Sauerwine
Practice Ben for Qualifying Exams
Thanks to Ben B. for this solution.
Near absolute temperature T0 , the tension F in a rubber band is given by:
F = aT 2 (L L0 )
where L is the stretched length, L0 is the unstretched length, and a is a constant.
When L = L0 , the heat capacity at constant length is:
C L = bT
(a) Find the entropy S (L, T ) for temperatures T near T0 in terms of its value at
T0 and L0 .
P F (T , L )
S S (T , L )
S
S
dS =
dT + dL
T L
L T
C
S
=L
T
T L
Maxwell :
F
F
dA =
dL = SdT PdL = SdT FdL
dT +
T L
L T
S F
0 =
L T T L
S
F
=
L T T L
Then,
F = 2aT (L L0 )
T L
C
dS = L dT + 2aT (L L0 )dL
T
Integrate,
(
S = S (T0 , L0 ) + bT + aT L2 2 LL0
)
(b) If one starts at T0 and L0 and stretches the insulated rubber band quasistatically to L, what will the temperature T be of the rubber band?
T=
CL
S
T L
T=
b
b + a L 2 LL0
(
2
)
(c) Calculate the heat capacity at constant tension C F (L, T ) .
S
CF T
T F
F = aT 2 (L L0 )
(
F 2 = a 2T 4 L2 2 LL0 + L2
0
(
)
F2
aTL2 = aT L2 2 LL0
0
aT 3
(
)
)
S = S (T0 , L0 ) + bT + aT L2 2 LL0 = S (T0 , L0 ) + bT +
4F 2
S
aL2
CF = T
= T b
0
4
aT
T F
F2
aTL2
0
3
aT