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

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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 \

Ýï2 /ð.Ò.Ï!

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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