Which one of the following has lowest value of paramagnetic behaviour?

Options

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

Explanation

To determine which compound has the lowest value of paramagnetic behavior, we need to analyze the number of unpaired electrons in each of the given coordination compounds. The magnetic moment (μ) can be calculated using the formula:

\[ \mu = \sqrt{n(n + 2)} \]

where \( n \) is the number of unpaired electrons. The more unpaired electrons there are, the higher the magnetic moment and thus the stronger the paramagnetic behavior.

### Step-by-Step Solution:

1. **Identify the Coordination Compounds**:
We need to analyze the following coordination compounds:
- \( \text{[Cr(CN)_6]}^{3-} \)
- \( \text{[Co(CN)_6]}^{3+} \)
- \( \text{[Fe(CN)_6]}^{3-} \)
- \( \text{[Co(H}_2\text{O})_6]^{3+} \)

2. **Determine the Oxidation States and Electron Configurations**:
- For \( \text{[Cr(CN)_6]}^{3-} \):
- Chromium in +3 state: \( \text{Cr}^{3+} \) has an electron configuration of \( [Ar] 3d^3 \).
- Number of unpaired electrons \( n = 3 \).

- For \( \text{[Co(CN)_6]}^{3+} \):
- Cobalt in +3 state: \( \text{Co}^{3+} \) has an electron configuration of \( [Ar] 3d^6 \).
- Since CN is a strong field ligand, it pairs up electrons: \( n = 0 \).

- For \( \text{[Fe(CN)_6]}^{3-} \):
- Iron in +3 state: \( \text{Fe}^{3+} \) has an electron configuration of \( [Ar] 3d^5 \).
- Again, CN is a strong field ligand, so \( n = 0 \) (all electrons are paired).

- For \( \text{[Co(H}_2\text{O})_6]^{3+} \):
- Cobalt in +3 state: \( \text{Co}^{3+} \) has an electron configuration of \( [Ar] 3d^6 \).
- Water is a weak field ligand, so \( n = 4 \) (two unpaired electrons).

3. **Calculate the Magnetic Moments**:
- For \( \text{[Cr(CN)_6]}^{3-} \):
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \text{ (paramagnetic)} \]
- For \( \text{[Co(CN)_6]}^{3+} \):
\[ \mu = \sqrt{0(0 + 2)} = 0 \text{ (diamagnetic)} \]
- For \( \text{[Fe(CN)_6]}^{3-} \):
\[ \mu = \sqrt{0(0 + 2)} = 0 \text{ (diamagnetic)} \]
- For \( \text{[Co(H}_2\text{O})_6]^{3+} \):
\[ \mu = \sqrt{4(4 + 2)} = \sqrt{24} \text{ (paramagnetic)} \]

4. **Conclusion**:
The compound with the lowest value of paramagnetic behavior is \( \text{[Co(CN)_6]}^{3+} \) and \( \text{[Fe(CN)_6]}^{3-} \) as both have zero unpaired electrons and are diamagnetic.

### Final Answer:
**The compound with the lowest value of paramagnetic behavior is \( \text{[Co(CN)_6]}^{3+} \) or \( \text{[Fe(CN)_6]}^{3-} \).**