Magnetic Moment Calculation in Coordination Complexes

The magnetic moment of a transition metal complex is primarily determined by the number of unpaired electrons present in the central metal ion. The spin-only magnetic moment ($\mu_{\text{s}}$) is calculated using the formula:

$$\mu_{\text{s}} = \sqrt{n(n+2)} \text{ BM}$$

where $n$ is the number of unpaired electrons and BM stands for Bohr Magnetons.

To determine the number of unpaired electrons ($n$), we need to:

1. Determine the oxidation state of the central metal atom.
2. Write the electronic configuration of the metal ion.
3. Consider the crystal field splitting of the d-orbitals (octahedral or tetrahedral).
4. Consider the strength of the ligands (weak field or strong field), which affects electron pairing according to Hund's rule and crystal field stabilization energy.

Let's analyze each pair provided in the options.

Analyzing Option 1: [CoCl₄]^2- and [Ni(H₂O)₆]²⁺

• [CoCl₄]^2-:
  • Cobalt oxidation state: +2. Configuration: $\text{Co}^{2+}$, $3\text{d}^7$.
  • Ligand (Cl^-): Weak field ligand. Geometry: Tetrahedral.
  • In a tetrahedral field, d-orbitals split into lower energy e set (2 orbitals) and higher energy t₂ set (3 orbitals).
  • Electron filling in weak field (Hund's rule followed):

    Level	Orbitals	Electrons
    t2	3	$\uparrow \uparrow \uparrow$
    e	2	$\uparrow\downarrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 3.
  • Magnetic moment: $\mu = \sqrt{3(3+2)} = \sqrt{15}$ BM.
• [Ni(H₂O)₆]²⁺:
  • Nickel oxidation state: +2. Configuration: $\text{Ni}^{2+}$, $3\text{d}^8$.
  • Ligand (H₂O): Weak field ligand. Geometry: Octahedral.
  • In an octahedral field, d-orbitals split into lower energy t_2g set (3 orbitals) and higher energy e_g set (2 orbitals).
  • Electron filling in weak field (Hund's rule followed):

    Level	Orbitals	Electrons
    eg	2	$\uparrow \uparrow$
    t2g	3	$\uparrow\downarrow \uparrow\downarrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 2.
  • Magnetic moment: $\mu = \sqrt{2(2+2)} = \sqrt{8}$ BM.
• The magnetic moments ($\sqrt{15}$ BM and $\sqrt{8}$ BM) are different.

Analyzing Option 2: [NiCl₄]^2- and [CoF₆]^3-

• [NiCl₄]^2-:
  • Nickel oxidation state: +2. Configuration: $\text{Ni}^{2+}$, $3\text{d}^8$.
  • Ligand (Cl^-): Weak field ligand. Geometry: Tetrahedral.
  • Electron filling in tetrahedral weak field ($3\text{d}^8$):

    Level	Orbitals	Electrons
    t2	3	$\uparrow \uparrow \uparrow$
    e	2	$\uparrow\downarrow \uparrow\downarrow$

    
    
  • Number of unpaired electrons ($n$) = 2.
  • Magnetic moment: $\mu = \sqrt{2(2+2)} = \sqrt{8}$ BM.
• [CoF₆]^3-:
  • Cobalt oxidation state: +3. Configuration: $\text{Co}^{3+}$, $3\text{d}^6$.
  • Ligand (F^-): Weak field ligand. Geometry: Octahedral.
  • Electron filling in octahedral weak field ($3\text{d}^6$):

    Level	Orbitals	Electrons
    eg	2	$\uparrow \uparrow$
    t2g	3	$\uparrow \uparrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 4.
  • Magnetic moment: $\mu = \sqrt{4(4+2)} = \sqrt{24}$ BM.
• The magnetic moments ($\sqrt{8}$ BM and $\sqrt{24}$ BM) are different.

Analyzing Option 3: [Co(H₂O)₆]²⁺ and [Cr(H₂O)₆]³⁺

• [Co(H₂O)₆]²⁺:
  • Cobalt oxidation state: +2. Configuration: $\text{Co}^{2+}$, $3\text{d}^7$.
  • Ligand (H₂O): Weak field ligand. Geometry: Octahedral.
  • Electron filling in octahedral weak field ($3\text{d}^7$):

    Level	Orbitals	Electrons
    eg	2	$\uparrow \uparrow$
    t2g	3	$\uparrow\downarrow \uparrow\downarrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 3.
  • Magnetic moment: $\mu = \sqrt{3(3+2)} = \sqrt{15}$ BM.
• [Cr(H₂O)₆]³⁺:
  • Chromium oxidation state: +3. Configuration: $\text{Cr}^{3+}$, $3\text{d}^3$.
  • Ligand (H₂O): Weak field ligand. Geometry: Octahedral.
  • For $3\text{d}^3$ configuration in octahedral complexes, regardless of ligand field strength (weak or strong), the electrons will occupy the lower energy t_2g orbitals unpaired before pairing or occupying the higher energy e_g orbitals.
  • Electron filling in octahedral field ($3\text{d}^3$):

    Level	Orbitals	Electrons
    eg	2
    t2g	3	$\uparrow \uparrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 3.
  • Magnetic moment: $\mu = \sqrt{3(3+2)} = \sqrt{15}$ BM.
• The magnetic moments ($\sqrt{15}$ BM and $\sqrt{15}$ BM) are the same.

Analyzing Option 4: [Ni(NH₃)₆]²⁺ and [CrCl₄]^2-

• [Ni(NH₃)₆]²⁺:
  • Nickel oxidation state: +2. Configuration: $\text{Ni}^{2+}$, $3\text{d}^8$.
  • Ligand (NH₃): Strong field ligand, but for $\text{d}^8$ octahedral complexes, pairing energy is less than crystal field splitting energy ($\text{P} < \Delta_{\text{o}}$) for most strong field ligands, resulting in a high spin complex.
  • Electron filling in octahedral field ($3\text{d}^8$):

    Level	Orbitals	Electrons
    eg	2	$\uparrow \uparrow$
    t2g	3	$\uparrow\downarrow \uparrow\downarrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 2.
  • Magnetic moment: $\mu = \sqrt{2(2+2)} = \sqrt{8}$ BM.
• [CrCl₄]^2-:
  • Chromium oxidation state: +2. Configuration: $\text{Cr}^{2+}$, $3\text{d}^4$.
  • Ligand (Cl^-): Weak field ligand. Geometry: Tetrahedral.
  • Electron filling in tetrahedral weak field ($3\text{d}^4$):

    Level	Orbitals	Electrons
    t2	3	$\uparrow \uparrow$
    e	2	$\uparrow \uparrow$

    
    
  • Number of unpaired electrons ($n$) = 4.
  • Magnetic moment: $\mu = \sqrt{4(4+2)} = \sqrt{24}$ BM.
• The magnetic moments ($\sqrt{8}$ BM and $\sqrt{24}$ BM) are different.

Based on the analysis, the pair $[\text{Co(H}_2\text{O})_6]^{2+}$ and $[\text{Cr(H}_2\text{O})_6]^{3+}$ both have 3 unpaired electrons and thus the same magnetic moment ($\sqrt{15}$ BM).