Detailed Explanation of the Line Spectrum of Hydrogen
The line spectrum observed in the case of the hydrogen atom results from the transitions of electrons between quantized energy levels described by Bohr's model. In a hydrogen atom, electrons can exist in specific energy states or orbits, which are identified by their principal quantum numbers (n). When energy is absorbed, an electron moves to a higher orbit (n_f > n_i), while energy is emitted when it transitions from a higher energy state to a lower one (n_i > n_f). The difference in energy for transitions between these orbits is given by the equation:
\[ \Delta E = E_f - E_i \]
where \(E_f\) and \(E_i\) are the energies of the final and initial orbits, respectively.
Using this principle, the emitted or absorbed energy can also be related to frequency \((
u)\) using Planck's equation,
\[ E = h
u \]
where \(h\) is Planck's constant. Furthermore, Rydberg’s formula empirically describes the wavelengths of transitions:
\[ \nu = R_H \left( \frac{1}{{n_1}^2} - \frac{1}{{n_2}^2} \right) \]
where \(R_H\) is the Rydberg constant and n1 and n2 are integers representing the principal quantum numbers. The line spectrum emerges from several transitions in the hydrogen atom, resulting in bright lines at specific wavelengths, thus allowing for different series of spectral lines such as the Balmer series, which is uniquely visible.