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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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(8.8)
:
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 -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 :
\ ln aj
-4-) = RT -
(8-17)
/,1,2
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
;
3 ln a= = ----- = ln {
(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 and 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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