Physical Chemistry Third Edition

1202 28 The Structure of Solids, Liquids, and Polymers

The Helmholtz energy is defined in the standard way:

AU−TS (definition) (28.9-4)

We denote the analogue to the Gibbs energy byJ:

JK−TSU−fL−TS (definition) (28.9-5)

Just as in Section 26.1 we can write differential expressions:

dKTdS−Ldf (28.9-6a)

dA−SdT+fdL (28.9-6b)

dJ−SdT−Ldf (28.9-6c)

We can write Maxwell relations similar to the relation in Eq. (4.2-18) from these

equations. For example, from Eq. (28.9-6b),

(

∂S

∂L

)

T

−

(

∂f

∂T

)

L

(28.9-7)

Exercise 28.15

Write the other three Maxwell relations from Eqs. (28.9-2), (28.9-6a), and (28.9-6c).

Using Eqs. (28.9-2) and (28.9-7), we can derive a useful equation:

(

∂U

∂L

)

T

T

(

∂S

∂L

)

T

+fT

(

∂f

∂T

)

L

+f (28.9-8)

This is anequation of statefor an ideal rubber. We can now show that property (3)

of an ideal rubber follows from property (2). Sincefis proportional toTfor an ideal

rubber,

fTφ(L) (28.9-9)

whereφis some function ofLthat is independent ofT. We now have

T

(

∂f

∂T

)

L

Tφf (28.9-10)

so that

(

∂U

∂L

)

T

−f+f 0 (28.9-11)

Equation (28.9-11) shows the difference between a rubber band and a spring. When

a spring is stretched at constant temperature, the energy increases as work is done on

the spring. When a rubber band is stretched at constant temperature, doing work on the

rubber band, the energy remains constant, so that heat must flow out. Stretching a rubber

band at constant temperature must decrease its entropy. This fact seems reasonable

from a molecular point of view, because the polymer molecules will be more nearly

parallel and more nearly ordered in the stretched state than in the relaxed state. From

Eqs. (28.9-8) and (28.9-11) we can derive a relation for this decrease in entropy:

f−T

(

∂S

∂L

)

T

(28.9-12)