4. Hydrogen Atom Spectrum and Bohr’s Atomic Model

Concept Explanation:

Niels Bohr developed his atomic model to explain the discrete lines observed in the hydrogen spectrum. According to Bohr, electrons orbit the nucleus in fixed energy levels or shells, and they can only occupy these specific orbits. When an electron jumps from a higher energy level to a lower one, it emits a photon with energy equal to the difference between the two energy levels. This energy corresponds to the wavelength of light observed in the hydrogen spectrum.

The energy levels in the hydrogen atom are given by:

Eₙ = −13.6 eV / n²

where n is the principal quantum number (1, 2, 3, ...).

Example Problem:

Calculate the wavelength of light emitted when an electron in a hydrogen atom falls from the n=3 level to the n=2 level.

Solution:

The energy difference between the two levels is:

ΔE = E₂ − E₃ = −13.6 eV / 2² − (−13.6 eV / 3²)

The energy of the photon emitted corresponds to this energy difference. To find the wavelength, we use the relation:

E = hc / λ

Substitute the known values (h = 6.626 × 10⁻³⁴ J·s, c = 3 × 10⁸ m/s):

λ = hc / ΔE = 656 nm