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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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model, and models are bound to oversimplify. Nevertheless, we can learn a
great deal from the attempt to evaluate thermodynamic properties from
molecular models, even if the effort falls short of quantitative success.
There is probably no area of science that is as rich in mathematical
relationships as thermodynamics. This makes thermodynamics very powerful,
but such an abundance of riches can also be intimidating to the beginner.
This chapter assumes that the reader is familiar with basic chemical and
statistical thermodynamics at the level that these topics are treated in
physical chemistry textbooks. In spite of this premise, a brief review of
some pertinent relationships will be a useful way to get started.
Notation frequently poses problems in science, and this chapter is an
example of such a situation. Our problem at present is that we have too
many things to count: They cannot all be designated n. We have
consistently used n to designate the degree of polymerization and shall
continue with this notation. In thermodynamics n is widely used to
designate the number of moles. Since we deal with (at least) two-
component systems in this chapter, any count of the number of moles will
always carry a subscript to indicate the component under consideration.
We shall use the subscript 1 to designate the solvent, and 2 to designate
the solute. The degree of polymerization is represented by n without a
subscript.
To describe the state of a two-component system at equilibrium, we
must specify the number of moles nx and n2 of each component, as well as-
ordinarily- the pressure p and the absolute temperature T. It is the
Gibbs free energy that provides the most familiar access to a discussion
of equilibrium. The increment in G associated with increments in the
independent variables mentioned above is given by the equation
dG = V dp - S dT + X jU| dnj
(8.1)
i=l, 2
where is the chemical potential of component i. An important aspect of
thermodynamics is the fact that the state variables (in the present
context, this applies especially to the internal energy U, the enthalpy
H, and the Gibbs
508
Thermodynamics of Polymer Solutions
free energy G) can be expanded as partial derivatives of fundamental
variables. Hence we can also write
The chemical potential is an example of a partial molar quantity:
is the
partial molar Gibbs free energy with respect to component i. Other
partial molar quantities exist and share the following features:
1. We may define, say, partial molar volume, enthalpy, or entropy by
analogy with Eq. (8.5):
where Y = V, H, or S, respectively. JExcept for the partial molar
Gibbs free energy, we shall use the notation Y. to signify a partial
molar quantity, where Y stands for the symbol of the appropriate
variable.
2. Partial molar quantities have "per mole" units, and for Yj this is
understood to mean "per mole of component i." The value of this
coefficient depends on the overall composition of the mixture. Thus VH2q
is not the same for a water-alcohol mixture that is 10% water as for one
that is 90% water.
T, nj ,n2
P> nl > n2
dT +
P,T,n2
dn
+
3G \
2 /..!
dn2
(8.2)
Comparing Eqs. (8.1) and (8.2) gives
(8.3)
S = -
P>n 1 > n2
(8.4)
and

(8.5)
P.T-nj#i
(8.6)
Classical and Statistical Thermodynamics
509
3. For a pure component the partial molar quantity is identical to the
molar (superscript ) value of the pure substance. Thus for pure
component i
4. A useful feature of the partial molar properties is that the
property of a mixture (subscript mix) can be written as the sum of the
mole-weighted contributions of the partial molar properties of the
components:
In this expression nt and n2 are the numbers of moles of components
1 and 2 in the mixture under consideration.
5. To express the value of property Ymix on a per mole basis, it is
necessary to divide Eq. (8.8) by the total number of moles, nt + n2. The
mole fraction Xj of component i is written
6. Relationships which exist between ordinary thermodynamic variables
also apply to the corresponding partial molar quantities. Two such
relation-
As noted above, all of the partial molar quantities are concentration
dependent. It is convenient to define a thermodynamic concentration
called the activity aj in terms of which the chemical potential is
correctly given by the relationship
The quantity is called the standard state (superscript 0) value of //;
it is the value of /ij when aj = 1. Neither nor G (nor U, H, etc.) can be
measured

(8.7)

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