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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
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);
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,
Pi = *iPi° = (1 " x2)Pi' or
Pi° - Pi = Ap = X2P!°
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
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
where the last approximation requires dilute conditions as well.
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
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
which is the definition of the number average molecular weight [Eq.