Is the Internal Energy of a Rubber Band Dependent on Temperature?

[XCBRA]

Homework Statement

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

(a) the internal Energy U is a function of temperature;

(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 Attempt at a Solution

Start with the fundamental equation for a thermoelastic system

du = T ds - f dl.

Then I am stuck as to how to continue from here.

I have tried tp then take the total differential of U:

$$du = \frac{\partial U}{\partial S}_Lds +\frac{\partial U}{\partial L}_SdL$$

but that doesn't seem to help. I think I need to use a maxwell relation but I unable to figure out a suitable relationship to do the firs part. Any help will be greatly appreciated.

 

[rude man]

Homework Statement

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

(a) the internal Energy U is a function of temperature;

(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 Attempt at a Solution

Start with the fundamental equation for a thermoelastic system

du = T ds - f dl.

Then I am stuck as to how to continue from here.

I have tried tp then take the total differential of U:

$$du = \frac{\partial U}{\partial S}_Lds +\frac{\partial U}{\partial L}_SdL$$

but that doesn't seem to help. I think I need to use a maxwell relation but I unable to figure out a suitable relationship to do the firs part. Any help will be greatly appreciated.

(a)
Start with your equation du = T ds - f dl.
What is dl here? Therefore, what is dW?
OK, so then can you rewrite the first law in terms of U and Q, where dQ = Cl*dt?

(b)
First law again! dQ = 0, so how is U affected when W is added to the system?
And what did part (a) say?

(c)
Go back to you 1st equation, now df = 0. How is W, and therefore l, affected?