Question

Hydrogen (H), deuterium (D), singly ionized helium (He⁺) and doubly ionized lithium (Li⁺+) all have one electron around the nucleus. Consider n = 2 to n = 1 transition. The wavelengths of emitted radiations are lambda₁, lambda₂, lambda₃ and lambda₄ respectively. Then approximately

• A. lambda₁ = lambda₂ = 4lambda₃
• B. 4lambda₁ = 2lambda₂ = 2lambda₃ = lambda₄
• C. lambda₁ = 2lambda₂ = 2lambda₃ = 3√(2)lambda₄
• D. lambda₁ = lambda₂ = 2lambda₃ = 3√(2)lambda₄

Hint:

For hydrogen-like species: λ ∝ (1)/(Z²) Mass of nucleus does not affect wavelength significantly.

Answer 1

The Correct Option is A

Step 1: For hydrogen-like atoms: (1)/(λ) ∝ Z² ⟹ λ ∝ (1)/(Z²) Step 2: Nuclear charges: ZH = 1, ZD = 1, ZHe⁺ = 2, ZLᵢ⁺+ = 3 Step 3: Wavelength ratios: lambda₁ : lambda₂ : lambda₃ : lambda₄ = (1)/(1²) : (1)/(1²) : (1)/(2²) : (1)/(3²) = 1 : 1 : (1)/(4) : (1)/(9) ⟹ lambda₁ = lambda₂ = 4lambda₃