Chapter 4 · Chemistry
A Balance That Counts Electrons: Paramagnetism and the Spin-only Magnetic Moment
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Every electron is a tiny magnet; in a pair the spins cancel, so only unpaired electrons make an ion paramagnetic. Count them from the ion's d configuration (singly up to d⁵, then pairing) and the spin-only formula gives the magnetic moment: μ = √(n(n+2)) BM, from 1.73 for one unpaired electron to 5.92 for Mn²⁺ (d⁵). A measured μ (Gouy balance) counts the unpaired electrons.
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Key points
- Paramagnetic = attracted by a field (unpaired electrons); diamagnetic = weakly repelled (all paired).
- Count n: write the ion's configuration (4s leaves first), fill the 3d orbitals singly first, then pair: d¹–d⁵ → n = d; d⁶–d¹⁰ → n = 10 − d.
- μ = √(n(n+2)) BM: 1.73, 2.84, 3.87, 4.90, 5.92 for n = 1 to 5. It does not double: n = 2 gives 2.84, not 3.46.
- μ peaks at d⁵ (Mn²⁺ 5.92 calc., 5.96 observed); a measured μ tells you the number of unpaired electrons.
Spin-only values
| n | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| μ (BM) | 0 | 1.73 | 2.84 | 3.87 | 4.90 | 5.92 |
For 3d ions the orbital contribution is effectively quenched, so μ depends only on n. One unpaired electron: 1.73 BM.
Calculated vs observed (NCERT Table 4.7)
| Ion | Config. | n | Calc. | Obs. |
|---|---|---|---|---|
| Ti³⁺ | 3d¹ | 1 | 1.73 | 1.75 |
| V²⁺ | 3d³ | 3 | 3.87 | 3.86 |
| Cr²⁺ | 3d⁴ | 4 | 4.90 | 4.80 |
| Mn²⁺ | 3d⁵ | 5 | 5.92 | 5.96 |
| Fe²⁺ | 3d⁶ | 4 | 4.90 | 5.3–5.5 |
| Co²⁺ | 3d⁷ | 3 | 3.87 | 4.4–5.2 |
| Ni²⁺ | 3d⁸ | 2 | 2.84 | 2.9–3.4 |
| Cu²⁺ | 3d⁹ | 1 | 1.73 | 1.8–2.2 |
| Zn²⁺ | 3d¹⁰ | 0 | 0 | 0 |
Fe²⁺ and Co²⁺ read higher than spin-only because their orbital contribution is not fully quenched.
Worked example (NCERT Example 4.8)
Z = 25, divalent ion in water: Mn²⁺ = 3d⁵, n = 5, μ = √(5 × 7) = √35 = 5.92 BM.
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