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

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(8.8)

Ï:

X;

(8.9)

therefore

Y •

mix

= XlYt + x2 Y2

(8.10)

ships are

(8.11)

and

(8.12)

Mi = Mi6 + RT In aj

(8.13)

510

Thermodynamics of Polymer Solutions

absolutely; we deal with differences in these quantities and the standard

state value disappears when differences are taken. Although the standard

state is defined differently in various situations, we shall generally

take the pure component (superscript °) as the standard state,

so/i16=/i10 = G1°. Equation (8.13) defines activity in terms of the

difference ju. - \ an obvious question is how

aj relates to the concentration of the solution as prepared or as

determined by analytical chemistry.

There are two ways to arrive at the relationship between aj and the

concentration expressed as, say, a mole fraction. One is purely

thermodynamic and involves experimental observations; the other involves

a model and is based on a statistical approach. We shall examine both.

The first point in developing the thermodynamic method is the

observation that for equilbrium between two phases-say, a and Ä-the

chemical potential must be equal in both phases for all components:

= rf

(8-14)

This becomes apparent if we consider the increment in G associated with

transferring a small number of moles of component i from phase a to phase

/3 at constant pressure and temperature. For equilibrium, dG = 0 = dGa +

dG^, and for each phase dG = 2^ dn^ Since dn^ = -dn^, it follows that

dG = 0 = dGQ + dG0 = 2 Mia dnjQ + 2 /if dnfi = 2 (MiQ - rf) dn*

i i i

(8.15)

from which Eq. (8.14) is obtained.

Next we apply this result to liquid-vapor equilibrium. The

following steps outline the argument:

1. Equation (8.14) applies to both components in the liquid and the

vapor: = Miv

(8.16)

The equality also holds if we take the partial derivative of both

sides of Eq. (8.16) with respect to p.

2. We use Eq. (8.13) to take the partial with respect to p of jUj1 :

ä \ Ý ln aj

-4-) = RT -

(8-17)

ÝÐ /ò,ï1,ï2 ÝÐ

3. We use Eq. (8.12) to evaluate the partial with respect to p of :

Classical and Statistical Thermodynamics

511

where the second version treats the vapor as an ideal gas, an

assumption we can make without loss of generality concerning the

solution.

4. Equations (8.16)-(8.18) can be combined to give

Ýð;

3 ln a= = ----- = Ý ln ð{

(8.19)

Pi

5. Equation (8.19) can be integrated using the convention that = 1 for

the pure component, which has the vapor pressure p.°:

a, = 4

<8-2°)

Pi

6. Finally, we note that Raoult's law (see Example 7.1) is a limiting

law that is observed to apply in all solutions in the limit Xj -* 1:

4 =

(8-2i>

Pi

7. Thus it is always observed in the limit of the pure component that

xj = aj

(8.22)

A solution which obeys Raoult's law over the full range of

compositions is called an ideal solution (see Example 7.1). Equation

(8.22) describes the relationship between activity and mole fraction for

ideal solutions. In the case of nonideal solutions, the nonideality may

be taken into account by introducing an activity coefficient as a factor

of proportionality into Eq. (8.22).

A second way of dealing with the relationship between aj and the

experimental concentration requires the use of a statistical model. We

assume that the system consists of Nt molecules of type 1 and N2

molecules of type 2. In addition, it is assumed that the molecules, while

distinguishable, are identical to one another in size and interaction

energy. That is, we can replace a molecule of type 1 in the mixture by

one of type 2 and both ÄÓ and ÄÍ are zero for the process. Now we

consider the placement of these molecules in the Nt + N2 = N sites of a

three-dimensional lattice. The total number of arrangements of the N

molecules is given by N!, but since interchanging any of the l's or 2's

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