When an electric discharge is passed through gaseous hydrogen, the $\mathrm{H}_2$ molecules are dissociated and energetically excited $\mathrm{H}$ atoms are produced. If the electron in an excited $\mathrm{H}$ atom makes a transition from $n=2$ to $n=1$, calculate the wavenumber of the corresponding line in the emission spectrum.
The wavenumber of the photon emitted when an electron makes a transition from $n_2=2$ to $n_1=1$ is given by
$$
\begin{aligned}
\tilde{\boldsymbol{v}} & =-\tilde{R}_{\mathrm{H}}\left(\frac{1}{n_2^2}-\frac{1}{n_1^2}\right) \\
& =-\left(109677 \mathrm{~cm}^{-1}\right) \times\left(\frac{1}{2^2}-\frac{1}{1^2}\right) \\
& =82258 \mathrm{~cm}^{-1}
\end{aligned}
$$