Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The values of the crystal field stabilization energies for a high spin \(d^{6}\) metal ion in octahedral and tetrahedral fields, respectively, are : (a) \(-0.4 \Delta_{0}\) and \(-0.6 \Delta_{\mathrm{t}}\) (b) \(-2.4 \Delta_{\mathrm{o}}\) and \(-0.6 \Delta_{\mathrm{t}}\) (c) \(-1.6 \Delta_{0}\) and \(-0.4 \Delta_{t}\) (d) \(-0.4 \Delta_{\mathrm{o}}\) and \(-0.27 \Delta_{\mathrm{r}}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(-0.4 \Delta_{0}\) and \(-0.6 \Delta_{\mathrm{t}}\).

Step 1: Understanding Crystal Field Stabilization Energy (CFSE)

Crystal Field Stabilization Energy (CFSE) is the energy difference between the energy of a metal ion in a crystal field and a hypothetical metal ion with the same electronic configuration without the crystal field. For high spin octahedral and tetrahedral complexes, the CFSE formulae differ due to the splitting pattern of d-orbitals.

Step 2: Calculate CFSE for High Spin Octahedral Complex

For a high spin octahedral complex with 6 d-electrons (\(d^6\)), the CFSE is calculated as: \[\text{CFSE} = (-0.4 \times n_{t2g} + 0.6 \times n_{eg}) \Delta_o\]Given that there are 4 \(t_{2g}\) electrons and 2 \(e_g\) electrons in a \(d^6\) high spin configuration:\[\text{CFSE} = (-0.4 \times 4 + 0.6 \times 2) \Delta_o = (-1.6 + 1.2)\Delta_o = -0.4\Delta_o\]

Step 3: Calculate CFSE for High Spin Tetrahedral Complex

In a tetrahedral crystal field, there are 3 \(t_2\) orbitals and 2 \(e\) orbitals. The splitting is opposite to octahedral, so:\[\text{CFSE} = (-0.6 \times n_{t2} + 0.4 \times n_{e}) \Delta_t\]For a \(d^6\) high spin configuration, there are usually 3 \(t_2\) electrons and 3 \(e\) electrons:\[\text{CFSE} = (-0.6 \times 3 + 0.4 \times 3) \Delta_t = (-1.8 + 1.2) \Delta_t = -0.6 \Delta_t\]

Step 4: Selecting the Correct Option

Compare the calculated values to the given options:- Octahedral: \(-0.4 \Delta_o\)- Tetrahedral: \(-0.6 \Delta_t\)Option (a) matches both calculated values: (a) \(-0.4 \Delta_{0}\) and \(-0.6 \Delta_{\mathrm{t}}\).

Key Concepts

High Spin Complexes

High spin complexes occur when the energy gap between the split d-orbitals in a crystal field is small enough, allowing the electrons to minimize repulsions by spreading across more orbitals while maintaining higher unpaired electrons. These complexes are typical of weak field ligands, which do not split the d-orbitals significantly. Hence, the electrons occupy the higher energy electron orbitals before pairing in lower ones.
Advantages of high spin configurations include:

• Increased magnetic properties since more unpaired electrons result in higher magnetic moments.
• Lower energy cost because pairing energies are higher than the splitting energy offered by weak field ligands.

High spin complexes are often found in 3d transition metals due to smaller ligand field strengths.

Octahedral Field

In an octahedral field, ligands approach the central metal ion along the axes, causing the d-orbitals to split into two sets. The three lower energy orbitals are termed as and the two higher energy orbitals as . The energy difference between these sets is known as the crystal field splitting energy, denoted as Delta_o.
This splitting of orbitals leads to crystal field stabilization energy (CFSE) in high spin octahedral complexes, contributing significantly to the stability of the complex.
Key points about octahedral fields include:

• Typically strong ligand fields promote low spin configurations.
• A higher value of Delta_o suggests that the ligands in the complex significantly affect the energy levels of the metal ion.

Tetrahedral Field

Contrary to octahedral fields, in a tetrahedral field, the ligands approach the metal ion between the axes. This arrangement results in a reverse crystal field splitting pattern where the lower energy states are the orbitals, and the higher are the orbitals. The crystal field splitting energy here is denoted as Delta_t, which is typically smaller compared to Delta_o.
In high spin tetrahedral complexes:

• The smaller Delta_t leads to a high spin state as electrons prefer not to pair up, opting instead to occupy the higher energy orbitals.
• Most tetrahedral complexes are high spin because Delta_t is usually less than the electron pairing energy.

This results in tetrahedral complexes often exhibiting more vibrant colors and weaker magnetic properties compared to their octahedral counterparts.