The de-Broglie wavelength of an electron in a particular orbit of a hydrogen-like helium ion ( He ⁺ ) is 4 a₀ , where a₀ is the Bohr radius of a hydrogen atom. What is the principal quantum number of this orbit?

Step-by-step solution

According to Bohr's quantization condition, the circumference of the orbit is an integral multiple of the de-Broglie wavelength: 2 r_n = n The radius of the n^ th orbit for a hydrogen-like species is given by: r_n = a₀ n^2 Z Substituting r_n into the quantization condition: 2 ( a₀ n^2 Z ) = n = 2 a₀ n Z For a He ⁺ ion, the atomic number is Z = 2 . We are given that = 4 a₀ . Substituting these values: 4 a₀ = 2 a₀ n 2 4 a₀ = a₀ n n = 4 Thus, the principal quantum number of the orbit is 4 . Answer: 4