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** Parameters for the virial coefficient equations at **

For true compatability of solute and solvent, matching of all these partial solubility parameters (i.e., 8, Bp, 8 ) is necessary. The total solubility parameter can be easily calculated [1, p. 307] finm the material s enthalpy of vaporization, vapor pressure as a function of temperature, surface tension, thermal expansion coefficient, critical pressure, and second virial coefficient of its vapor, as well as by calculating its value for the chemical structure of the material. For the calculation of the Hildebrand solubility parameter fi om chemical structure, we use Small s [58] equation [Pg.404]

Parameters for mixed electrolytes with the virial coefficient equations (at 25°C) [Pg.463]

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

These manipulations may appear to add little except for needless complication to an interpretation of the second virial coefficient for random coils. Recall, however, that Equation (81) allows the variation of solvent goodness caused by temperature changes to be described quantitatively. Thus the interaction parameter x is used to describe how B changes when a polymer is dissolved in different solvents. By contrast, 9 is used to describe the variation in B when a given polymer-solvent system is examined at different temperatures. This has been done for the polystyrene-cyclohexane system at three different temperatures the results are discussed in Example 3.4. [Pg.132]

Here B, C,... are called the second, third,... virial coefficients these parameters depend only on temperature (and composition for mixtures). An alternative expression for the virial equation is a power series expansion in pressure [Pg.240]

The acentric factor, CO, was the third parameter used (20) in an equation based on the second virial coefficient. This equation was further modified and is suitable for reduced temperatures above 0.5. [Pg.240]

The standard form of the Virial EOS formulates the compressibility as an infinite series of the inverse molar volume or pressure, as shown by the equations (5.3) and (5.4). In the low-pressure region, up to 15 bar, Virial EOS is the most accurate. The formulation known as second virial coefficient is sufficient for technical computations. Virial EOS is able to handle a variety of chemical classes, including polar species. Hayden and O Connell have proposed one of the best correlations for the second virial parameter. The method is predictive, because considers only physical data, as dipole moment, critical temperature, critical pressure, and the degree of association between the interacting components. Usually Virial EOS it is an option to describe the vapour phase [Pg.163]

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. [Pg.39]

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, [Pg.29]

The experimentally fitted hydrate guest Kihara parameters in the cavity potential uj (r) of Equation 5.25 are not the same as those found from second virial coefficients or viscosity data for several reasons, two of which are listed here. First, the Kihara potential itself does not adequately fit pure water virials over a wide range of temperature and pressure, and thus will not be adequate for water-hydrocarbon mixtures. Second, with the spherical Lennard-Jones-Devonshire theory the point-wise potential of water molecules is smeared to yield an averaged spherical shell potential, which causes the water parameters to become indistinct. As a result, the Kihara parameters for the guest within the cavity are fitted to hydrate formation properties for each component. [Pg.274]

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are [Pg.531]

Table 4 compares different theoretical approaches with respect to the equations of state and the second and third virial coefficients (B2, B3) for a hard rod solution in the isotropic state B2 and B3 are the parameters appearing in the expansion [Pg.100]

To make the basic Pitzer equation more useful for data correlation of aqueous strong electrolyte systems, Pitzer modified it by defining a new set of more directly observable parameters representing certain combinations of the second and third virial coefficients. The modified Pitzer equation is [Pg.63]

Mixing mles for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 9 for mixtures are usually given by the following, where, as for the second virial coefficient, = 0-. [Pg.486]

When i =j, all equations reduce to the appropriate values for a pure species. When i these equations define a set of interaction parameters having no physical significance. For a mixture, values of B and dB /dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In for the component fugacity coefficients are given by Eq. (4-196). [Pg.356]

Since Lennard-Jones (6-12) potential has been widely used for calcn of properties of matter in the gaseous, liquid, and solid states, Hirschfelder et al (Ref 8e, pp 162ff) discuss it in detail. They show that the parameters o and ( of the potential function may be determined by analysis of the second virial coefficient of the LJD equation of state [Pg.282]

Express the compressibility factor Z = PV td/iRT) for a gas that follows the Redlich-Kwong equation. Convert the resulting equation into one in which the independent variable is (l/Vm)- Obtain a McLaurin series for Z as a polynomial in (1 /Vad, and express the virial coefficients for that equation in terms of the parameters of the Redlich-Kwong equation. [Pg.107]

This is a virial equation, the word virial being taken from the Latin word for force and thus indicating that forces between the molecules are having an effect. It turns out that statistical mechanical models also give equations that can be written in this form with the virial coefficients, B C > etc., being related to various interaction parameters. [Pg.358]

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. [Pg.458]

One may sometimes have access to the parameters required for the Pitzer approaches, e.g., for some hydrolysis equilibria and for some solubility product data, cf. Baes and Mesmer [3] and Pitzer [4]. In this case, the reviewer should perform a calculation using both the B-G-S and the P-B equations and the full virial coefficient methods and compare the results. [Pg.259]

In this study the Pitzer equation is also used, but a different, more straightforward approach is adopted in which the drawbacks just discussed do not arise. First, terms are added to the basic virial form of the Pitzer equation to account for molecule-ion and molecule-molecule interactions. Then, following Pitzer, a set of new, more observable parameters are defined that are functions of the virial coefficients. Thus, the Pitzer equation is extended, rather than modified, to account for the presence of molecular solutes. The interpretation of the terms and parameters of the original Pitzer equation is unchanged. The resulting extended Pitzer equation is [Pg.65]

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. [Pg.222]

** Parameters for the virial coefficient equations at **

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