The wavelength of the first line of Lyman series for H - atom is equal to that of the second line of Balmer series for a H -like ion. The atomic number Z of H -like ion is

Step-by-step solution

The wavelength for a transition in a hydrogen-like ion is given by the Rydberg formula: 1 = R Z^2 ( 1 n₁^2 - 1 n₂^2 ) . For the first line of the Lyman series in H-atom ( Z=1 ): n₁ = 1 , n₂ = 2 . 1 ₁ = R(1)^2 ( 1 1^2 - 1 2^2 ) = R ( 1 - 1 4 ) = 3R 4 . For the second line of the Balmer series in a H-like ion with atomic number Z : n₁ = 2 , n₂ = 4 . 1 ₂ = R Z^2 ( 1 2^2 - 1 4^2 ) = R Z^2 ( 1 4 - 1 16 ) = R Z^2 ( 4-1 16 ) = 3R Z^2 16 . Equating the wavelengths ₁ = ₂ , we have 1 ₁ = 1 ₂ . 3R 4 = 3R Z^2 16 . Simplifying