The hydrogen spectrum consists of several spectral lines in Lyman series ( L ₁, ~L ₂ , L ₃ ; L ₁ has lowest energy among Lyman series). Similarly it consists of several spectral lines in Balmer series ( B ₁, ~B ₂, ~B ₃ ; B ₁ . has lowest energy among Balmer lines). The energy of L₁ is x times the energy of B₁ . The value of x is _ _ _ _ 10⁻¹ . (Nearest integer)

Step-by-step solution

The energy of a spectral line in the hydrogen spectrum is given by E = 13.6 ( 1 n₁^2 - 1 n₂^2 ) eV. For the Lyman series, the lowest energy line L₁ corresponds to the transition from n=2 to n=1 . E(L₁) = 13.6 ( 1 1^2 - 1 2^2 ) = 13.6 ( 1 - 1 4 ) = 13.6 3 4 eV. For the Balmer series, the lowest energy line B₁ corresponds to the transition from n=3 to n=2 . E(B₁) = 13.6 ( 1 2^2 - 1 3^2 ) = 13.6 ( 1 4 - 1 9 ) = 13.6 5 36 eV. Given E(L₁) = x E(B₁) , we have: 13.6 3 4 = x 13.6 5 36 3 4 = x 5 36 x = 3 4 36 5 = 3 9 5 = 27