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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2020 · Paper I

Q8(e). Magnetic moment of Gd³⁺

  • 10 marks
  • Answered
Question

Calculate the magnetic moment of gadolinium (III) ion (in BM units).

Answer in brief

Gd³⁺ is [Xe]4f⁷: S=72, L=0, J=72, g_J=2, so μ=g_J√(J(J+1))=√(63)=7.94 BM, the same as the spin-only value for seven unpaired electrons.

Key points

  • Gd (Z=64) loses 6s² and 5d¹: Gd³⁺ = [Xe]4f⁷, a half-filled 4f shell, one electron in each of the seven orbitals (mₗ=+3 to -3).
  • S=7×12=72; L=Σ mₗ=3+2+1+0-1-2-3=0; J=L+S=72. Term symbol ^(2S+1)L_J= ⁸S_(7/2).
  • g_J=1+(J(J+1)+S(S+1)-L(L+1))/(2J(J+1))=1+(63/4+63/4-0)/(2×63/4)=2
  • Moment: μ=g_J√(J(J+1))=2√(72×92)=√(63)=7.94 BM

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

  1. Configuration: Gd (Z=64) loses 6s² and 5d¹: Gd³⁺ = [Xe]4f⁷, a half-filled 4f shell, one electron in each of the seven orbitals (mₗ=+3 to -3).
  2. Quantum numbers: S=7×12=72; L=Σ mₗ=3+2+1+0-1-2-3=0; J=L+S=72. Term symbol ^(2S+1)L_J= ⁸S_(7/2).
  3. Landé factor: g_J=1+(J(J+1)+S(S+1)-L(L+1))/(2J(J+1))=1+(63/4+63/4-0)/(2×63/4)=2
  4. Moment: μ=g_J√(J(J+1))=2√(72×92)=√(63)=7.94 BM
  5. Check: with L=0 there is no orbital contribution, so the spin-only formula gives the same: √(n(n+2))=√(7×9)=7.94 BM. The measured value is 7.9–8.0 BM.
  6. Why J is used for lanthanides: 4f electrons are shielded by 5s²5p⁶, so the ligand field does not quench L and spin–orbit coupling is strong; μ follows g_J√(J(J+1)), not the spin-only formula. Gd³⁺ is the one ion where both agree.

Application. seven unpaired electrons with no orbital moment give Gd³⁺ a long electron-spin relaxation time: its chelates (with DTPA or DOTA ligands) are the contrast agents of magnetic resonance imaging.

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Model answer to Q8(e), 2020 Paper I: solution page 44
Solution page 44 of 45
Read the whole 2020 solution2020 question paper

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