The ratio of spin-only magnetic moment values $$\mu_{eff}[Cr(CN)_6]^{3-}$$ / $$\mu_{eff}[Cr(H_2O)_6]^{3+}$$ is _______

Both complexes contain Cr$$^{3+}$$, which has the electronic configuration [Ar]3d$$^3$$.

[Cr(CN)$$_6$$]$$^{3-}$$: CN$$^-$$ is a strong field ligand. The d$$^3$$ electrons occupy the three t$$_{2g}$$ orbitals with one electron each.

Number of unpaired electrons = 3

[Cr(H$$_2$$O)$$_6$$]$$^{3+}$$: H$$_2$$O is a weak field ligand. However, for d$$^3$$ configuration, even with weak field ligands, the three electrons go into the three t$$_{2g}$$ orbitals one each (since there are only 3 electrons, no pairing occurs regardless of crystal field strength).

Number of unpaired electrons = 3

The spin-only magnetic moment is: $$\mu = \sqrt{n(n+2)}$$ BM

Since both complexes have the same number of unpaired electrons (n = 3):

$$\mu_{eff}[Cr(CN)_6]^{3-} = \mu_{eff}[Cr(H_2O)_6]^{3+} = \sqrt{3(3+2)} = \sqrt{15}$$ BM

$$\frac{\mu_{eff}[Cr(CN)_6]^{3-}}{\mu_{eff}[Cr(H_2O)_6]^{3+}} = 1$$