$($a$)$ Using the Helmholtz free energy A$,$ derive the well$-$known "equation of state" for rubber elasticity $($f is the elastic force, E is internal energy, S is entropy and L is length$).$

$($b$)$ An ideal rubber exhibits purely entropic elasticity. Describe an experiment you could do to confirm the entropic nature of rubber elasticity. Show how you would plot the data to check the entropic and elastic nature of the rubber, and indicate the kind of curve you would expect for an ideal rubber. Explain how the curve can change if you use a real rubber.