The de Broglie wavelength of an electron present in the 4^ th Bohr orbit of a He ⁺ ion is x a₀ , where a₀ is the Bohr radius. The value of x is ________.

Step-by-step solution

According to Bohr's quantization condition, the angular momentum of an electron in the n^ th orbit is: mvr = nh 2 Rearranging this gives the relation between the orbit circumference and the de Broglie wavelength : 2 r = n h mv = n The radius of the n^ th Bohr orbit for a hydrogen-like ion is given by: r = a₀ n^2 Z Substituting the expression for r into the quantization condition: 2 ( a₀ n^2 Z ) = n = 2 a₀ n Z For the 4^ th orbit of a He ⁺ ion, n = 4 and Z = 2 . Substituting these values: = 2 a₀ (4) 2 = 4 a₀ Compari