Mechanical Properties of Polymer Solids and Liquids - Polymer Science and Engineering - page 179

(This is a useful thermodynamic quantity to characterize changes at constant
volume of the working substance.) For a change at constant temperature, from
Eq. (4-26) ,

dA 5 dU 2 TdS (4-10)

Combining Eqs. (4-25) and (4-26)
dA 5 dw (4-11)

That is, the change in A in an isothermal process equals the work done on the sys-
tem by its surroundings. Conventionally, when gases and liquids are of major
interest the work done on the system is written dw PdV . When we consider

elastic solids, the work done by the stress is important. If tensile force is f and l is
the initial length of the elastic specimen in the direction of the force, the work
done in creating an elongation dl is

52

dw5fdl (4-12)

If a hydrostatic pressure P is acting in addition to the tensile force f , the total
work on the system is

dw 5 fdl 2 PdV (4-13)

In the case of rubbers, dV is very small and if P 5 1 atm, PdV is less than

10 2 3 fdl . Thus we can neglect PdV and use Eq. (4-12) .
From Eqs. (4-11) and (4-12) ,

ð@ A =@ l Þ T 5 ð@ w =@ l Þ T 5 f (4-14)

That is, the tension is equal to the change in Helmoholtz free energy per unit
extension. From Eqs. (4-10) and (4-12) ,

ð@ A =@ l Þ T 5 ð@ U =@ l Þ T 2 ð@ S =@ l Þ T 5 f (4-15)

Thus, the force consists of an internal energy component and an entropy com-
ponent [compare (iii) of Table 4.1 for the pressure of a gas].
To evaluate Eq. (4-15) experimentally, we proceed in an analogous fashion to
the method used to estimate the entropy component of the pressure of a gas
( Table 4.3 ). From Eq. (4-9) , for any change,
dA 5 dU 2 TdS 2 SdT (4-16)

For a reversible change, from Eqs. (4-8) and (4-12) ,

dU 5fdl1TdS (4-17)

Combining the last two equations,
dA 5 fdl 2 SdT (4-18)

Thus, by partial differentiation,

ð@ A =@ l Þ T 5 f