Adiabatic stretching of a rubber band

[Toby_phys]

Homework Statement

For a stretched rubber band, it is observed experimentally that the tension $f$ is proportional to the temperature $T$ if the length $L$ is held constant. Prove that:

(b) adiabatic stretching of the band results in an increase in temperature;
(c) the band will contract if warmed while kept under constant tension.

Homework Equations

the first law:
$$
dU=Tds+fdL=C_L dT
$$
$$
f=\left (\frac{\partial f}{\partial T}\right )_L T
$$
$$
\left (\frac{\partial L}{\partial f}\right )_T>0
$$

The Attempt at a Solution

(b)[/B]

For an adiabatic process, entropy doesn't increase and so:
$$
dU=fdL=C_LdT
$$

The force is always positive and so temperature is positively increased by length.

This feels too simple so i doubt I am correct. I have no idea for part (c).

 

[Chestermiller]

How do you know that dU=CdT?

 

[Toby_phys]

There was a part A that was to show this is the case
$$dU=C_vdT+\left[f-T\left(\frac{\partial f}{\partial T}\right)_L\right]dL$$

The second term drops out

 

[Chestermiller]

Homework Statement

For a stretched rubber band, it is observed experimentally that the tension $f$ is proportional to the temperature $T$ if the length $L$ is held constant. Prove that:

(b) adiabatic stretching of the band results in an increase in temperature;
(c) the band will contract if warmed while kept under constant tension.

Homework Equations

the first law:
$$
dU=Tds+fdL=C_L dT
$$
$$
f=\left (\frac{\partial f}{\partial T}\right )_L T
$$
$$
\left (\frac{\partial L}{\partial f}\right )_T>0
$$

The Attempt at a Solution

(b)[/B]

For an adiabatic process, entropy doesn't increase and so:
$$
dU=fdL=C_LdT
$$

The force is always positive and so temperature is positively increased by length.

This feels too simple so i doubt I am correct. I have no idea for part (c).

In part (a) it was shown that f = Tg(L), where g is an increasing function of L. This is the equation of state of the rubber. So if T increases at constant f, what happens to L?