Question

The total number of unpaired electrons present in the $d^3$, $d^4$ (low spin), $d^5$ (high spin), $d^6$ (high spin) and $d^7$ (low spin) octahedral complex systems is ______.

Answer 1

Correct Answer: 15

Step 1: Understanding the Concept:
In an octahedral crystal field, the five d-orbitals split into two sets: the lower energy $t_{2g}$ set (consisting of $d_{xy}, d_{yz}, d_{zx}$) and the higher energy $e_g$ set (consisting of $d_{x^2-y^2}, d_{z^2}$).
The distribution of electrons depends on the crystal field splitting energy ($\Delta_o$) relative to the pairing energy (P).
: Key Formula or Approach:
For high spin (weak field), electrons fill all orbitals singly before pairing starts (Hund's rule).
For low spin (strong field), electrons fill the lower energy $t_{2g}$ orbitals completely before moving to $e_g$.
Step 2: Detailed Explanation:
We analyze each electronic configuration:
1. $d^3$: Electrons fill $t_{2g}$ orbitals singly.
Configuration: $t_{2g}^3 e_g^0$. Unpaired electrons ($n$) = 3.
2. $d^4$ (low spin): Electrons stay in $t_{2g}$ and pair up.
Configuration: $t_{2g}^4 e_g^0$. Two electrons are unpaired, one pair is formed ($n$) = 2.
3. $d^5$ (high spin): Electrons fill all five orbitals singly.
Configuration: $t_{2g}^3 e_g^2$. Unpaired electrons ($n$) = 5.
4. $d^6$ (high spin): Five electrons are unpaired, the sixth pairs in $t_{2g}$.
Configuration: $t_{2g}^4 e_g^2$. Unpaired electrons ($n$) = 4.
5. $d^7$ (low spin): $t_{2g}$ is completely filled (6 electrons), the 7th is in $e_g$.
Configuration: $t_{2g}^6 e_g^1$. Unpaired electrons ($n$) = 1.

Total unpaired electrons = $3 + 2 + 5 + 4 + 1 = 15$.
Step 3: Final Answer:
The total number of unpaired electrons is 15.