(a) Starting from the canonical ensemble, derive the Maxwell-Boltzmann distribution for the velocities of particles in a classical gas of atoms of mass $m$. Derive also the distribution of speeds $v$ of the particles. Calculate the most probable speed.

(b) A certain atom emits photons of frequency $\omega_{0}$. A gas of these atoms is contained in a box. A small hole is cut in a wall of the box so that photons can escape in the positive $x$-direction where they are received by a detector. The frequency of the photons received is Doppler shifted according to the formula

$$\omega=\omega_{0}\left(1+\frac{v_{x}}{c}\right)$$

where $v_{x}$ is the $x$-component of the velocity of the atom that emits the photon and $c$ is the speed of light. Let $T$ be the temperature of the gas.

(i) Calculate the mean value $\langle\omega\rangle$ of $\omega$.

(ii) Calculate the standard deviation $\sqrt{\left\langle(\omega-\langle\omega\rangle)^{2}\right\rangle}$.

(iii) Show that the relative number of photons received with frequency between $\omega$ and $\omega+d \omega$ is $I(\omega) d \omega$ where

$$I(\omega) \propto \exp \left(-a\left(\omega-\omega_{0}\right)^{2}\right)$$

for some coefficient $a$ to be determined. Hence explain how observations of the radiation emitted by the gas can be used to measure its temperature.