# Thermodynamics of a rubber band

1. Dec 23, 2011

### XCBRA

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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.

2. Dec 24, 2011

### rude man

(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?