An electron is present in the 3^ rd Bohr orbit of a He ⁺ ion. Its de Broglie wavelength is expressed as x a₀ , where a₀ is the Bohr radius. The value of x is _ _ _ _ _ .

Step-by-step solution

According to de Broglie's hypothesis and Bohr's quantization condition, the circumference of the n^ th orbit is equal to n times the de Broglie wavelength of the electron: 2 r = n The radius of the n^ th Bohr orbit for a hydrogen-like species is given by: r = a₀ n^2 Z For the He ⁺ ion, atomic number Z = 2 . For the 3^ rd orbit, n = 3 . Substituting these values into the radius formula: r = a₀ 3^2 2 = 4.5 a₀ Now, substitute r and n into the quantization condition: 2 (4.5 a₀) = 3 9 a₀ = 3 = 3 a₀ Comparing this with t