Question

The crystal field splitting energy of $[Co(oxalate)_3]^3-$ complex is 'n' times that of the $[Cr(oxalate)_3]^3-$ complex. Here 'n' is_______ (Assume $\Delta_0 > P$)}

Hint:

While $Co^{3+}$ usually has a slightly higher splitting energy than $Cr^{3+}$ due to higher effective nuclear charge, in many numerical problems of this type, they are treated as having comparable splitting magnitudes ($n=1$) unless specific values are provided.

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
The crystal field splitting energy ($\Delta_0$) depends on the oxidation state of the metal and the nature of the metal ion itself. For the same ligand (oxalate) and same oxidation state (+3), we compare the $3d^6$ Cobalt ion and the $3d^3$ Chromium ion.
Step 2: Detailed Explanation:
In octahedral complexes, for metals in the same oxidation state from the same period, $\Delta_0$ generally increases with the atomic number due to the increase in effective nuclear charge ($Z_{eff}$), which pulls ligands closer. However, for $Co^{3+}$ and $Cr^{3+}$, experimental data shows that $Co^{3+}$ (a $d^6$ system) typically has a higher $\Delta_0$ than $Cr^{3+}$ (a $d^3$ system) with the same ligand. Specifically, $Co^{3+}$ is known for having a very high CFSE, often resulting in low-spin complexes even with intermediate ligands like oxalate.
Step 3: Calculating the Ratio:
Based on standardized values for these specific complexes: - For $[Cr(ox)_{3}]^{3-}$, $\Delta_0 \approx 17,400 \text{ cm}^{-1}$. - For $[Co(ox)_{3}]^{3-}$, $\Delta_0 \approx 18,000 \text{ cm}^{-1}$. Given the context of such problems (where $n$ is often a simple integer or ratio), and noting that the question asks for $n$, we look at the specific electronic contribution. If the question implies the ratio of CFSE (Crystal Field Stabilization Energy) rather than just $\Delta_0$: - $Cr^{3+} (d^3)$: $CFSE = 3 \times (-0.4\Delta_0) = -1.2\Delta_0$. - $Co^{3+} (d^6, \text{Low Spin}): CFSE = 6 \times (-0.4\Delta_0) + 2P = -2.4\Delta_0 + 2P$. In many competitive exam contexts for this specific pair, the value of $n$ is taken as 1.5 or treated as 1 depending on the specific data provided in the original paper's key. Assuming standard $\Delta_0$ comparison, $n \approx 1$.
Step 4: Final Answer:
The value of n is 1.