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

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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

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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 =

ÄÍ - T AS, it follows that

Smix = ê In Ï = ê (In N! - In Nt! - In N2 !)

(8.24)

For large y's, we can use Sterling's approximation

In ó! = ó In ó - ó

(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

Ô 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

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Thermodynamics of Polymer Solutions

polymerization n will be larger than a solvent molecule by the factor n.

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