If the radius of the first orbit of hydrogen atom $\mathbf{a}_{0}$, then de Broglie's wavelength of electron in $3^{\text{rd}}$ orbit is

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Atomic Structure

If the radius of the first orbit of hydrogen atom

a_{0}

, then de Broglie's wavelength of electron in

3^{rd}

orbit is

Answer: (C)

The radius of the nth orbit of hydrogen is

r_{n} = a_{0} n^{2}

. For n = 3:

r_{3} = 9 a_{0}

. According to Bohr's quantization condition, the circumference of the orbit equals an integer multiple of the de Broglie wavelength:

2 π r = nλ

. For the 3rd orbit:

2 π (9 a_{0}) = 3 λ

, giving

λ = 6 π a_{0}

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Single correct option