Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.34.
We can model a rubber band as a one-dimensional chain of polymer links each of length , where each link can point either to the left or right. The total length of the rubber band is therefore
where is the number of links pointing to the right or left, so that . We’ll assume that , although the converse is just the mirror image and gives the same behaviour.
The entropy can be found by noting that the system is equivalent to a coin-flipping experiment, so for a given , the multiplicity is
Using Stirling’s approximation, the entropy is
Since this is a one-dimensional system, the role of the pressure in a 3-d system is taken here by the tension force generated by stretching the rubber band. If when the band is pulling inward, then work is done on the band when it is stretched through a distance .
The thermodynamic identity for this system is therefore
If the band is stretched in such a way that its energy remains constant (e.g. by losing heat), then and the force is given by
so from 3
If , the band is almost fully contracted since its length is much less than the maximum length. In this case we can get an estimate of the force:
That is, which is Hooke’s law for a spring, with spring constant .
The formula 12 for says that should increase as the temperature is increased, that is, a rubber band should contract when heated (up to a point, obviously; after a while it will just melt). Although this relation was derived in the special case of constant energy , it does seem to make sense. A higher temperature would increase the entropy, meaning that gets closer to its maximum entropy value of , which means that gets smaller. By the same token, we’d expect a rubber band to get warmer if it is suddenly stretched. I did try the experiment suggested, although I couldn’t find a heavy rubber band, and it did seem to warm up a bit when it was stretched.