Describe briefly why a low density gas can be investigated using classical statistical mechanics.

Explain why, for a gas of $N$ structureless atoms, the measure on phase space is

$$\frac{1}{N !} \frac{d^{3 N} q d^{3 N} p}{(2 \pi \hbar)^{3 N}}$$

and the probability density in phase space is proportional to

$$e^{-E(q, p) / T}$$

where $T$ is the temperature in energy units.

Derive the Maxwell probability distribution for atomic speeds $v$,

$$\rho(v)=\left(\frac{m}{2 \pi T}\right)^{3 / 2} 4 \pi v^{2} e^{-m v^{2} / 2 T}$$

Why is this valid even if the atoms interact?

Find the mean value $\bar{v}$ of the speed of the atoms.

Is $\frac{1}{2} m(\bar{v})^{2}$ the mean kinetic energy of the atoms?