CBSE · Class 12 · Chemistry · Coordination CompoundsCalculate the magnetic moment (spin-only) of $Mn(H2O)6^{2+}$ complex ion. Given that the atomic number of Manganese ($Mn$) is $25$. Also, explain the crystal field splitting in octahedral complexes with a neat labeled description.

Part 1: Calculation of Magnetic Moment

1. Find the Electronic Configuration of Manganese ($Mn$):

   • Atomic number of $Mn = 25$
   • Ground state electronic configuration of neutral $Mn$ atom: $[Ar] 3d^5 4s^2$

2. Determine the Oxidation State of $Mn$ in $[Mn(H_2O)_6]^{2+}$:

   • Let the oxidation state of $Mn$ be $x$.
   • Water ($H_2O$) is a neutral ligand, so its charge is $0$.
   • $x + 6(0) = +2 \implies x = +2$
   • Electronic configuration of $Mn^{2+}$ ion: $[Ar] 3d^5 4s^0$ or simply $3d^5$.

3. Determine the Number of Unpaired Electrons ($n$):

   • $H_2O$ is a weak field ligand, so it does not cause pairing of electrons.
   • In a $3d^5$ configuration, according to Hund's rule, all five $d$-orbitals will singly occupy one electron each.
   • Therefore, the number of unpaired electrons ($n$) = $5$.

4. Calculate Spin-Only Magnetic Moment ($\mu$):

   • The formula for spin-only magnetic moment is: $$\mu = \sqrt{n(n + 2)} \text{ BM}$$ (Bohr Magneton)
   • Substitute $n = 5$ into the formula: $$\mu = \sqrt{5(5 + 2)}$$ $$\mu = \sqrt{5(7)}$$ $$\mu = \sqrt{35}$$ $$\mu \approx 5.92 \text{ BM}$|

Part 2: Crystal Field Splitting in Octahedral Complexes

• Degeneracy of $d$-Orbitals: In an isolated, gaseous transition metal ion, all five $d$-orbitals ($dxy$, $dyz$, $dxz$, $dx^2-y^2$, and $dz^2$) are degenerate, meaning they possess the exact same energy.
• Approach of Ligands: When six ligands approach the central metal ion along the Cartesian axes ($x, y, z$) to form an octahedral complex, the electrostatic field of the negatively charged or dipolar ligands destroys the spherical symmetry.
• Barycenter and Splitting: The average energy of these $d$-orbitals rises due to repulsion, forming a hypothetical state called the barycenter. From this barycenter, the orbitals split into two distinct sets due to directional preferences:
  • $e_g$ Set: The $dx^2-y^2$ and $dz^2$ orbitals lie directly along the axes where the ligands approach. Consequently, they experience maximum electrostatic repulsion and are raised in energy.
  • $t_{2g}$ Set: The $dxy$, $dyz$, and $dxz$ orbitals lie between the axes. They experience lesser repulsion from the ligands and are thus lowered in energy relative to the barycenter.
• Energy Distribution: The $t_{2g}$ orbitals are stabilized by an energy of $-0.4 \Delta_o$ (or $-2/5 \Delta_o$), and the $e_g$ orbitals are destabilized by an energy of $+0.6 \Delta_o$ (or $+3/5 \Delta_o$), where $\Delta_o$ represents the crystal field splitting energy for octahedral complexes.