A hydrogen atom emits photons due to electron transitions. The transition from an unknown energy level n to the level n=2 emits a photon of frequency ₁ . The transition from the level n=3 to the level n=2 emits a photon of frequency ₂ . If the ratio ₁ ₂ = 27 20 , the value of the principal quantum number n is

Step-by-step solution

The frequency of a photon emitted during an electron transition in a hydrogen atom is given by Rydberg's formula: = cR ( 1 n_f^2 - 1 n_i^2 ) For the first transition ( n 2 ): ₁ = cR ( 1 2^2 - 1 n^2 ) = cR ( 1 4 - 1 n^2 ) For the second transition ( 3 2 ): ₂ = cR ( 1 2^2 - 1 3^2 ) = cR ( 1 4 - 1 9 ) = cR ( 5 36 ) Given the ratio ₁ ₂ = 27 20 , we have: 1 4 - 1 n^2 5 36 = 27 20 1 4 - 1 n^2 = 27 20 5 36 1 4 - 1 n^2 = 3 9 5 4 5 4 9 = 3 16 1 n^2 = 1 4 - 3 16 = 4 16 - 3 16 = 1 16 n^2 = 16 n = 4 Answer: 4