Arrange the following complexes in increasing order of crystal field splitting energy (Δ₀)

(a) $\left[\mathrm{Cr}\right(\mathrm{CN})6{]}^{3-}$

(b) $[\mathrm{CrF}6{]}^{3-}$

(c) $\left[\mathrm{Cr}\right(\mathrm{NH}3)6{]}^{3+}$

(d) $\left[\mathrm{Cr}\right(\mathrm{en})3{]}^{3+}$

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The crystal field splitting energy in octahedral complexes (${\Delta }_o$) depends primarily on the strength of the ligands coordinates to the central metal ion.

According to the spectrochemical series, ligands are arranged in increasing order of their field strength (ability to split d-orbitals):
Halide donors < Oxygen donors < Nitrogen donors < Carbon donors
Specifically, for the ligands present in the given complexes, the order of ligand strength is:
$F^{-}$ < ${\mathrm{NH}}_3$ < en (ethylenediamine) < ${\mathrm{CN}}^{-}$

Since the central metal ion in all the given complexes is chromium in the same oxidation state (+3), the crystal field splitting energy (${\Delta }_o$) is directly proportional to the strength of the coordinating ligands:
• (a) $\left[\mathrm{Cr}\right(\mathrm{CN}{)}_6{]}^{3-}$ contains the very strong-field ligand ${\mathrm{CN}}^{-}$.
• (b) $[{\mathrm{CrF}}_6{]}^{3-}$ contains the weak-field ligand $F^{-}$.
• (c) $\left[\mathrm{Cr}\right({\mathrm{NH}}_3{)}_6{]}^{3+}$ contains the moderate-field ligand ${\mathrm{NH}}_3$.
• (d) $\left[\mathrm{Cr}\right(\mathrm{en}{)}_3{]}^{3+}$ contains ethylenediamine (en), which is a stronger nitrogen donor than ammonia (${\mathrm{NH}}_3$).

Therefore, the increasing order of crystal field splitting energy (${\Delta }_o$) is:
(b) < (c) < (d) < (a)