In a hydrogen-like atom model, if the radius of the n₁^ th Bohr orbit of a Be ³⁺ ion is exactly equal to the radius of the n₂^ th Bohr orbit of a hydrogen atom, then the ratio n₁ : n₂ is

Step-by-step solution

The radius of the n^ th Bohr orbit for a hydrogen-like species is given by r n^2 Z , where Z is the atomic number. For the Be ³⁺ ion, Z = 4 . Thus, the radius of its n₁^ th orbit is r₁ n₁^2 4 . For the hydrogen atom, Z = 1 . The radius of its n₂^ th orbit is r₂ n₂^2 1 . Given that r₁ = r₂ , we have: n₁^2 4 = n₂^2 1 n₁^2 n₂^2 = 4 Taking the square root on both sides, we get: n₁ n₂ = 2 Therefore, the ratio n₁ : n₂ is 2 : 1 . Answer: 2 : 1