Second virial coefficient experimental

Scattering methods, 10 Second virial coefficient experimental data, 419 experimental methods, 9 Segment number, 12 Sorption, 2, 8 System pressure, 2... 

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. 

Intermolecular potential functions have been fitted to various experimental data, such as second virial coefficients, viscosities, and sublimation energy. The use of data from dense systems involves the additional assumption of the additivity of pair interactions. The viscosity seems to be more sensitive to the shape of the potential than the second virial coefficient hence data from that source are particularly valuable. These questions are discussed in full by Hirschfelder, Curtiss, and Bird17 whose recommended potentials based primarily on viscosity data are given in the tables of this section. 

If p is not high, terms beyond the second and third virial coefficients in equation (A3.3) and (A3.5) are usually small and can be neglected. This is fortunate, since experimental data are usually not accurate enough to give reliable values for the higher order terms. At low pressures, equation (A3.5) is often used and truncated after the second virial coefficient so that... 

This expression is called the virial equation. The coefficients B, C,. . . are called the second virial coefficient, third virial coefficient, and so on. The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. 

The theoretical foundations of these rules are, however, rather weak the first one is supposed to result from a formula derived by London for dispersion forces between unlike molecules, the validity of which is actually restricted to distances much larger than r the second one would only be true for molecules acting as rigid spheres. Many authors tried to check the validity of the combination rules by measuring the second virial coefficients of mixtures. It seems that within the experimental accuracy (unfortunately not very high) both rules are roughly verified.24... 

Other dilute solution properties depend also on LCB. For example, the second virial coefficient (A2) is reduced due to LCB. However, near the Flory 0 temperature, where A2 = 0 for linear polymers, branched polymers are observed to have apparent positive values of A2 [35]. This is now understood to be due to a more important contribution of the third virial coefficient near the 0 point in branched than in linear polymers. As a consequence, the experimental 0 temperature, defined as the temperature where A2 = 0 is lower in branched than in linear polymers [36, 37]. Branched polymers have also been found to have a wider miscibility range than linear polymers [38], As a consequence, high MW highly branched polymers will tend to coprecipitate with lower MW more lightly branched or linear polymers in solvent/non-solvent fractionation experiments. This makes fractionation according to the extent of branching less effective. 

The calculation of compressibility factors of gaseous ethanol can be made with equation 2.18, because the second virial coefficient (B) is available at different temperatures [20] and the saturation vapor pressures can be interpolated or extrapolated from the experimental data (figure 2.4). One obtains Z = 0.991 at... 

Hall, K.R. Eubank, P.T. "Experimental Technique for Direct Measurement of Interaction Second Virial Coefficients,"... 

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

To compile quantitatively reliable information, we need a source of experimental measurements. One way to determine the nature of inter-molecular forces between biopolymer molecules in a solvent medium is to measure the so-called osmotic second virial coefficient A2. Expressed in molar (biopolymer) terms, the quantity A2 can be related to the two-body potential of mean force W(r) by the following equation (Vrij, 1976 de Kruif, 1999 Prausnitz, 2003 de Kruif and Tuinier, 2005) ... 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

Let us now consider some actual numerical data for specific mixed biopolymer systems. Table 5.1 shows a set of examples comparing the values of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated theoretically on the basis of various simple excluded volume models using equations (5.32) to (5.35). For the purposes of this comparison, the experimental data were obtained under conditions of relatively high ionic strength (/ > 0.1 mol dm- ), i.e., under conditions where the contribution of the electrostatic term (A if1) is expected to be relatively insignificant. 

Table 5.1 Comparison of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated from theory on the basis of the excluded volume contribution only.

We can compare experimentally measured values of the cross second virial coefficient (Ay ) with theoretical ones (Ay exc) based on the contri-... 

Some of the experimental details of osmometry and the use of osmometry for determining molecular weights and second virial coefficients are discussed in Section 3.3. 

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