Through a small window in a furnace, which contains a gas at a high temperature \(T\), the spectral lines emitted by the gas molecules are observed. Because of molecular motions, each spectral line exhibits Doppler broadening. Show that the variation of the relative intensity \(I(\lambda)\) with wavelength \(\lambda\) in a line is given by

\[
I(\lambda) \propto \exp \left\{-\frac{m c^{2}\left(\lambda-\lambda_{0}ight)^{2}}{2 \lambda_{0}^{2} k T}ight\}
\]

where \(m\) is the molecular mass, \(c\) the speed of light, and \(\lambda_{0}\) the mean wavelength of the line.