Electronic configuration of Ti (Z=22):
Ti: [Ar] 3d² 4s²
Ti²⁺: Remove 2 electrons from 4s first → [Ar] 3d²
3d² — apply Hund's rule:
5 d-orbitals: ↑ ↑ _ _ _
Both electrons go to separate orbitals with parallel spins
n = 2 unpaired electrons
Spin-only magnetic moment formula:
μ = √n(n+2) BM
μ = √2(2+2) = √(2×4) = √8 = 2√2 ≈ 2·84 BM
Ti²⁺: [Ar] 3d² → n = 2 unpaired electrons
μ = √n(n+2) = √(2×4) = √8 ≈ 2·83 BM ≈ 2·84 BM
The magnetic moment of a transition metal ion depends on the number of unpaired electrons (n). The spin-only formula: μ = √n(n+2) BM (Bohr Magnetons). This formula assumes only spin contribution (no orbital contribution). For first-row transition metals, orbital contribution is often quenched, making the spin-only formula approximately valid. Each unpaired electron has a spin magnetic moment of approximately 1·73 BM (= √3 BM = √1(1+2) BM). Diamagnetic species (n=0): μ = 0, repelled by magnetic field. Paramagnetic species (n≥1): μ > 0, attracted to magnetic field.
| Unpaired e⁻ (n) | μ = √n(n+2) | μ (BM) | Example ions |
|---|---|---|---|
| 0 | √0 | 0 | Zn²⁺, Cu⁺, Sc³⁺, Ti⁴⁺ |
| 1 | √3 | 1·73 | Ti³⁺, Cu²⁺, V⁴⁺ |
| 2 | √8 | 2·84 | Ti²⁺, V³⁺, Ni²⁺ |
| 3 | √15 | 3·87 | V²⁺, Cr³⁺, Co³⁺(hs) |
| 4 | √24 | 4·90 | Cr²⁺, Mn³⁺, Fe²⁺(hs) |
| 5 | √35 | 5·92 | Mn²⁺, Fe³⁺(hs) |
📌 Ti (Z=22): [Ar]3d²4s² → Ti²⁺: [Ar]3d² (n=2, μ=2·84 BM)
📌 V (Z=23): [Ar]3d³4s² → V²⁺: [Ar]3d³ (n=3, μ=3·87 BM)
📌 Cr (Z=24): [Ar]3d⁵4s¹ → Cr³⁺: [Ar]3d³ (n=3, μ=3·87 BM)
📌 Mn (Z=25): [Ar]3d⁵4s² → Mn²⁺: [Ar]3d⁵ (n=5, μ=5·92 BM) ← maximum!
📌 Fe (Z=26): [Ar]3d⁶4s² → Fe²⁺: [Ar]3d⁶ (n=4, μ=4·90 BM); Fe³⁺: [Ar]3d⁵ (n=5, μ=5·92 BM)
📌 Co (Z=27): [Ar]3d⁷4s² → Co²⁺: [Ar]3d⁷ (n=3, μ=3·87 BM)
📌 Ni (Z=28): [Ar]3d⁸4s² → Ni²⁺: [Ar]3d⁸ (n=2, μ=2·84 BM) — same as Ti²⁺!
📌 Cu (Z=29): [Ar]3d¹⁰4s¹ → Cu²⁺: [Ar]3d⁹ (n=1, μ=1·73 BM)
📌 Zn (Z=30): [Ar]3d¹⁰4s² → Zn²⁺: [Ar]3d¹⁰ (n=0, μ=0, diamagnetic)
In neutral atoms, 4s fills before 3d (Aufbau principle — 4s is lower energy than 3d for neutral atoms). But in ion formation, 4s electrons are removed first because once the atom starts losing electrons, the 3d orbitals become lower in energy than 4s (due to increased effective nuclear charge). So Ti (3d²4s²) loses 2 electrons from 4s to give Ti²⁺ (3d²), NOT from 3d. This is a highly tested NEET concept — always remove ns electrons before (n-1)d electrons when forming positive ions.
Cr (Z=24): Expected [Ar]3d⁴4s² but actual [Ar]3d⁵4s¹. Reason: half-filled d⁵ configuration (all 5 d orbitals singly occupied) has extra stability due to symmetry and exchange energy. Cu (Z=29): Expected [Ar]3d⁹4s² but actual [Ar]3d¹⁰4s¹. Reason: completely filled d¹⁰ configuration has extra stability. These two exceptions are extremely important for NEET — must be memorised. Their ions: Cr³⁺ = [Ar]3d³; Cu²⁺ = [Ar]3d⁹; Cu⁺ = [Ar]3d¹⁰ (diamagnetic).
Transition metal compounds are coloured because of d-d electronic transitions. When ligands coordinate to the metal, the d orbitals split into two groups (in octahedral field: t₂g and eg). The energy difference between these groups (Δ = crystal field splitting energy) falls in the visible light range (~400–700 nm). The compound absorbs certain wavelengths (complementary colour is observed). Species with d⁰ (Sc³⁺, Ti⁴⁺) or d¹⁰ (Zn²⁺, Cu⁺) configurations are colourless — no d-d transitions possible. Ti³⁺ (d¹) is purple, Cu²⁺ (d⁹) is blue, Fe³⁺ (d⁵ high spin) is yellow.
In octahedral complexes, d orbitals split by crystal field: lower t₂g (3 orbitals) and upper eg (2 orbitals). Electrons can pair up in t₂g (low spin — strong field ligands: CN⁻, CO, NO⁺, en, NH₃) or fill all orbitals before pairing (high spin — weak field ligands: F⁻, Cl⁻, Br⁻, I⁻, OH⁻, H₂O). High spin → more unpaired electrons → higher magnetic moment. Low spin → more paired electrons → lower magnetic moment. Example: Fe²⁺ (d⁶): high spin has 4 unpaired (μ = 4·90 BM); low spin has 0 unpaired (μ = 0, diamagnetic).
Magnetic moment measurements help determine: (1) Number of unpaired electrons → electronic configuration of the ion. (2) Oxidation state of the metal. (3) Whether a complex is high spin or low spin. (4) Geometry of complex (square planar complexes of d⁸ metal ions like Ni²⁺, Pd²⁺, Pt²⁺ are diamagnetic — all electrons paired; tetrahedral Ni²⁺ complexes are paramagnetic with 2 unpaired electrons). Practical: MRI (magnetic resonance imaging) uses magnetic properties of hydrogen nuclei (protons) in body tissues.