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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
Download (direct link): polymerchemistry1984.djvu
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makes no difference, we divide by the number of ways of doing the latter-
Nt! and N2 !, respectively-to obtain the total number of different ways
the system can come about. This is called the thermodynamic probabilty SI
of the system, and we saw in Sec. 3.3 that SI is the basis for the
statistical calculation of entropy. For this specific model
512
Thermodynamics of Polymer Solutions
Nt !N2!
(8.23)
The thermodynamic probability is converted to an entropy through the
Boltzmann equation [Eq. (3.20)] so we can write for the entropy of the
mixture (subscript mix)
Multiplying and dividing the right-hand side of this expression by N
converts the Nj's to mole fractions, and, if N is taken to be NA,
Avogadro's number, kNA becomes R. Accordingly, we write for 1 mol of
mixture
If either one of the mole fractions in Eq. (8.27) is unity, that is, for
a pure component, Smix becomes either St or S2. For the mixing process 1
+ 2 mix,
Although the right-hand sides of Eqs. (8.27) and (8.28) are the same, the
former applies to the mixture (subscript mix), while the latter applies
to the mixing process (subscript m). The fact that these are identical
emphasizes that in Eq. (8.27) we have calculated only that part of the
total entropy of the mixture which arises from the mixing process itself.
This is called the configurational entropy and is our only concern in
mixing problems. The possibility that this mixing may involve other
entropy effects-such as an entropy of solvation-is postponed until Sec.
8.12.
The model system for which this value of ASm has been calculated is
one for which AHm has been specified to equal zero. Therefore, since AG =
- T AS, it follows that
Smix = In = (In N! - In Nt! - In N2 !)
(8.24)
For large y's, we can use Sterling's approximation
In ! = In -
(8.25)
in terms of which Eq. (8.24) becomes
= -k

(8.26)
Smix = _R(xi Inxi +x2 h x2)
(8.27)
- St - S2 = -R 2 (xj In Xj)
(8.28)
i=l, 2
AGm = RT 2 (xjlnxj) i=l, 2
(8.29)
The Flory-Huggins Theory: The Entropy of Mixing
513
for this system. We can also use Eqs. (8.7), (8.10), and (8.13) to write
AGm in terms of chemical potentials:
AGm = Gmi - - G2 = RT 2 (Xj In aj)
(8.30)
i=l, 2
Comparing Eqs. (8.29) and (8.30) also leads to the conclusion expressed
by Eq. (8.22): aj = Xj. Again we emphasize that this result applies only
to ideal solutions, but the statistical approach gives us additional
insights into the molecular properties associated with ideality in
solutions:
1. The energy of interaction between a pair of solvent molecules, a
pair of solute molecules, and a solvent-solute pair must be the same so
that the criterion that AHm = 0 is met. Such a mixing process is said to
be athermal.
2. The solvent and solute molecules must be the same size so that the
criterion AVm = 0 is met.
3. Solutions can deviate from ideality because they fail to meet either
one or both of these criteria. In reference to polymers in solutions of
low molecular weight solvents, it is apparent that nonideality is present
because of a failure to meet criterion (2), whether the mixing is
athermal or not.
In the next section we shall examine the mixing process for molecules
which differ greatly in size, building on the principles reviewed in this
section. The reader who desires additional review of these ideas will
find this material discussed in detail in textbooks of physical
chemistry.
8.3 The Flory-Huggins Theory: The Entropy of Mixing
We concluded the last section with the observation that a polymer
solution is expected to be nonideal on the grounds of entropy
considerations alone. A nonzero value for AHm would exacerbate the
situation even further. We therefore begin our discussion of this problem
by assuming a polymer-solvent system which shows athermal mixing. In the
next section we shall extend the theory to include systems for which AHm
0. The theory we shall examine in the next few sections was developed
independently by Flory and Huggins and is known as the Flory-Huggins
theory.
We assume that the mixture contains Nt solvent molecules, each of
which occupies a single site in the lattice we propose to fill. The
system also contains N2 polymer molecules, each of which occupies n
lattice sites. The polymer molecule is thus defined to occupy a volume n
times larger than the solvent molecules. Strictly speaking, this is the
definition of n in the derivation which follows. We shall adopt the
attitude that the repeat units in the polymer are equal to solvent
molecules in volume, however, so a polymer of degree of
514
Thermodynamics of Polymer Solutions
polymerization n will be larger than a solvent molecule by the factor n.
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