The ratio of the wavelengths of the light absorbed by a Hydrogen atom when it undergoes n = 2 → n = 3 and n = 4 → n = 6 transitions, respectively, is

CONCEPT:

Energy Levels and Wavelengths in Hydrogen Atom

• Hydrogen atom transitions occur when an electron moves between quantized energy levels.
• The energy difference between levels is given by the formula:

  E = -13.6 × (1/n²_final - 1/n²_initial) eV

• The wavelength of the light absorbed or emitted during a transition is related to the energy difference:

  λ = hc/E

  Where h is Planck's constant, c is the speed of light, and E is the energy difference.

EXPLANATION:

• For the transition n = 2 → n = 3:
  • Energy difference:

    E₁ = -13.6 × (1/3² - 1/2²)

    E₁ = -13.6 × (1/9 - 1/4) = -13.6 × (-5/36) = 13.6 × 5/36

  • Corresponding wavelength:

    λ₁ = hc/E₁

    λ₁ is inversely proportional to E₁.

• For the transition n = 4 → n = 6:
  • Energy difference:

    E₂ = -13.6 × (1/6² - 1/4²)

    E₂ = -13.6 × (1/36 - 1/16) = -13.6 × (-5/144) = 13.6 × 5/144

  • Corresponding wavelength:

    λ₂ = hc/E₂

    λ₂ is inversely proportional to E₂.

• Ratio of wavelengths:
  • λ₁/λ₂ = E₂/E₁
  • E₁ = 13.6 × 5/36, E₂ = 13.6 × 5/144
  • λ₁/λ₂ = (5/144) / (5/36) = 36/144 = 1/4

Therefore, the ratio of the wavelengths is 1/4.