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JEE Advanced 2024 Paper 1 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

Consider the Li^2+ ion (Z=3) in the Bohr model, with its single electron in the stated state. Here a0 is the Bohr radius and the ground-state energy…

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Consider the Li2+\mathrm{Li^{2+}} ion (Z=3Z=3) in the Bohr model, with its single electron in the stated state. Here a0a_0 is the Bohr radius and the ground-state energy of hydrogen is −13.6 eV-13.6\ \mathrm{eV}. Match List-I with List-II.

List-I
(P) Radius of the orbit with n=2n=2
(Q) Total energy of the electron for n=3n=3
(R) de Broglie wavelength of the electron for n=3n=3
(S) Minimum energy needed to remove the electron from the n=2n=2 state

List-II
(1) 2πa02\pi a_0
(2) 122.4 eV122.4\ \mathrm{eV}
(3) −13.6 eV-13.6\ \mathrm{eV}
(4) 43a0\dfrac43a_0
(5) 30.6 eV30.6\ \mathrm{eV}
  1. A. P→4, Q→3, R→1, S→5
  2. B. P→4, Q→3, R→1, S→2
  3. C. P→1, Q→3, R→4, S→5
  4. D. P→4, Q→5, R→1, S→3
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Answer: (A) P→4, Q→3, R→1, S→5

In the Bohr model rn=n2a0Zr_n=\dfrac{n^2a_0}{Z} and En=−13.6 Z2n2 eVE_n=-13.6\,\dfrac{Z^2}{n^2}\ \mathrm{eV}. (P) r2=4a03r_2=\dfrac{4a_0}{3}, which is (4). (Q) E3=−13.6×99=−13.6 eVE_3=-13.6\times\dfrac{9}{9}=-13.6\ \mathrm{eV}, which is (3). (R) The orbit circumference holds nn de Broglie wavelengths, 2πrn=nλ2\pi r_n=n\lambda, so λ=2πr33=2π3⋅9a03=2πa0\lambda=\dfrac{2\pi r_3}{3}=\dfrac{2\pi}{3}\cdot\dfrac{9a_0}{3}=2\pi a_0, which is (1). (S) Removing the electron from n=2n=2 needs ∣E2∣=13.6×94=30.6 eV|E_2|=13.6\times\dfrac94=30.6\ \mathrm{eV}, which is (5). The 122.4 eV122.4\ \mathrm{eV} in (2) is the ground-state ionisation energy and is a distractor. All four entries follow from rn=n2a0/Zr_n=n^2a_0/Z, En=−13.6Z2/n2E_n=-13.6Z^2/n^2 and the quantisation of the orbit circumference in de Broglie wavelengths. Hence P→4, Q→3, R→1, S→5.
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