Second viscosity virial coefficient

This theory covers the complete density range from dilute gas to solidification. However, the hard sphere model is inappropriate for a real gas at low and intermediate densities where specific effects of intermolecular forces are significant. It is therefore necessary to modify the theory this has been done in different ways. In Section 5.2, a method is described for determination of the second viscosity virial coefficient and the translational part of the thermal conductivity coefficient. Although this approach is not rigorous, it does provide a useful estimate for these coefficients - especially for low reduced temperatures. 

Since the existence of internal degrees of fteedom and their interaction with translational modes of motion has a negligible effect on viscosity, the second viscosity virial coefficient of polyatomic fluids can be formally identified with that of monatomic gases (57" = 5 ). 

Fig. 5.3. The reduced second viscosity virial coefficient as a function of reduced temperature. Curves 1 - Rainwater-Friend theory (S = 1.04 and 9 = 1.25) 2 - Rainwater-Friend theory (5 = 1.02 and 6> = 1.15) 3 - MET-I.

For application to real gases, this theory has been modified (Enskog 1922 Hanley et al. 1972 Hanley Cohen 1976 Vogel etal. 1986 Ross etal. 1986). Although this has no rigorous theoretical basis, it does provide an alternative rqiresentation of the second viscosity virial coefficient and the translational part of the second thermal conductivity virial coefficient, which is particularly useful at reduced temperatures below T = 0.5, the lower limit of the coefficients in Table 5.1. On the basis that a real fluid differs from a hard sphere fluid mainly in the temperature dependence of the collision frequency, the pressure P of the hard-sphere fluid is replaced by the thermal pressure T(dP/dT)p of... 

The latter can be predicted from the universal expression (5.19) for the reduced second viscosity virial coefficient by using... 

As a result of both theoretical and experimental studies of the second viscosity virial coefficient, it has been possible to develop a generalized representation of its temperature dependence based on the Lennard-Jones (12-6) potential (Rainwater Friend 1987 Bich Vogel 1991). A particular advantage of this s proach is that it is possible to estimate the coefficient BrjiT) for a gas for which no experimental viscosity data as a function of density exist, given a knowledge of the Lennard-Jones (12-6) potential parameters as derived by an analysis of dilute-gas viscosity data. Such an estimation of Bfi has been performed for ethane by use of the reconunended parameters / b = 251.1 K and cr = 0.4325 nm (Hendl etal. 1994). 

This procedure based on the Rainwater-Friend theory is valid only for reduced temperatures T > 0.7 (T = 175 K for ethane). This lower limit will not be exceeded by the viscosity representation of ethane in the vapor phase, since the range of validity of its zero-density contribution has a lower limit of T = 2(X) K. Nevertheless, experimental data in the liquid phase are available at much lower temperatures extending to T = 100 K. In order to use a single overall viscosity correlation it must be ensured that the initial-density contribution extrapolates satisfactorily to low temperatures. For this purpose, for temperatures below T = 0.7, the second viscosity virial coefficient has been estimated by use of the modified Enskog theory (see Chapter 5), which relates B,f to the second and third pressure virial coefficients. Although this method enables Brj to be evaluated, it is cumbersome for practical applications. Therefore, the calculated B, values using both methods have been fitted to the functional form... 

Fig. 14.20. Second viscosity virial coefficient as a function of temperature.

Essentially exact values for V2 and have been reported by Wajnryb and Dahler [42] for both stick and slip boundary conditions. These values are recorded in Table V.4, along with estimates obtained by several previous investigators. In Figure 5.16, predictions based on the formula (5.297) are compared with the available experimental data. The solid curve is based on the stick boundary condition values V2 = 2.5 (Einstein) and = 5.9147 (Wajnryb and Dahler). To obtain the dashed curve, which agrees much better with (some of) the experimental data and for which virial coefficients with V2 = 5.0781. 

M weight average molar mass M number average molar mass A second osmotic virial coefficient R radius of gyration Rj, hydrodynamic radius p ratio of and R [rj] intrinsic viscosity. 

The first density correction for viscosity and for the translational part of the thermal conductivity is best predicted by the Rainwater-Friend model, for which values for the reduced second transport virial coefficients are given in Table 5.1. For computer codes the tabulated values can be approximated using the correlation... 

Bich Vogel 1991) to express the coefficient in terms of the second viscosity virial... 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

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. 

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

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. 

Intrinsic viscosity measurements revealed a conformational transition upon heating from 26 to 40 °C, while the UV absorbance of the solution was insensitive to the change. The entropy parameters for PA were also discussed in light of the Flory-Krigbaum correlation between the second virial coefficient and theta temper-... 

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... 

Critical data Second virial coefficient Viscosity Average value... 

The next step consists of the determination of the size of the macromolecules in space. Two equivalent sphere radii can be measured directly by means of static and dynamic LS. Another one can be determined from a combination of the molar mass and the second virial coefficient A2. Similarly, an equivalent sphere radius is obtained from a combination of the molar mass with the intrinsic viscosity. This is outlined in the following sections. 

In principle all combinations of universal ratios of the four radii can be formed. A useful combination, however, is the ratio of Rj/Rwhere the two radii are related to the second virial coefficient and the intrinsic viscosity as outlined in Eqs. (21) and (22). Likewise one could form the ratio [6,142-145]... 

Another uncertainty arises from the influence of polydispersity. Intrinsic viscosity data were mostly obtained from fractions but the second virial coefficient data were chosen from unfractionated samples. The resulting error is probably not large since A2 depends only slightly on the width of the distribution [183, 184]. 

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. 

The properties of solutions of macromolecular substances depend on the solvent, the temperature, and the molecular weight of the chain molecules. Hence, the (average) molecular weight of polymers can be determined by measuring the solution properties such as the viscosity of dilute solutions. However, prior to this, some details have to be known about the solubility of the polymer to be analyzed. When the solubility of a polymer has to be determined, it is important to realize that macromolecules often show behavioral extremes they may be either infinitely soluble in a solvent, completely insoluble, or only swellable to a well-defined extent. Saturated solutions in contact with a nonswollen solid phase, as is normally observed with low-molecular-weight compounds, do not occur in the case of polymeric materials. The suitability of a solvent for a specific polymer, therefore, cannot be quantified in terms of a classic saturated solution. It is much better expressed in terms of the amount of a precipitant that must be added to the polymer solution to initiate precipitation (cloud point). A more exact measure for the quality of a solvent is the second virial coefficient of the osmotic pressure determined for the corresponding solution, or the viscosity numbers in different solvents. 

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