# Adiabatic stretching of a rubber band

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1. Aug 10, 2017

### Toby_phys

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 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.

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

3. The attempt at a solution

(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).

2. Aug 10, 2017

### Staff: Mentor

How do you know that dU=CdT?

3. Aug 10, 2017

### 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

4. Aug 10, 2017

### Staff: Mentor

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?