The first three spectral lines of H-atom in the Balmer series are
given λ₁, λ₂, λ₃ considering the Bohr atomic model, the wave lengths of first and third spectral lines ( λ₁/λ₃ ) are related by a factor of approximately 'x' × 10⁻¹.
The value of x, to the nearest integer, is _________.

Solution

Physics sir · 4 steps

Every Balmer line ends on n = 2, so the first and third lines start from n = 3 and n = 5.

1. Pick the transitions. In the Balmer series every line ends on n = 2. The first line starts from n = 3, the second from n = 4 and the third from n = 5.

   1/λ = R(1/2²-1/n²)

2. Each line. First line: 1/λ₁ = R(1/4-1/9) = 5R/36. Third line: 1/λ₃ = R(1/4-1/25) = 21R/100.
3. Take the ratio. Wavelength is the reciprocal, so λ₁/λ₃ = (21/100)/(5/36) = (21 × 36)/500 = 756/500 = 1.512.
4. So. 1.512 = 15.12 × 10⁻¹, so x to the nearest integer is 15.

The common slip: 14. Taking the third line as n = 4 → 2 (that is the second line) gives a ratio of 1.35, so x comes out as 14, not 15.

The idea to keep. Fix the lower level for the series first, then count lines upward from the next level. Wavelength ratios are the inverse of the bracket ratios, and the Rydberg constant cancels.

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