Step 1: Understanding the Concept:
Magnetic properties of transition metal complexes arise primarily from the unpaired electrons in their d-orbitals.
Paramagnetism is the property where an ion is attracted by an external magnetic field, and its strength is quantified by the magnetic moment (\(\mu\)).
The "spin-only" magnetic moment assumes that the orbital angular momentum is "quenched" (cancelled out) by the electric field of the surrounding ligands, leaving only the electron spin to contribute to the magnetic field.
To calculate this, we must first determine the electronic configuration and then the number of unpaired electrons (\(n\)).
Step 2: Key Formula or Approach:
The spin-only magnetic moment (\(\mu\)) is calculated using the formula:
\[ \mu = \sqrt{n(n+2)} \text{ Bohr Magnetons (BM)} \]
Where:
\( n \) = total number of unpaired electrons in the valence d-subshell.
Step 3: Detailed Explanation:
1. Find the configuration of neutral Mn:
Manganese has an atomic number \( Z = 25 \). Following the Aufbau principle, its ground-state configuration is:
\( Mn = [Ar] 3d^5 4s^2 \)
2. Determine the configuration of the \(Mn^{2+}\) ion:
Transition metals lose electrons from the highest principal quantum number shell (the 4s orbital) before losing electrons from the 3d shell.
To form the \(Mn^{2+}\) ion, we remove the two 4s electrons:
\( Mn^{2+} = [Ar] 3d^5 4s^0 \)
3. Count unpaired electrons using Hund's Rule:
The d-subshell consists of five degenerate (equal energy) orbitals. According to Hund's Rule of Maximum Multiplicity, electrons fill these orbitals singly first with parallel spins.
Distributing 5 electrons across 5 orbitals gives exactly one electron per orbital.
Thus, the number of unpaired electrons \( n = 5 \).
4. Calculate the magnetic moment:
Substitute \( n = 5 \) into the spin-only formula:
\[ \mu = \sqrt{5(5 + 2)} = \sqrt{5 \times 7} = \sqrt{35} \text{ BM} \]
5. Approximate the value:
We know that \( \sqrt{36} = 6.0 \). Since 35 is slightly less than 36, the square root will be slightly less than 6.
Calculating the value precisely: \( \sqrt{35} \approx 5.916 \).
Rounding to the nearest provided option gives 5.92 BM.
Step 4: Final Answer:
The spin-only magnetic moment for a divalent manganese ion with 5 unpaired electrons is 5.92 BM.
Hence, the correct answer is (D).