Chemistry Reference
In-Depth Information
ð@A=@TÞ
t
52S
(4-19)
Since
t
5
@
@
@
@
A
@
@A
@
(4-20)
l
T
@
T
l
T
we can substitute
Eqs. (4-14) and (4-19)
into
Eq. (4-20)
to obtain
ð@S=@lÞ
T
52ð@f =@TÞ
t
(4-21)
T
)
T
, which occurs in
Eq. (4-15)
, in terms of the temperature coefficient of tension at constant length
(
This gives the entropy change per unit extension, (
@
S/
@
@
f/
@
T
)
l
, which can be measured. With
Eq. (4-21)
,
Eq. (4-15)
becomes
ð@
U
=@
l
Þ
T
5
f
2
T
ð@
f
=@
T
Þ
l
(4-22)
where (
l
)
T
is the internal energy contribution to the total force. [Compare
Eq. (iv)
in
Table 4.3
for a gas.]
Figure 4.13
shows how experimental data can be used with
Eqs. (4-21) and
(4-22)
to determine the internal energy and entropy changes accompanying defor-
mation of an elastomer. Such experiments are simple in principle but difficult in
practice because it is hard to obtain equilibrium values of stress.
For an ideal elastomer (
@
U/
@
@
U/
@
l
)
T
is zero and
Eq. (4-22)
reduces to
f
5
T
ð@
f
=@
T
Þ
l
(4-23)
in complete analogy to
Eq. (vii)
of
Table 4.3
for an ideal gas. In real elastomers,
chain uncoiling must involve the surmounting of bond rotational energy barriers
T
1
Absolute temperature,
T
FIGURE 4.13
Experimental measurement of (
@f/@l)
T
and (
@U/@l)
T
. The slope of the tangent to the curve
at temperature T
1
5
@ S/@l)
T
, which equals entropy change per
unit extension when the elastomer is extended isothermally at T
1
. The intercept on the
force axis equals (
(
@ f/@T)
t
at T
1
. This equals (
@U/@l)
T
since this corresponds to T5
0in
Eq. (4-22)
. The intercept is
the internal energy change per unit extension.
