The ratio of the difference between the radii of 3^ rd and 4^ th orbits of the He ⁺ and those of Li ²⁺ is:

Step-by-step solution

The radius of the n^ th orbit of a hydrogen-like ion is given by r_n = a₀ n^2 Z , where a₀ is the Bohr radius and Z is the atomic number. For He ⁺ ( Z=2 ), the radii of the 3^ rd and 4^ th orbits are r₃ = a₀ 3^2 2 = 9a₀ 2 and r₄ = a₀ 4^2 2 = 16a₀ 2 = 8a₀ . The difference for He ⁺ is r_ He ⁺ = r₄ - r₃ = 8a₀ - 4.5a₀ = 3.5a₀ = 7a₀ 2 . For Li ²⁺ ( Z=3 ), the radii of the 3^ rd and 4^ th orbits are r₃ = a₀ 3^2 3 = 3a₀ and r₄ = a₀ 4^2 3 = 16a₀ 3 . The difference for Li ²⁺ is r_ Li ²⁺ = r₄ - r₃ = 16a₀ 3 - 3a₀ = 7a₀ 3 . Th