Virial Coefficient Prediction

The nonideality of polymer solutions is incorporated in the virial coefficients. Predicting nonideality of polymer solutions means, in reality, predictions of the second virial coefficient, because this is the coefficient which can be measured most accurately. Better solvents generally produce greater swelling of macromolecules and result in higher virial coefficients. 

Limitations of the RK methods which have been mentioned (or assumed) in previous literature include (a) poor second virial coefficient prediction, especially for compounds having nonzero acentric factors (b) poor prediction of component liquid densities (this is a disadvantage only of the Soave form the RKJZ method is fit to component liquid densities) and (c) inability to represent all PVT properties at the component critical simultaneously the Soave form fails to reproduce the critical density while the RKJZ form gives nonzero values of (dP/dV)r and (d2P/dV2)T at the critical point. 

Because F(Y) is always less than unity, the value of the second virial coefficient predicted by the Flory-Krigbaum theory is necessarily lower than that predicted by the theory of concentrated solutions. 

Making a Taylor expansion in powers of the density, the first and second virial coefficients predicted by TPTl are found to be ... 

In the absence of experimental values for the second virial coefficients, predicted ones through the Tsonopoulos (1974 1975) or the Hayden-O Connell (1975) correlations, provide reliable estimates of the fugacity coefficients. 

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE SECOND VIRIAL COEFFICIENTS. 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

Second Virial Coefficient. A group contribution method including polar and nonpolar contributions has been proposed for second virial coefficients (241). This method has been appHed to both pure components and mixtures, the latter through prediction of cross-second virial coefficients. 

Second virial coefficients, B, are a fnncBon of temperature and are available for about 1500 compounds in the DIPPR compilaOond The second virial coefficient can be regressed from experimental PX T data or can be reasonably and accurately predicted. Tsonoponlos proposed a predicOon method for nonpolar compounds that requires the criOcal temperature, critical pressure, and acentric factor Equations (2-68) through (2-70) describe the method. 

Binary interaction parameters are determined for each pq pair p q) from experimental data. Note that = k and k = k = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

Although the virial equation can be used to make accurate predictions about the properties of a real gas, provided that the virial coefficients are known for the temperature of interest, it is not a source of much insight without a lot of advanced analysis. An equation that is less accurate bur easier to interpret was proposed by the Dutch scientist Johannes van der Waals. The van der Waals equation is... 

Its precise basis in statistical mechanics makes the virial equation of state a powerful tool for prediction and correlation of thermodynamic properties involving fluids and fluid mixtures. Within the study of mixtures, the interaction second virial coefficient occupies an important position because of its relationship to the interaction potential between unlike molecules. On a more practical basis, this coefficient is useful in developing predictive correlations for mixture properties. 

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... 

It is shown that the properties of fully ionized aqueous electrolyte systems can be represented by relatively simple equations over wide ranges of composition. There are only a few systems for which data are available over the full range to fused salt. A simple equation commonly used for nonelectrolytes fits the measured vapor pressure of water reasonably well and further refinements are clearly possible. Over the somewhat more limited composition range up to saturation of typical salts such as NaCl, the equations representing thermodynamic properties with a Debye-Hiickel term plus second and third virial coefficients are very successful and these coefficients are known for nearly 300 electrolytes at room temperature. These same equations effectively predict the properties of mixed electrolytes. A stringent test is offered by the calculation of the solubility relationships of the system Na-K-Mg-Ca-Cl-SO - O and the calculated results of Harvie and Weare show excellent agreement with experiment. 

For a theta solvent (V2 = 0) the relevant interaction is described by the third virial coefficient using a simple Alexander approach similar to the one leading to Eq. 13, the brush height is predicted to vary with the grafting density as h pa in agreement with computer simulations [65]. 

If the average molecular weight of the sample is known its second virial coefficient can be predicted using the Kok-Rudin method (6). Input parameters for this calculation are M,... 

The intrinsic viscosity of PVB is shown as a function of solvent composition for various MIBK/MeOH mixtures in Figure 6. Since [ij] increases with a (see Equation 8), the higher [ly] the better the solvent. Apparently, most mixtures of MIBK and MeOH are better solvents for PVB than either pure solvent. Based on Figure 6, PVB should have a weak selective adsorption of MIBK in a 1 1 solvent mixture and weak adsorption of MeOH in a 3 1 MIBK/MeOH solvent mix. These predictions are in accord with light scattering data discussed previously. The intrinsic viscosity data is also consistent with the second virial coefficient data in Table II in indicating that the 1 1 and 3 1 MIBK/MeOH mixtures are nearly equally good solvents for PVB, the 9 1 mix is a worse solvent, but still better than pure MeOH. 

In practice, from a knowledge of measured values of the osmotic second virial coefficients it is rather easy to calculate the spinodal curve. It is worthy of note here to observe that reciprocal values of m, for biopolymers of rather high molecular weight (> 104 g/mol) are often comparable with the magnitude of A 24. This requires that, as well as values of the osmotic second virial coefficients, the molecular weight should also be taken into account in the prediction of the boundary conditions relating to phase separation. 

A2 from equation (5.16) or the cross second virial coefficient from equation (5.17). In turn, this knowledge of the second virial coefficients and their temperature dependence allows calculation of the values of the chemical potentials of all components of the biopolymer solution or colloidal system, as well as enthalpic and entropic contributions to those chemical potentials. On the basis of this information, a full description and prediction of the thermodynamic behaviour can be realised (see chapter 3 and the first paragraph of this chapter for the details). 

The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. The virial equation is much more general than the van der Waals equation, but it is more difficult to use to make predictions. 

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. 

The possibility of occurrence of instability of colloidal dispersions in the presence of free polymer was first predicted by Asakura and Oosawa (5), who have shown that the exclusion of the free polymer molecules from the interparticle space generates an attractive force between particles, DeHek and Vrij (1) have developed a model in which the particles and the polymer molecules are treated as hard spheres and rederived in a simple and illuminating way the interaction potential proposed by Asakura and Oosawa. Using this potential, they calculated the second virial coefficient for the particles as a function of the free polymer concentration and have shown that... 

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