In a hydrogen atom the electron makes a transition from (n + 1)ᵗʰ level to the nᵗʰ level. If n >> 1, the frequency of radiation emitted is proportional to :

Solution

Physics sir · 3 steps

Write the energy gap for adjacent levels, then let n be huge: the gap shrinks like 1/n³.

1. Energy gap. In hydrogen, Eₙ = -13.6/n² eV. The emitted photon carries the difference between levels n+1 and n, and hν equals that difference.

   ν = Rc(1/n²-1/(n+1)²) = Rc (2n+1)/n²(n+1)²

2. Use n ≫ 1. For large n, the numerator 2n+1 ≈ 2n and the denominator n²(n+1)² ≈ n⁴. So ν ≈ Rc 2n/n⁴ = 2Rc/n³.
3. So. The frequency of the emitted radiation is proportional to 1/n³, which is option (c).

The idea to keep. For neighbouring levels at large n, the gap behaves like a derivative: d/dn(1/n²) ∝ 1/n³. Always simplify the difference before you approximate.

Answer from the NTA final answer key. Solution by PrepWiser.