Q6(b). Crystal field stabilisation energy and the factors that fix it

What CFSE is

1. In a gaseous ion the five d orbitals are degenerate. Six ligands at the corners of an octahedron repel electrons in d_(x²-y²) and d_(z²) (pointing at them) more than in d_(xy), d_(xz), d_(yz): the set splits into e_g (+0.6 Δₒ) and t_(2g) (-0.4 Δₒ), keeping the barycentre.
2. For t_(2g)^xe_gʸ: CFSE=(-0.4x+0.6y) Δₒ (+ pP) where p is the number of extra pairs formed relative to the gaseous ion and P the pairing energy.
3. Examples: d³ [Cr(NH₃)₆]³+: -1.2 Δₒ; Δₒ=21 500 cm⁻¹, so CFSE =-25 800 cm⁻¹=-308.6 kJ mol⁻¹. d⁶ low spin: -2.4 Δₒ+2P; d⁵ high spin: 0.

Strong-field values exclude the pairing energy.

Factors that affect the magnitude

1. Oxidation state of the metal: a higher charge pulls the ligands closer: Δₒ(aqua) for Co²+ 8400, for Co³+ 16 750 cm⁻¹ (about double); Fe²+ 9350, Fe³+ 14 000 cm⁻¹.
2. Period (4d, 5d metals): larger d orbitals overlap better with the ligands, so Δₒ rises down a group: [Co(NH₃)₆]³+ 22 900, [Rh(NH₃)₆]³+ 34 000 cm⁻¹ (×1.48). Pairing energy also falls, so 4d and 5d complexes are almost always low spin.
3. Nature of the ligand (spectrochemical series): I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < bpy < NO₂⁻ < CN⁻ < CO. For Cr³+: Cl⁻ 13 700, H₂O 17 400, NH₃ 21 500, en 21 900, CN⁻ 26 600 cm⁻¹. π-donors lower Δ, π-acceptors raise it.
4. Geometry: fewer ligands, not pointing at the orbitals: Δₜ=49Δₒ, so tetrahedral complexes are always high spin; square planar gives the largest splitting.
5. Number of d electrons and spin state: CFSE follows the double-humped pattern of the panel; whether the strong-field values are reached depends on Δₒ against P.
6. Metal–ligand distance: Δ falls steeply as the M–L distance grows, so shorter bonds (higher charge, compression) raise it.

Application. CFSE explains the double-humped curves of hydration and lattice enthalpies of the M²⁺ ions from Ca²+ to Zn²+ (d⁰, d⁵, d¹⁰ on the base line), the preference of Ni²+ and Cr³+ for octahedral sites in spinels, and the normal/inverse spinel structures (Fe₃O₄ inverse).