Osmotic second virial coefficient

The osmotic second virial coefficient for a dilute solution of solid hard cylinders of length L and diameter d can be evaluated from geometric considerations alone. The presence of another rod whose center of mass is within a distance L restricts the available rotational freedom of the first rod. Calculations by Zimm et al. in a work by Tanford ° all agree that the second virial coefficient for these objects is given by  

For a fixed molar mass and partial specific volume, the second virial coefficient is inversely proportional to the axial ratio. For a fixed diameter, the second virial coefficient is independent of rod length, since L scales as M. 

Actual rodlike polymers in solution are influenced by intermolecular attractions of longer range due to van der Waals and electrolyte effects. An attempt to deal with solvent quality is presented below in connection with the Flory theory of rodlike solutions. Electrolyte solutions are considered in Chapter 10. 

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Section 2 brings the cluster development for the osmotic pressure. Section 3 generalizes the approach of Section 2 to distribution functions, including a new and simple derivation of the cluster expansion of the pair distribution function. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 contains an application of our general solution theory to compact macromolecular molecules. Section 6 contains the second osmotic virial coefficient of flexible macromokcules, followed in Section 7 by concluding remarks. 

Generally speaking, positive values of A2 mean net repulsion between the particles, while negative values of A2 correspond to attraction. Eor more detailed analysis of the values of the second osmotic virial coefficient, the use of other definitions of the particle concentration is more convenient. The common virial expansion"... 

The term B1 (vfMf) is called the second osmotic virial coefficient and is denoted by B. Then... 

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 same behavior as that described for the second osmotic virial coefficients has been reported for the radii of gyration determined by static light scattering. At the theta temperature of the system PMAC dissolved in toluene, Kuhn s law (< Kg A4) was verified [13, 15, 20]. 

There are about 850 newly published referenees containing about 150 new vapor-liquid equilibrium data sets and some new tables containing classical Henry s coefficients, about 600 new liquid-liquid equilibrium data sets and some new high-pressure fluid phase equilibrium data, 10 new enthalpic data sets, 20 new data sets describing PVT-properties of polymers, and 120 new data sets with densities or excess volumes. There are also new results on second osmotic virial coefficients of about 45 polymers in aqueous solution. So, in comparison to the original handbook, the new supplementary volume contains even a larger amoimt of data and will be a useful as well as necessary completion of the origirral handbook. 

Now rib is the bulk concentration of (small) hard spheres. The quantity B2 is the second osmotic virial coefficient... 

The theory of the second osmotic virial coefficient in moderate salt solutions was developed in detail by Orofino and Flory. ° An extension of the Flory-Krigbaum approach was also developed. The chemical potential of the solvent in a nonionic solution was expressed as ... 

The second osmotic virial coefficient for a polyelectrolyte in a salt solution is then given approximately by Equation 10.17, with the new expression for Xj. The value of the second virial coefficient is raised by the inclusion of the Donnan effect. 

The second osmotic virial coefficient of the expansion, A2, is a measure of the thermodynamic quality of the solvent for the polymer and accounts for binary... 

The second osmotic virial coefficient. The change in the chemical potential on mixing is made up of an ideal term and an excess term such that ... 

The thermodynamic relations discussed so far were, above all, formulated for the description of moderately to highly concentrated polymer solutions. The information acquired in the context of the determination of molar masses, on the other hand, refers to dilute solution and is usually expressed in terms of second osmotic virial coefficients and higher members of a series expansion of the chemical potential of the solvent with respect to the polymer concentration... 

Measurements performed to determine the molar masses of polymers yield - as a valuable byproduct - information on the pair interaction between the macromolecules [30]. The composition dependence of the osmotic pressure Tiosm observed via membrane osmometry is directly related to the chemical potential of the solvent [cf. (14) of Sect. 2] and provides the second osmotic virial coefficient A2, from which Xo> Ihe Flory-Huggins interaction parameter in the limit of high dilution becomes accessible [cf. (15)]. Such data are particularly valuable because they can be measured with higher accuracy than the x values for concentrated polymer solutions and because they represent a solid starting point for the sometimes very complex function xiV )- In principle, membrane osmometry can also be operated with polymer solutions of different composition in the two chambers (differential osmometry) to gain data for higher polymer concentrations however, little use is made of this option. 

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