For hydrogen atom, ₁ and ₂ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of ₁ and ₂ is x/32 . The value of x is

Step-by-step solution

The wavelength of a photon emitted during a transition from n_i to n_f in a hydrogen atom is given by the Rydberg formula: 1 = R ( 1 n_f^2 - 1 n_i^2 ) For transition 1, the electron jumps from n=3 to n=1 . 1 ₁ = R ( 1 1^2 - 1 3^2 ) = R ( 1 - 1 9 ) = 8R 9 ₁ = 9 8R For transition 2, the electron jumps from n=2 to n=1 . 1 ₂ = R ( 1 1^2 - 1 2^2 ) = R ( 1 - 1 4 ) = 3R 4 ₂ = 4 3R The ratio of ₁ to ₂ is: ₁ ₂ = 9 8R 4 3R = 9 8 3 4 = 27 32 Given that ₁ ₂ = x 32 , we have: 27 32 = x 32 x = 27 Answer: 27