Crystal Field TheoryJEE Advanced

Concept

Ligands approaching a metal ion split its five degenerate d-orbitals. Octahedral: orbitals pointing AT ligands (e_g) rise; between them (t_2g) fall — gap Δ₀. Strong-field ligands (CN⁻) make Δ₀ > pairing energy → electrons pair up low spin; weak-field (halides) → high spin. Tetrahedral splits the other way with Δ_t ≈ 4/9 Δ₀ — always high spin.

Key formula

Derivation

d_{x²−y²} and d_{z²} point directly at octahedral ligands → destabilised (+0.6Δ₀); d_{xy}, d_{yz}, d_{xz} point between → stabilised (−0.4Δ₀). Barycentre preserved: 2(0.6) = 3(0.4).

Filling d⁴–d⁷ poses the choice: pay P to pair in t_2g, or pay Δ₀ to occupy e_g. The spectrochemical series I⁻<Br⁻<Cl⁻<F⁻<H₂O<NH₃<en<CN⁻≈CO ranks Δ₀.

Scenarios to explore

• Crystal Field Theory — t₂g/e_g splitting, high vs low spin, CFSE & magnetism.

Real-world applications

• Colours of transition-metal complexes (d–d transitions absorb ~Δ₀).
• Magnetic moments: μ = √(n(n+2)) BM identifies spin states.
• [Fe(H₂O)₆]²⁺ paramagnetic vs [Fe(CN)₆]⁴⁻ diamagnetic — same d⁶!

JEE exam tips

• d⁴–d⁷ octahedral are the ONLY cases where spin state depends on the ligand.
• Magnetic moment √(n(n+2)): 1.73, 2.83, 3.87, 4.90, 5.92 BM for n = 1..5.
• d⁵ high spin: CFSE = 0 (half-filled symmetric) — why Mn²⁺/Fe³⁺ complexes are pale.

Common mistakes

• Applying low-spin logic to tetrahedral complexes (Δ_t too small — always high spin).
• CFSE sign errors — t2g electrons stabilise (negative energy).
• Confusing the ligand order of the spectrochemical series.

Exam traps to avoid

• CO and CN⁻ are strong-field DESPITE being neutral/anionic — π-acceptor character, not charge.
• Pairing energy is per PAIR formed beyond the free-ion configuration.