Virial expansion nonideal solutions

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... 

In the treatment of nonideal solutions, expansions of a form similar to the virial expansion for gases [Eq. (7-14)] are introduced. The osmotic coefficient of 1 can be expanded in a power series in concentration of component 2 in the form... 

For nonideal dilute solutions we can study the osmotic pressure using virial expansion. For simplicity, let us specifically assume that the solute is one component. The virial expansion of the osmotic pressure. 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

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