The radius of the first Bohr orbit of a hydrogen atom is a₀ . The de Broglie wavelength of an electron in the third Bohr orbit of a He ^+ ion is:

Step-by-step solution

The radius of the n^ th Bohr orbit is given by r_n = a₀ n^2 Z . For the 3^ rd orbit of He ^+ ( n = 3, Z = 2 ): r₃ = a₀ 3^2 2 = 4.5 a₀ According to de Broglie's hypothesis and Bohr's quantization condition, the circumference of the orbit is equal to an integral multiple of the de Broglie wavelength ( ): 2 r_n = n Substituting the values: 2 (4.5 a₀) = 3 9 a₀ = 3 = 3 a₀ Alternatively, using = 2 a₀ n Z , we get = 2 a₀ (3) 2 = 3 a₀ . Answer: 3 a₀