Which one of the cyano complexes would exhibit the lowest value of para magnetic behaviour ?
(At. No. Cr = 24, Mn = 25, Fe = 26, Co = 27)

Options

A. `[Co(CN)_6]^(3-)`
B. `[Fe(CN)_6]^(3-)`
C. `[Mn(CN)_6]^(3-)`
D. `[Cr(CN)_6]^(3-)`

Answer

To determine which cyano complex exhibits the lowest value of paramagnetic behavior, we need to analyze the oxidation states and electronic configurations of the metal ions in each complex, as well as the effect of the cyanide ligand (CN⁻) on electron pairing.

### Step-by-Step Solution:

1. **Identify the complexes and their oxidation states**:
- The complexes in question are:
1. \( \text{Co(CN)}_6^{3-} \)
2. \( \text{Fe(CN)}_6^{3-} \)
3. \( \text{Mn(CN)}_6^{3-} \)
4. \( \text{Cr(CN)}_6^{3-} \)

- For each complex, the oxidation state of the metal can be determined:
- For \( \text{Co(CN)}_6^{3-} \): Cobalt is in +3 oxidation state.
- For \( \text{Fe(CN)}_6^{3-} \): Iron is in +3 oxidation state.
- For \( \text{Mn(CN)}_6^{3-} \): Manganese is in +3 oxidation state.
- For \( \text{Cr(CN)}_6^{3-} \): Chromium is in +3 oxidation state.

2. **Determine the electronic configurations**:
- The electronic configurations of the metals in their +3 oxidation states are:
- Cobalt (Co): Atomic number 27 → \( \text{[Ar]} 3d^6 \) → \( 3d^6 \) in +3 state becomes \( 3d^6 \) with all paired electrons (due to strong field ligand).
- Iron (Fe): Atomic number 26 → \( \text{[Ar]} 3d^6 4s^2 \) → \( 3d^5 \) in +3 state has 5 electrons, 1 unpaired electron.
- Manganese (Mn): Atomic number 25 → \( \text{[Ar]} 3d^5 4s^2 \) → \( 3d^4 \) in +3 state has 4 electrons, 2 unpaired electrons.
- Chromium (Cr): Atomic number 24 → \( \text{[Ar]} 3d^5 4s^1 \) → \( 3d^3 \) in +3 state has 3 electrons, 3 unpaired electrons.

3. **Count unpaired electrons**:
- \( \text{Co(CN)}_6^{3-} \): 0 unpaired electrons (paramagnetic behavior = 0).
- \( \text{Fe(CN)}_6^{3-} \): 1 unpaired electron (paramagnetic behavior).
- \( \text{Mn(CN)}_6^{3-} \): 4 unpaired electrons (paramagnetic behavior).
- \( \text{Cr(CN)}_6^{3-} \): 3 unpaired electrons (paramagnetic behavior).

4. **Determine the magnetic moments**:
- The magnetic moment (μ) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
- Where \( n \) is the number of unpaired electrons:
- \( \text{Co(CN)}_6^{3-} \): \( \mu = 0 \)
- \( \text{Fe(CN)}_6^{3-} \): \( \mu = \sqrt{1(1 + 2)} = \sqrt{3} \)
- \( \text{Mn(CN)}_6^{3-} \): \( \mu = \sqrt{4(4 + 2)} = \sqrt{24} \)
- \( \text{Cr(CN)}_6^{3-} \): \( \mu = \sqrt{3(3 + 2)} = \sqrt{15} \)

5. **Conclusion**:
- The complex with the lowest value of paramagnetic behavior is \( \text{Co(CN)}_6^{3-} \) because it has no unpaired electrons.

### Final Answer:
The cyano complex that exhibits the lowest value of paramagnetic behavior is \( \text{Co(CN)}_6^{3-} \).