(a) A macroscopic system has volume $V$ and contains $N$ particles. Let $\Omega(E, V, N ; \delta E)$ denote the number of states of the system which have energy in the range $(E, E+\delta E)$ where $\delta E \ll E$ represents experimental uncertainty. Define the entropy $S$ of the system and explain why the dependence of $S$ on $\delta E$ is usually negligible. Define the temperature and pressure of the system and hence obtain the fundamental thermodynamic relation.

(b) A one-dimensional model of rubber consists of a chain of $N$ links, each of length a. The chain lies along the $x$-axis with one end fixed at $x=0$ and the other at $x=L$ where $L<N a$. The chain can "fold back" on itself so $x$ may not increase monotonically along the chain. Let $N_{\rightarrow}$ and $N_{\leftarrow}$ denote the number of links along which $x$ increases and decreases, respectively. All links have the same energy.

(i) Show that $N_{\rightarrow}$ and $N_{\leftarrow}$ are uniquely determined by $L$ and $N$. Determine $\Omega(L, N)$, the number of different arrangements of the chain, as a function of $N_{\rightarrow}$ and $N_{\leftarrow}$. Hence show that, if $N_{\rightarrow} \gg 1$ and $N_{\leftarrow} \gg 1$ then the entropy of the chain is

$$\begin{aligned} S(L, N)=k N & {\left[\log 2-\frac{1}{2}\left(1+\frac{L}{N a}\right) \log \left(1+\frac{L}{N a}\right)\right.} \\ &\left.-\frac{1}{2}\left(1-\frac{L}{N a}\right) \log \left(1-\frac{L}{N a}\right)\right] \end{aligned}$$

where $k$ is Boltzmann's constant. [You may use Stirling's approximation: $n$ ! $\approx$ $\sqrt{2 \pi} n^{n+1 / 2} e^{-n}$ for $\left.n \gg 1 .\right]$

(ii) Let $f$ denote the force required to hold the end of the chain fixed at $x=L$. This force does work $f d L$ on the chain if the length is increased by $d L$. Write down the fundamental thermodynamic relation for this system and hence calculate $f$ as a function of $L$ and the temperature $T$.

Assume that $N a \gg L$. Show that the chain satisfies Hooke's law $f \propto L$. What happens if $f$ is held constant and $T$ is increased?