Virial Coef cients

Once the analytical expression for AGmix is known, the calculation of chemical potentials and other thermodynamic functions (activities, activity coef cients, virial coef cients, etc.) is straightforward. For polymer solutions, we must apply Eq. (3.3) to Flory-Huggins equation (3.45), keeping in mind that volume fractions 0, are fimctions of the number of moles, as given by... 

Problem 3.5 The osmotic pressure data for polystyrene of molecular weight 1.6x10 yielded, according to Eq. (3.67), the following values for the second virial coef cient (a) 2.88x10 mol cm g in dichloroethane and (b) - 0.37x10 mol cm g in cyclohexane, both at 22°C. Determine x for the polymer-solvent systems. Which is a better solvent ... 

Virial coef cients are often reported in the literature without specifying the equation to which they apply. However, this can be deduced by inspecting the units of the virial coef cient. 

In terms of the new symbols, the second virial coef cient is then given by the expression [cf. Eq. (3.78)] ... 

Thus, at a special temperature T = 6, A2 becomes equal to zero and the solution therefore becomes pseudoideal. Such solutions are also called theta solutions. The second virial coef cient is positive at temperatures higher than 0 and negative at lower temperatures. 

Figure 3.4 Plot of second virial coef cient vs. temperature (data of Problem 3.7).

Several methods can be used to determine theta solvents. These include phase equilibria studies (see Phase Equilibria in Poor Solvents), determination of second virial coef cient (see Problem 3.7), viscosity-molecular weight relationship, and cloud point titration. 

Expansion Factor The second virial coef cient, A2, is a measure of solvent-polymer compatibility. Thus, a large positive value ofAz indicates a good solvent for the polymer favoring expansion of its size, while a low value (sometimes even negative) shows that the solvent is relatively poor. The value of Ai will thus tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating an expansion factor a. As de ned by Eqs. (3.86) and (3.87), a represents the ratio of perturbed dimension of the polymer coil to its unperturbed dimension. 

In an ideal solvent (i.e., a poor solvent at the theta temperature for which the second virial coef cient vanishes), also known as theta solvent, = 1 and Eq. (3.150) reduces to... 

The second virial coef cient can be obtained from the slope of the straight line portion of the (IT/c) versus c plot by removing the terms in Eqs. (4.41)-(4.43). When plotted according to... 

Eq. (4.41), the osmotic pressures of solutions of the same polymer in different solvents should yield plots with the same intercept (at c = 0) but with different slopes (see Fig. 4.5), since the second virial coef cient, which re ects polymer-solvent interactions, will be different in solvents of differing solvent power. For example, the second virial coef cient can be related to the Flory-... 

Estimate (a) polymer molecular weight, M , (b) second virial coef cients Ai and F2, and (c) polymer-solvent interaction parameter... 

Problem 4,16 For the polystyrene sample in Problem 4.15 calculate (a) the second virial coef cient, (b) the root mean square end-to-end distance, and (c) the root-mean-square radius of gyration. 

A polyisobutylene sample has M = 400,000. The second virial coef cient of the polymer in chlorobenzene solution at 25°C is F2 = 94.5 cm /g. Calculate the osmotic pressure in g/cm of 0.30 g/dL solution of this polymer in chlorobenzene at 25°C. Compare this with the value calculated for an ideal solution. [Chlorobenzene density at25°C is 1.11 g/cm. ]... 

Make a Zimm plot using the data and determine the weight-average molecular weight of the polymer. Determine also the second virial coef cient F2 and the radius of gyration of the polymer in solution. (For toluene, h = 1.4976 and for polystyrene-toluene solutions dhldc = 0.112 cm /g.)... 

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