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

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n = 100 2.25 2.62 2.65 2.53 2.31 1.99 1.66 1.25 0.69

n = 500 3.63 3.87 3.77 3.49 3.11 2.66 2.11 1.52 0.85

Table 8.1 Values of ô Corresponding to the Indicated Mole Fractions for

Values of n = 50, 100, and 500 (These Quantities are Used in Example

8.1.)

x2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Xl 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

For n = 50 02 0.847 0.926 0.955 0.971 0.980 0.987 0.992 0.995 0.998

01 0.153 0.074 0.045 0.029 0.020 0.013 0.008 0.005 0.002

For n = 100 02 0.917 0.962 0.977 0.985 0.990 0.993 0.996 0.998 0.999

01 0.083 0.038 0.023 0.015 0.010 0.007 0.004 0.002 0.001

For n = 500 02 0.982 0.992 0.995 0.997 0.998 0.9987 0.9991 0.9995 0.9998

01 0.018 0.008 0.005 0.003 0.002 0.0013 0.0009 0.0005 0.0002

The Flory-Huggins Theory: The Entropy of Mixing 519

520

Thermodynamics of Polymer Solutions

Figure 8.1 The entropy of mixing (in units of R) as a function of mole

fraction solute for ideal mixing and for the Flory-Huggins lattice model

with n = 50, 100, and 500. Values are calculated in Example 8.1.

A plot of these values is shown in Fig. 8.1. Note the increase in the

entropy of mixing over the ideal value with increasing n value. Also note

that the maximum occurs at decreasing mole fractions of polymer with

increasing degree of polymerization.

•

Our ultimate goal is to develop expressions for the free energy of

mixing according to the lattice model of the Flory-Huggins theory. The

next step toward realization of this goal is to relax the restriction

which limits the result to athermal solutions. In the next section we

shall develop the theory for the enthalpy of mixing. Then we combine the

equations for entropy and enthalpy to obtain an expression for AGm

according to the Flory-Huggins model.

The Flory-Huggins Theory: The Enthalpy of Mixing

521

8.4 The Flory-Huggins Theory: The Enthalpy of Mixing

In the liquid state molecules are in intimate contact, so the energetics

of molecular interactions generally make a contribution to the overall

picture of the mixing process. There are several aspects of the situation

that we should be aware of before attempting to formulate a theory for

ÄÍò :

1. Our immediate goal is an expression for AHm and we must

remember that

this is the difference in the enthalpies of the solution and the

pure components. We need not worry about the absolute values of

the enthalpies of

the individual states.

2. Enthalpies of mixing have their origin in the forces that operate

between individual molecules. Intermolecular forces drop off rapidly with

increasing distance of separation between molecules. This means that only

nearest neighbors need be considered in the model.

3. Until surface contact, the force between molecules is always one of

attraction, although this attraction has different origins in different

systems. London forces, dipole-dipole attractions, acid-base

interactions, and hydrogen bonds are some of the types of attraction we

have in mind. In the foregoing list, London forces are universal and also

the weakest of the attractions listed. The interactions increase in

strength and also in specificity in the order listed.

4. Since London forces are universal and nonspecific, we shall

eventually emphasize these, realizing that stronger forces may outweigh

the London contribution in some systems. We anticipate the best results

in nonpolar systems where London forces account for the interactions.

The lattice model that served as the basis for calculating ASm in

the last section continues to characterize the Flory-Huggins theory in

the development of an expression for AHm. Specifically, we are concerned

with the change in enthalpy which occurs when one species is replaced by

another in adjacent lattice sites. The situation can be represented in

the notation of a chemical reaction:

(1,1) + (2,2) -*2(1,2)

(8.A)

where 1 and 2 refer to solvent molecules and polymer repeat units,

respectively, and the parentheses enclose the particular pair under

consideration. Suppose we indicate the pairwise interaction energy

between species i and j as w^; then for reaction (8.A),

Aw = 2w12 - wu - w22

(8.39)

522

Thermodynamics of Polymer Solutions

The change in interaction energy per 1,2 pair is thus Vl Aw. Next we

must consider how this scales up for a large array of molecules, and

particularly how to describe the concentration dependence of the result.

Each lattice site is defined to have z nearest neighbors, and ôi and

ô2, respectively, can be used to describe the fraction of sites which are

occupied by solvent molecules and polymer segments. The following

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