Ben Sauerwine
Practice for Qualifying Exams
Thanks to Ben B. for this solution.
Near absolute temperature
, the tension F in a rubber band is given by:
0
T
()
0
2
L
L
aT
F
−
=
where L is the stretched length,
is the unstretched length, and a is a constant.
When
, the heat capacity at constant length is:
0
L
0
L
L
=
bT
C
L
=
(a)
Find the entropy
for temperatures T near
in terms of its value at
and
.
(
T
L
S
,
)
0
T
0
T
0
L
L
T
L
T
T
L
L
L
T
L
T
F
L
S
T
F
L
S
FdL
SdT
PdL
SdT
dL
L
F
dT
T
F
dA
Maxwell
T
C
T
S
dL
L
S
dT
T
S
dS
L
T
S
S
L
T
F
P
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
=
−
−
=
−
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
≈
≈
0
:
,
,
Then,
() ()
0
2
0
0
0
0
2
,
,
2
2
LL
L
aT
bT
L
T
S
S
Integrate
dL
L
L
aT
dT
T
C
dS
L
L
aT
T
F
L
L
−
+
+
=
−
+
=
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
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If one starts at
and
and stretches the insulated rubber band quasi
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 Spring '12
 Dr.Jaouni
 Thermodynamics, Entropy, ∂t, rubber band, Ben Sauerwine

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