(i) Write down the first law of thermodynamics for the change $d U$ in the internal energy $U(N, V, S)$ of a gas of $N$ particles in a volume $V$ with entropy $S$.

Given that

$$P V=(\gamma-1) U,$$

where $P$ is the pressure, use the first law to show that $P V^{\gamma}$ is constant at constant $N$ and

Write down the Boyle-Charles law for a non-relativistic ideal gas and hence deduce that the temperature $T$ is proportional to $V^{1-\gamma}$ at constant $N$ and $S$.

State the principle of equipartition of energy and use it to deduce that

$$U=\frac{3}{2} N k T$$

Hence deduce the value of $\gamma$. Show that this value of $\gamma$ is such that the ratio $E_{i} / k T$ is unchanged by a change of volume at constant $N$ and $S$, where $E_{i}$ is the energy of the $i$-th one particle eigenstate of a non-relativistic ideal gas.

(ii) A classical gas of non-relativistic particles of mass $m$ at absolute temperature $T$ and number density $n$ has a chemical potential

$$\mu=m c^{2}-k T \ln \left(\frac{g_{s}}{n}\left(\frac{m k T}{2 \pi \hbar^{2}}\right)^{\frac{3}{2}}\right)$$

where $g_{s}$ is the particle's spin degeneracy factor. What condition on $n$ is needed for the validity of this formula and why?

Thermal and chemical equilibrium between two species of non-relativistic particles $a$ and $b$ is maintained by the reaction

$$a+\alpha \leftrightarrow b+\beta$$

where $\alpha$ and $\beta$ are massless particles with zero chemical potential. Given that particles $a$ and $b$ have masses $m_{a}$ and $m_{b}$ respectively, but equal spin degeneracy factors, find the number density ratio $n_{a} / n_{b}$ as a function of $m_{a}, m_{b}$ and $T$. Given that $m_{a}>m_{b}$ but $m_{a}-m_{b} \ll m_{b}$ show that

$$\frac{n_{a}}{n_{b}} \approx f\left(\frac{\left(m_{a}-m_{b}\right) c^{2}}{k T}\right)$$

for some function $f$ which you should determine.

Explain how a reaction of the above type is relevant to a determination of the neutron to proton ratio in the early universe and why this ratio does not fall rapidly to zero as the universe cools. Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Let

$$Y_{H e}=\frac{\rho_{H e}}{\rho}$$

be the fraction of the universe that ends up in helium. Compute $Y_{H e}$ as a function of the ratio $r=n_{a} / n_{b}$ at the time of nucleosynthesis.