Explain why [Fe(CN)₆]⁴⁻ is diamagnetic but [Fe(H₂O)₆]²⁺ is paramagnetic

In both complexes, iron is in the +2 oxidation state. Iron (atomic number 26) has the electronic configuration [Ar] 3d⁶ 4s².

Fe²⁺ loses 2 electrons (from 4s first): configuration becomes [Ar] 3d⁶.

So in both complexes, Fe²⁺ has 6 d-electrons to distribute among the d-orbitals.

When 6 ligands approach the central metal in an octahedral arrangement, the 5 d-orbitals split into two sets:

• t₂g (lower energy): $d_{xy}$, $d_{xz}$, $d_{yz}$ — 3 orbitals
• e_g (higher energy): $d_{x^{2}-y^{2}}$, $d_{z^{2}}$ — 2 orbitals

The energy gap between these sets is called the crystal field splitting energy, Δ_o.

Whether electrons pair up in t₂g orbitals or occupy the higher e_g orbitals depends on the relative magnitudes of Δ_o (crystal field splitting) vs P (pairing energy).

This is the key step. Ligands are classified by the spectrochemical series based on how large a Δ_o they produce:

CN⁻ is a strong field ligand — it produces a very large Δ_o, so Δ_o > P. H₂O is a weak field ligand — it produces a small Δ_o, so Δ_o < P.

With CN⁻ (strong field), Δ_o > P. Electrons prefer to pair in the lower t₂g orbitals rather than jump to e_g.

Filling 6 electrons into t₂g with maximum pairing:

• t₂g: ↑↓ ↑↓ ↑↓ (all 6 electrons paired)
• e_g: empty

Configuration: t₂g⁶ e_g⁰

Number of unpaired electrons = 0 → Diamagnetic ✓

With H₂O (weak field), Δ_o < P. Electrons prefer to spread out (Hund’s rule takes precedence — it costs less energy to occupy e_g than to pair in t₂g).

Filling 6 electrons following Hund’s rule:

• t₂g: ↑↓ ↑ ↑ (4 electrons, one pair + two singles)
• e_g: ↑ ↑ (2 electrons, both unpaired)

Configuration: t₂g⁴ e_g²

Number of unpaired electrons = 4 → Paramagnetic ✓

The magnetic moment: $\mu = \sqrt{n(n+2)}$ BM $= \sqrt{4 \times 6} = \sqrt{24} \approx 4.9$ BM