MP Board · Class 12 · Physics · AtomsWhat are spectral series of hydrogen atom? Explain the different series (Lyman, Balmer, Paschen, Brackett, and Pfund) in detail, specifying their region in the electromagnetic spectrum and the energy level transitions involved.

Introduction to Hydrogen Spectral Series\nWhen an excited hydrogen atom undergoes transitions from higher energy levels ($n_2$) to lower energy levels ($n_1$), it emits radiation of specific frequencies. When these emitted frequencies are sorted in order of wavelength or frequency, they form a line spectrum known as the hydrogen spectrum. The spectral lines are grouped into distinct families or series, each named after its discoverer.

Detailed Classification of Spectral Series\nThe spectral series of hydrogen are systematically categorized based on the lower energy level ($n_1$) to which the electrons drop:

• 1. Lyman Series:

  • Lower Energy Level: $n_1 = 1$
  • Higher Energy Levels: $n_2 = 2, 3, 4, \dots$
  • Spectral Region: Ultraviolet (UV) region.
  • Description: Discovered by Theodore Lyman, this series represents the highest energy transitions in the hydrogen atom, resulting in shorter wavelengths falling strictly within the ultraviolet spectrum.

• 2. Balmer Series:

  • Lower Energy Level: $n_1 = 2$
  • Higher Energy Levels: $n_2 = 3, 4, 5, \dots$
  • Spectral Region: Visible and near-ultraviolet region.
  • Description: Discovered by Johann Balmer, this is the most historically significant series because its first few lines lie entirely within the visible region of the electromagnetic spectrum, making it easily observable with optical spectrometers.

• 3. Paschen Series:

  • Lower Energy Level: $n_1 = 3$
  • Higher Energy Levels: $n_2 = 4, 5, 6, \dots$
  • Spectral Region: Infrared (IR) region.
  • Description: Discovered by Friedrich Paschen, transitions terminating at the third orbit emit lower energy photons, shifting the wavelengths into the infrared region of the spectrum.

• 4. Brackett Series:

  • Lower Energy Level: $n_1 = 4$
  • Higher Energy Levels: $n_2 = 5, 6, 7, \dots$
  • Spectral Region: Near-infrared region.
  • Description: Discovered by Frederick Sumner Brackett, these spectral lines correspond to transitions ending at the fourth energy level and are found further deep in the infrared spectrum.

• 5. Pfund Series:

  • Lower Energy Level: $n_1 = 5$
  • Higher Energy Levels: $n_2 = 6, 7, 8, \dots$
  • Spectral Region: Far-infrared region.
  • Description: Discovered by August Herman Pfund, this series involves transitions ending at the fifth energy level, producing even longer wavelengths situated in the far-infrared region.

Mathematical Formulation\nThe wavelength ($\lambda$) of any spectral line in these series is given by the Rydberg formula:

$$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$\nwhere $R$ is the Rydberg constant ($R \approx 1.097 \times 10^7 \text{ m}^{-1}$).