Consider an electron in the second stationary orbit of a He ⁺ ion and an electron in the first stationary orbit of a hydrogen atom. What is the ratio of the area enclosed by the orbit of the He ⁺ ion to the area enclosed by the orbit of the hydrogen atom?

Step-by-step solution

The radius of the n^ th Bohr orbit for a hydrogen-like species is given by: r n^2 Z For the He ⁺ ion in the second orbit ( n=2, Z=2 ): r_ He ⁺ 2^2 2 = 2 For the hydrogen atom in the first orbit ( n=1, Z=1 ): r_ H 1^2 1 = 1 The ratio of their radii is: r_ He ⁺ r_ H = 2 1 Assuming the orbits to be circular, the area enclosed by an orbit is A = r^2 . Therefore, the ratio of the areas is proportional to the square of the ratio of their radii: A_ He ⁺ A_ H = ( r_ He ⁺ r_ H )^2 = ( 2 1 )^2 = 4 1 The required ratio is 4:1