Total Number of Unpaired Electrons in Octahedral Complex Systems

To find the total number of unpaired electrons in the given octahedral complex systems, we analyze the crystal field splitting of $d$-orbitals in an octahedral field, where the $d$-orbitals split into a lower-energy triply degenerate set ($t_{2g}$) and a higher-energy doubly degenerate set ($e_g$).

1. $\text{d}^3$ System:

   • Electrons enter the $t_{2g}$ set singly according to Hund's rule.
   • Electronic configuration: $t_{2g}^3 \, e_g^0$
   • Number of unpaired electrons ($n_1$) = $3$

2. $\text{d}^4$ (low spin) System:

   • For a low-spin configuration, electron pairing energy is less than the crystal field splitting energy ($\Delta_o > P$), so electrons pair up in the $t_{2g}$ orbitals before occupying the $e_g$ orbitals.
   • Electronic configuration: $t_{2g}^4 \, e_g^0$
   • Number of unpaired electrons ($n_2$) = $2$

3. $\text{d}^5$ (high spin) System:

   • For a high-spin configuration ($\Delta_o < P$), maximum multiplicity is maintained by placing electrons in both $t_{2g}$ and $e_g$ orbitals prior to pairing.
   • Electronic configuration: $t_{2g}^3 \, e_g^2$
   • Number of unpaired electrons ($n_3$) = $5$

4. $\text{d}^6$ (high spin) System:

   • Electrons fill the subshells to maximize total spin: five electrons singly occupy all five $d$-orbitals, and the sixth electron pairs up in a $t_{2g}$ orbital.
   • Electronic configuration: $t_{2g}^4 \, e_g^2$
   • Number of unpaired electrons ($n_4$) = $4$

5. $\text{d}^7$ (low spin) System:

   • Six electrons fully pair up to fill the $t_{2g}$ set, and the seventh electron occupies an $e_g$ orbital.
   • Electronic configuration: $t_{2g}^6 \, e_g^1$
   • Number of unpaired electrons ($n_5$) = $1$

Total Number of Unpaired Electrons: $\text{Total} = n_1 + n_2 + n_3 + n_4 + n_5$ $\text{Total} = 3 + 2 + 5 + 4 + 1 = 15$