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:
(a) the internal energy $U$ is a function of temperature only;
So we have the first law:
$$ dU=T dS+fdL=dQ+dW $$
The work done on the elastic band is $0$ as it is held at a fixed length. $$ \implies dU=dQ=dT/C $$
where $C$ is heat capacity. By integrating we show that $U=T/C+A$ where $A$ is an arbitrary constant of integration.
However I feel my method implies that $TdS=dQ$ as both the length and work are held constant. However I don't think heating up an elastic band is reversible so this cannot be true, which would imply work is done.