12.5 Line Spectra of Hydrogen Atom

According to the third postulate of Bohr’s model, when an atom makes a transition from the higher energy state with quantum number \(n_{i}\) to the lower energy state with quantum number \(n_{f}\) (\(n_{f}\) < \(n_{i}\) ), the difference of energy is carried away by a photon of frequency \( ν_{if} \) such that

\[ h v_{i f}=E_{n_{i}}-E_{n_{f}} \]

Using Eq. \((12.16),\) for \(E_{n_{f}}\) and \(E_{n_{i}},\) we get

\[ \(h v_{i f}=\frac{m e^{4}}{8 \varepsilon_{o}^{2} h^{2}}\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{t}^{2}}\right)\) or \(v_{i f}=\frac{m e^{4}}{8 \varepsilon_{0}^{2} h^{3}}\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right)\) \]

Equation (12.21) is the Rydberg formula, for the spectrum of the hydrogen atom. In this relation, if we take \(n_{f}\)= 2 and \(n_{i}\) = 3, 4, 5..., it reduces to a form similar to Eq. (12.10) for the Balmer series. The Rydberg constant R is readily identified to be

\[ \(R=\frac{m e^{4}}{8 \varepsilon_{\theta}^{2} h^{3} c}\) \]

If we insert the values of various constants in Eq. \((12.23),\) we get \[ R=1.03 \times 10^{7} \mathrm{~m}^{-1} \]

This is a value very close to the value \(\left(1.097 \times 10^{7} \mathrm{~m}^{-1}\right)\) obtained from the empirical Balmer formula. This agreement between the theoretical and experimental values of the Rydberg constant provided a direct and striking confirmation of the Bohr's model.

Since both \(n_{f}\) and \(n_{i}\) are integers. this immediately shows that in transitions between different atomic levels, light is radiated in various discrete frequencies. For hydrogen spectrum, the Balmer formula corresponds to \(n_{f}=2\) and \(n_{i}=3,4,5\) etc. The results of the Bohr's model suggested the presence of other series spectra for hydrogen atom-those corresponding to transitions resulting from \(n_{f}=1\) and \(n_{i}=2.3,\) etc.: \(n_{f}=3\) and \(n_{i}=4,5,\) etc., and so on. Such series were identified in the course of spectroscopic investigations and are known as the Lyman, Balmer, Paschen, Brackett, and Pfund series. The electronic transitions corresponding to these series are shown in Fig. 12.9

The various lines in the atomic spectra are produced when electrons jump from higher energy state to a lower energy state and photons are emitted. These spectral lines are called emission lines. But when an atom absorbs a photon that has precisely the same energy needed by the electron in a lower energy state to make transitions to a higher energy state, the process is called absorption. Thus if photons with a continuous range of frequencies pass through a rarefied gas and then are analysed with a spectrometer, a series of dark spectral absorption lines appear in the continuous spectrum. The dark lines indicate the frequencies that have been absorbed by the atoms of the gas.

The explanation of the hydrogen atom spectrum provided by Bohr’s model was a brilliant achievement, which greatly stimulated progress towards the modern quantum theory. In 1922, Bohr was awarded Nobel Prize in Physics.