in computer model calculations and from the magnitude of the change in the volume-ratio value determined with nonideal and nearly ideal gases. For these reasons, we include such information along with our virial coefficient results for ethylene. 

We now show predictions of the compressibility factor for pure carbon dioxide along two isotherms, one supercritical and the other subcritical. All results shown here used values of a and b computed from and P. Figure 4.14 shows the results for the supercritical isotherm, T = 350 K. Up to about 75 bar, the three equations are all in good agreement with experiment, indicating that, at least at this temperature, all three satisfactorily estimate the second virial coefficient. However, for P > 100 bar, errors in... 

The same equations are used for mixtures of gases. In this case the coefficients are combinations of the coefficients of the pure components. They are combined according to mixing rules, which are also specified by statistical mechanics for any number of components. The second virial coefficient is, for a mixture of m components,... 

The most general approach (Pitzer and Brewer, 1961, p. 326) would be to start with the D-H limiting law (Equation 15.26) and add a power series of virial coefficients, as for gases. A simpler approach, begun by Scatchard in 1936, and used by Pitzer and Brewer (1961, pp. 326, 578 and Appendix 4), is to define a deviation function B (called B to distinguish it from the first virial coefficient, which in a sense it replaces) as the difference between observed and predicted activity coefficients for an electrolyte such as NaCl. This is... 

Table 2 Parameters for the virial coefficient equat ions at 25 C...

The quantities B, C, D,..., termed the second, third, fourth,..., virial coefficients, are for a given gas functions of temperature only. Equation (15) can be described as the virial equation of state in the volume-explicit (or Leiden) form. A related equation of state, that in the pressure-explicit (or Berlin) form, is also in common use ... 

Eq. (275) has been written here in the form of two terms. The first term on the right-hand side is the same as the ideal gas. One may think of the second term as a correction to the ideal gas. In Fig. 118 is a comparison of Eq. (275) with the virial equation derived by Ree and Hoover. It is apparent that this is a good approximation above a value of V/VJl) 2. Comparison to experimental data is difficult since, firstly, there is no such gas represented by hard spheres and, secondly, experimental virial coefficients even for gases such as argon are not readily available to the fifth term. This, however, seems to be a reasonable staring point for modeling. 

In addition to effects arising from local stiffness along the chain, monomers of real chains undergo excluded volume interactions among themselves mediated by solvent molecules as pointed out in Section 2.3. Since the chain connectivity has been parametrized in terms of Kuhn steps, we define the second virial coefficient (v) for a pair of Kuhn segments, in an equivalent way to Equation 2.17, as... 

SOLUTION We can sdve this problem by substituting the Lennard-Jones potential (Equation 4.10) into the molecular formulation for the second virial coefficient. Equation (4.29), as fiJlows ... 

Determine the second virial coefficient, B, for CS2 at 100°C from the following data. The saturation pressure of carbon disulfide (CS2) has been fit to the following equation ... 

For a pure vapor the virial coefficients are functions only of temperature for a mixture they are also functions of composition. An important advantage of the virial equation is that there are theoretically valid relations between the virial coefficients of a mixture and its composition. These relations are ... 

To use Equation (10b), we require virial coefficients which depend on temperature. As discussed in Appendix A, these coefficients are calculated using the correlation of Hayden and O Connell (1975). The required input parameters are, for each component critical temperature T, critical pressure P, ... 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. 

Equilibrium constants,, for all possible dimerization reactions are calculated from the metastable, bound, and chemical contributions to the second virial coefficients, B , as given by Equations (6) and (7). The equilibrium constants, K calculated using Equation (3-15). 

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