Why is Δo the double of Δt ?

The crystal field splitting in octahedral complexes (Δ_o) is double than that in tetrahedral complexes (Δ_t) (i.e., Δo is double of Δt) because:

(i) The number of ligands is 2/3 of the octahedral complex, therefore, the splitting of the d-orbitals is also two-thirds.

(ii) None of the ligands point directly towards any of the five d-orbitals. Therefore, the splitting is reduced by roughly two-third. In tetrahedral and octahedral complexes, the d_xy, d_yz and d_zx orbitals are at 45^o, i.e., at equal distance from the ligands. Therefore, in both the complexes, each ligand repels these orbitals in equal amounts. In octahedral complexes, the ligand approach directly to d_{x² – y²} and d_z² orbitals and repel them strongly. But in tetrahedral complexes, these orbitals are at (√2/2) l = 0.71 l (it approximately equal to the 2/3) as compared to that of octahedral complexes. Therefore, in tetrahedral complexes, the ligands repel these orbitals 2/3 times than that of octahedral complexes. Thus, due to this fact, splitting decreases by 2/3rd in tetrahedral complexes as compared to octahedral complexes.

Therefore, Δ_t = 2/3 x 2/3 x Δ_o or Δ_t = 4/9 Δ_o

Thus, Δo is double of Δt or we can say that the crystal field splitting in tetrahedral complexes is smaller than that in octahedral complexes.

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