Understanding Crystal Field Splitting Ratio ($n$)

The question asks for the ratio '$n$' between the crystal field splitting energies ($\Delta_o$) of two octahedral complexes: $[\text{Co(oxalate)}_3]^{3-}$ and $[\text{Cr(oxalate)}_3]^{3-}$. The ratio is defined as:

$ n = \frac{\Delta_o[\text{Co(oxalate)}_3]^{3-}}{\Delta_o[\text{Cr(oxalate)}_3]^{3-}} $

Factors Influencing Splitting Energy ($\Delta_o$)

The magnitude of $\Delta_o$ in an octahedral complex depends primarily on:

• The identity of the central metal ion (charge, position in the periodic table, $d$-electron count).
• The nature of the ligands (position in the spectrochemical series).
• The geometry of the complex (assumed octahedral).

Analyzing the Complexes

In this case:

• Ligand: The ligand is oxalate (ox), which is identical in both complexes. Oxalate is considered a moderately strong field ligand.
• Metal Ion & Oxidation State: Both complexes feature a metal ion in the +3 oxidation state: Cobalt (Co) and Chromium (Cr).
• Electron Configuration: Cr(III) has a $3d^3$ configuration, while Co(III) has a $d^6$ configuration.
• Geometry: Both are $[\text{M(ox)}_3]^{3-}$ type complexes, implying octahedral geometry.

Comparing $\Delta_o$ for Co(III) vs. Cr(III)

Since the ligand and geometry are the same, the difference in $\Delta_o$ arises from the central metal ions, Co(III) and Cr(III).

• Periodic Trend: Chromium (Cr) and Cobalt (Co) are both in the 4th period. Cr (atomic number 24) appears before Co (atomic number 27). For transition metals in the same oxidation state and period, the crystal field splitting energy ($\Delta_o$) generally increases from left to right.
• Trend Implication: Therefore, Co(III) is expected to exhibit a larger $\Delta_o$ than Cr(III) when coordinated by the same ligand.
• Empirical Observation: Experimental data and theoretical studies show that for many ligands, the crystal field splitting energy of Co(III) complexes is significantly higher than that of analogous Cr(III) complexes. This ratio is often found to be approximately 2.

Determining the Ratio '$n$'

Based on the established trends and empirical observations for $3d$ transition metal complexes:

$ \Delta_o[\text{Co(oxalate)}_3]^{3-} \approx 2 \times \Delta_o[\text{Cr(oxalate)}_3]^{3-} $

Thus, the ratio '$n$' is approximately 2.

The correct value lies between 2 and 2.