# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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of the distribution of molecular weights in a sample. The method is time-

consuming and involves large quantities of solvent, which must be

subsequently removed to obtain pure polymer.

With these remarks, we conclude our discussion of phase separation. In

the remainder of the chapter we consider another thermodynamic property

of

n

Figure 8.6 Effect of successive fractionations: weight fraction n-mer

versus n for seven successive precipitates and final residue (calculated

for R = 10~3). [Adapted from G. V. Schulz, Z. Phys. Chem. B46:137 (1940);

B47.155 (1940).]

542

Thermodynamics of Polymer Solutions

polymer solutions, namely the osmotic pressure. As in the previous

sections, we shall combine pure thermodynamics as well as statistical

models to understand osmostic pressure more fully and to maximize our

ability to interpret experimental osmometry.

8.8 Osmotic Pressure

Osmotic pressure is one of four closely related properties of solutions

that are collectively known as colligative properties. In all four, a

difference in the behavior of the solution and the pure solvent is

related to the thermodynamic activity of the solvent in the solution. In

ideal solutions the activity equals the mole fraction, and the mole

fractions of the solvent (subscript 1) and the solute (subscript 2) add

up to unity in two-component systems. Therefore the colligative

properties can easily be related to the mole fraction of the solute in an

ideal solution. The following review of the other three colligative

properties indicates the similarity which underlies the analysis of all

the colligative properties:

1. Vapor pressure lowering. Equation (8.20) shows that for any

component in a binary liquid solution as = Pj/Pi°. For an ideal solution,

this becomes

Pi = *iPi° = (1 " x2)Pi' or

Pi° - Pi = Ap = X2P!°

(8.68)

(8.69)

This is an expression of Raoult's law which we have used previously.

Freezing point depression. A solute which does not form solid solutions

with the solvent and is therefore excluded from the solid phase lowers

the freezing point of the solvent. It is the chemical potential of the

solvent which is lowered by the solute, so the pure solvent reaches the

same (lower) value at a lower temperature. At equilibrium

AHf° I 1 1 .

ln a'=' "ã ( t; ' v 1

(8 70)

where AHf° is the heat of fusion of the solvent, Tf is the freezing

point of the solution, and Tf° is the freezing point of the pure solvent.

For an ideal solution the following relationships apply :

In at = In Xj = In (1 - x2) = -x2

(8.71)

where the last approximation requires dilute conditions as well.

Osmotic Pressure

543

3. Boiling point elevation. A solute which does not enter the vapor

phase to any significant extent raises the boiling point of the solvent.

As above, the solute lowers the activity of the solvent, which, in turn,

lowers the vapor pressure. Therefore the solution must be raised to a

higher temperature before its vapor pressure reaches 1.0 atm. At

equilibrium

where ÄÍÓ° is the heat of vaporization of the solvent, Tb is the

boiling point of the solution, and Tb° is the boiling point of the pure

solvent. For ideal or ideal and dilute solutions, the approximations of

Eq. (8.71) can be applied here as well.

For each of these phenomena in the limit of dilute solutions, some

difference in the behavior of the solvent-Apt, ATf, or ÄÒÜ-is

proportional to the mole fraction of the solute. Since the solutions are

already assumed to be dilute, we note that

under these conditions. The number of moles of solvent in a solution is

generally known from its mass and molecular weight. Therefore measuring

the colligative properties Äð*, ATf, and ÄÒÜ and using the dilute, ideal

solution approximation provides a means for evaluating n2 , the number of

moles of solute in the sample. If the solute is unknown with respect to

molecular weight but its mass in the solution is known, then the

colligative properties can be used to determine the molecular weight of

the solute: M2 = m2/n2 .

One way to describe this situation is to say that the colligative

properties provide a method for counting the number of solute molecules

in a solution. In these ideal solutions this is done without regard to

the chemical identity of the species. Therefore if the solute consists of

several different components which we index i, then n2 = 5^ï2 j is the

number of moles counted. Of course, the total mass of solute in this case

is given by m2 = 5^ï2 jM2 so the molecular weight obtained for such a

mixture is given by

(8.72)

(8.74)

which is the definition of the number average molecular weight [Eq.

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