The elasticity of a rubber band can be described in terms of a onedimensional model of polymer involving $N$ molecules linked together endto-end. The angle between successive links is equally likely to be $0^{\circ}$ or $180^{\circ}$.
(a) Show that the number of arrangements that give an overall length of $L=2 m d$ is given by
$$
g(N, m)=\frac{2 N!}{\left(\frac{N}{2}+m\right)!\left(\frac{N}{2}-m\right)!},
$$
where $m$ is positive .
Indicate clearly the reasoning you used to get this result.
(b) For $m \ll N$, this expression becomes
$$
g(N, m) \approx g(N, 0) \exp \left(-2 m^2 / N\right) .
$$
Find the entropy of the system as a function of $L$ for $N \gg 1, L \ll N d$.
(c) Find the force required to maintain the length $L$ for $L \ll N d$.
(d) Find the relationship between the force and the length, without using the condition in (c), i.e., for any possible value of $L$, but $N \gg 1$.
FIGURE CAN'T COPY.