Osmotic pressure virial expansion

Hill, T. L. 1959. Theory of solutions. II. Osmotic pressure virial expansion and hght scattering in two component solutions. Journal of Chemical Physics. 30, 93. 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Coefficients in the virial expansion of the osmotic pressure as a power series in the concentration c (Chap. XII et seq.). 

Coefficients in the alternative virial expansion of the osmotic pressure (see Eqs. VII-13 and XII-76). 

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

In these equations ns is the solvent refractive index, dn/dc the refractive index increment, c the polymer concentration in g/ml, T the temperature in K, R the gas constant, NA Avogadro s number, and n the osmotic pressure. Equation (B.8) follows from Eq. (B.7) by using the familiar virial expansion of the osmotic pressure... 

What does this picture predict for the scaling functions Let us first consider the osmotic pressure. Clearly for small overlap the virial expansion holds... 

Some authors [59] call the region c < c the virial regime. Probably they consider that the virial expansion for osmotic pressure would diverge as c approaches c. ... 

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. 

We shall now derive the virial expansion of the osmotic pressure following McMillan and Mayer and Hill (P). but simplifying the derivation. The virial expansion plays an imjwrtant role in the theory of solutions. For our purpose we introduce the grand partition function of... 

Each polymer coil in a solution contributes to viscosity. In very dilute solutions, the contribution from different coils is additive and solution viscosity r] increases above the solvent viscosity t/s linearly with polymer concentration c. The effective virial expansion for viscosity at low concentration is of the same form as Eq. (1.76) for osmotic pressure and Eq. (1.96) for light scattering ... 

This virial expansion is analogous to that used for the osmotic pressure in Chapter 1 [Eq. (1.74)] and we will see in Section 3.3.4, how the excluded volume is related to the second virial coefficient. 

This expression of osmotic pressure can be written in the form of the virial expansion in terms of number density of A monomers c = 4>jb [see Eq. (3.8)]... 

Recall the mean-field virial expansion for the osmotic pressure of polymer solutions discussed in Section 4.5.1 [Eq. (4.67)]. 

The semidilute osmotic pressure has a stronger concentration dependence than predicted by the mean-field virial expansion [Eq. (5.41)] -------... 

The mean-field prediction for the osmotic pressure [Eq. (5.40)] in 0-sol-vents is the virial expansion with vanishing excluded volume (v = 0) ... 

Equations 13 and 14 have the same functional form as that postulated by Edmonds and Ogston (3) and later generalized by King et aL (4). The significance of the work presented here is that it enables us to give a fundamental interpretation of the coefficients and the reference potential in terms of forces between the species. It also allows us to relate these coefficients to the virial coefficients which appear in the McMillan-Meyer virial expansion (12) of the osmotic pressure. In the equations of Ogston and of King et the coefficients are set equal to the virial coefficients of the McMillan-Meyer virial expansion, but, as we shall see, these coefficients are equivalent only when certain assumptions are made. 

Here we have approximated V by Vq since the solution is assumed to be dilute. The Z s can be evaluated by relating them to the virial coefficients in the McMillan and Meyer theory expansion for the osmotic pressure, 11, in the density of each species, Pj = N. / V. For the case of three solutes in a solvent, its expansion is... 

In practice, the most important set of thermodynamic variables is of course T, P, pA, employed in (6.34). However, relation (6.33) is also useful and has enjoyed considerable attention in osmotic experiments where pB is kept constant. This set of variables provides relations which bear a remarkable analogy to the virial expansion of various quantities of real gases. We demonstrate this point by extracting the first-order expansion of the osmotic pressure n in the solute density pA. This can be obtained by the use of the thermodynamic relation... 

The virial expansion of the osmotic pressure, although formally exact, is not very useful beyond the first-order correction to the DI limiting case. Higher-order correction terms involve higher-order potentials of mean force about which very little is known. 

Under conditions of partly screened interactions in dilute solutions (high added salt concentration cs and low polymer concentration c), the solution osmotic pressure can be expressed via a virial expansion (Eq. 24). Then light scattering becomes a useful tool to obtain values of second virial coefficients characterizing interactions in solution. The second virial coefficient can be calculated from the slope of the dependence given by Eq. 25. The relation between the true and the apparent second virial coefficient is similar to the relation between the true and the apparent molecular weight (see the previous section for more details and the meaning of the symbols) ... 

The effect of concentration is often expressed in a virial expansion. For the osmotic pressure it reads... 

Nonideality of solutions is discussed in Section 2.2.5. It can be expressed as the deviation of the colligative properties from that of an ideal, i.e., very dilute, solution. Here we will consider the virial expansion of osmotic pressure. Equation (2.18) can conveniently be written for a neutral and flexible polymer as... 

Figure 8.1 gives some examples of the water activity versus mole fraction of water for some simple solutions. It is seen that Eq. (8.2) is poorly obeyed, except for sucrose at very small xs. Deviations from ideality are discussed in Section 2.2.5. They can be expressed in a virial expansion, as given in Eq. (2.18) for the osmotic pressure 17. The relation between the two properties is... 

Consequently, the first terms of the virial expansion are obtained by eliminating the fugacities in terms of which the osmotic pressure and the concentrations are expressed. In the monodisperse case, we set... 

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