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JEE Main 2022 · Chemistry · Atomic Structure · easy

In the Bohr model the de Broglie wavelength of the electron in the third orbit of Li^2+ is (where a0 is the radius of the first Bohr orbit of…

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In the Bohr model the de Broglie wavelength of the electron in the third orbit of Li2+\mathrm{Li^{2+}} is (where a0a_0 is the radius of the first Bohr orbit of hydrogen)
  1. A. 2πa02\pi a_0
  2. B. 6πa06\pi a_0
  3. C. 2πa03\dfrac{2\pi a_0}{3}
  4. D. 3πa03\pi a_0
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Answer: (A) 2πa02\pi a_0

Radius of the nn-th orbit: rn=a0n2/Z=a0(9)/3=3a0r_n=a_0n^2/Z=a_0(9)/3=3a_0. The standing-wave condition 2πrn=nλ2\pi r_n=n\lambda gives λ=2πrn/n=2π(3a0)/3=2πa0\lambda=2\pi r_n/n=2\pi(3a_0)/3=2\pi a_0. The circumference of the orbit (6πa06\pi a_0) holds three wavelengths, so λ\lambda is one third of it.
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