The shortest wavelengths of Paschen, Lymen and Balmer series are in the ratio

Step-by-step solution

The wavelength for a transition in a hydrogen-like atom is given by the Rydberg formula: 1 = R Z^2 ( 1 n₁^2 - 1 n₂^2 ) . For the shortest wavelength of any series, the transition occurs from n₂ = to n₁ . Thus, 1 _ min = R ( 1 n₁^2 ) , which implies _ min = n₁^2 R . For the Paschen series, n₁ = 3 , so _ Paschen = 3^2 R = 9 R . For the Lyman series, n₁ = 1 , so _ Lyman = 1^2 R = 1 R . For the Balmer series, n₁ = 2 , so _ Balmer = 2^2 R = 4 R . The ratio of the shortest wavelengths of Paschen, Lyman, and Balmer series