An atomic species emits light with a characteristic frequency $f_{0}$ while at a rest state. A diatomic gas of such atomic species has been heated to temperature $T^{\prime}$. Due to their random thermal motions, some atoms move toward or against an onlooker. Assuming that root mean square speed characterizes their thermal motions:

(i) Determine the expression for the fractional variation $\frac{\Delta f}{f_{0}}$ between Doppler shifted frequencies corresponding to those moving straightly toward or against the onlooker, Assume $m c^{2} \gg 3 k T^{\prime}$.

(ii) Compute $\Delta f / f_{0}$ for a gas of oxygen atoms at 300 K.