x/(x + 4) is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its
(i) third permitted energy level to the second level and
(ii) the highest permitted energy level to the second permitted level.
The value of x will be ____________.

Solution

Physics sir · 3 steps

Both photons end on n = 2, so only the brackets 1/4 - 1/n² matter. The 13.6 eV cancels.

1. Photon energy. For hydrogen, a jump from level nᵢ down to level n_f gives a photon of energy E = 13.6(1/n_f² - 1/nᵢ²) eV. The highest permitted level is n = ∞, where 1/n² = 0.

   E₁/E₂ = (1/2² - 1/3²)/(1/2² - 1/∞²) = (5/36)/(9/36) = 5/9

2. Match the form. Set x/(x+4) = 5/9. Cross-multiplying gives 9x = 5x + 20, so 4x = 20.
3. So. x = 5. As a check, 5/(5+4) = 5/9, and the answer is 5.

The idea to keep. When two transitions end on the same level, the ratio of photon energies is just the ratio of the (1/n_f² - 1/nᵢ²) brackets. The highest level means n = ∞.

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