Second virial coefficient table

Extensive tables and equations are given in ref. 1 for viscosity, surface tension, thermal conductivity, molar density, vapor pressure, and second virial coefficient as functions of temperature. 

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. 

From Table V we see that in general the values of bb/ aa deduced from critical constants and second virial coefficients agree rather well with each other while there seems to be a large discrepancy with the values obtained from viscosity. This tends... 

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...

The second virial coefficients (A2) shown in Table I do not show significant dependence upon solvent for most of the LB samples. The only exception is K1107 which yielded an anomalously low A2 value in toluene this sample exhibited marginal solubility in toluene, and the low A 2 suggests unfavorable polymer-solvent interaction. The A2 values for MA polymers, however, are consistently lower than their LB... 

The second virial coefficients measured in various solvents at room temperature after heating are given in Table I. In the absence of aggregation and selective adsorption, a ranking by quality of solvents for PVB would follow the order of A2 values. The higher A2, the better the solvent. However, since these effects were not completely absent in the data used to construct Table I (particularly in the 9 1 solvent mixture), the solvent ranking given in this table must be considered tentative. 

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. 

Tables VI and VII give results corresponding to two series of lignin fractions obtained with a flow-through reactor (3). (The units for dn/dc and A2 are respectively ml.g-1 and mole.ml.g-2). These results show that LALLS allows the determination of low Mw values. The dn/dc values differ from sample to sample but vary only slightly for a given set of fractions. The second virial coefficient exhibits no definite trend. Negative values indicate perhaps some association effects but light scattering alone is not sufficient to ascertain this point.

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.

Table 3.3 also includes an approximation for the case in which the concentration of the salt exceeds that of the colloid, but not to the swamping extent, mM > mP. Comparison of that case with the result given in Equation (34) suggests that the contribution of charge to the second virial coefficient of the solution is given by... 

For a multicomponent mixture, a virial coefficient is needed to account for each possible interaction. The second virial coefficients for a two component mixture are Bpi, Bp2> and B22 where B- represents the interaction between two molecules of component 1, Bp2 represents the interaction between a molecule of 1 and a molecule of 2, and B22 represents interaction between two molecules of 2. A tabulation of some compounds whose virial coefficients have been measured by GC is given in Table 11.6. 

Table 1.3 summarizes molecular weights (Mn) and second virial coefficient (A2) data for various fractions of this polymer. 

Table 1.2 Reduced osmotic pressures (tt/c)c=o, number average molecular weights Mn and osmotic second virial coefficient A2 for poly(pentachlorophenyl methacrylate) fractions in toluene at 25°C and benzene at 40°C (tt in cm of benzene or toluene) (c in g dl-1). (From ref. [44])...

Here the quantity PV/nRT is often called the virial and the quantities 1, B(T), C(7T), etc., the coefficients of its expansion in inverse powers of the volume per mole, F/n, are called the virial coefficients, so that B(T) is called the second virial coefficient, C(T) the third, etc. The experimental results for equations of state of imperfect gases are usually stated by giving B(T), C(T), etc., as tables of values or as power series in the temperature. It now proves possible to derive the second virial coefficient B T) fairly simply from statistical mechanics. 

Before comparing theory and experiment let us discuss the convergence of the semiclassical expansion of the dielectric second virial coefficient. In Table 1-15 the classical dielectric virial coefficient the first and second quantum corrections, and the full quantum result are reported. An inspection of this table shows that the quantum effects are small for temperatures larger than 100 K, and /it(/) can be approximated by the classical expression with an error smaller than 2.5%. At lower temperatures the dielectric virial coefficient of 4He starts to deviate from the classical value. Still, for T > 50 K the quantum effects can be efficiently accounted for by the sum of the first and second quantum corrections. Indeed, for T = 50, 75, and 100 K the series (7) + lli 1 (7) + (7) reproduces the exact results with errors... 

Table 12. Second virial coefficient At of polystyrene in toluene at 25° C...

Table 19. Values of Mo, scattering asymmetry and the second virial coefficient, A2, for silarylene carboorganocyclosiloxanes fraction [117]...

TABLE 1 Second virial coefficient of argon" (B in units cm mol )... 

Tables 8.2.2 and 8.2.3 give typical results for two series of polydisperse lignin fractions obtained from acidic organosolv delignification of black cottonwood (Pla et al. 1986) and from alkaline delignification of western hemlock (Dolk et al. 1986). In both cases, LALLS allows accurate determination of low molecular weight values. The nearly identical dn/dc values for a given series of lignin fractions indicate the good reproducibility and accuracy of the technique. However, the second virial coefficients, A2, vary considerably depending upon the fraction measured.

Figure 3. Dependence of the second virial coefficient on the concentration of cosolvent in the water (1)—lysozyme (2)—arginine (3) mixture. , experimental data solid line, values predicted using eq 10 with722 (see Table 2) as an adjustable parameter , protein—solvent interaction contribution B " calculated using eq IIB O, ideal mixture contribution B (eq 11 A) A, protein—protein interaction contribution B " P predicted by eq 11C with 722 as an adjustable parameter from Table 2. See details in Figure 1.

The following table gives the second virial coefficient as a function of temperature for several common gases. 

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