Arrange the following cyano complexes in decreasing order of their magnetic moment.

Options

A. `[Cr(CN)_(6)]^(3-)gt [Mn(CN)_(6)]^(3-)gt [Fe(CN)_(6)]^(3-) gt [Co(CN)_(6)]^(3-)`
B. `[Mn(CN)_(6)]^(3-) gt [Cr(CN)_(6)]^(3-) gt [Fe(CN)_(6)]^(3-) gt [Co(CN)_(6)]^(3-)`
C. `[Fe(CN)_(6)]^(3-) gt [Cr(CN)_(6)]^(3-) gt [Mn(CN)_(6)]^(3-) gt [Co(CN)_(6)]^(3-)`
D. `[Co(CN)_(6)]^(3-) gt [Cr(CN)_(6)]^(3-) gt [Mn(CN)_(6)^(3-) gt [Fe(CN)_(6)]^(3-)`

Answer

To arrange the given cyano complexes in decreasing order of their magnetic moment, we need to determine the number of unpaired electrons in each complex. The magnetic moment (μ) can be calculated using the formula:

\[ \mu = \sqrt{n(n + 2)} \text{ B.M.} \]

where \( n \) is the number of unpaired electrons.

### Step-by-Step Solution:

1. **Identify the oxidation states and electron configurations of the metal ions in the complexes:**
- **Cr(CN)₆³⁻**: Chromium (Cr) has an atomic number of 24. In the +3 oxidation state, its electron configuration is [Ar] 3d³.
- **Mn(CN)₆³⁻**: Manganese (Mn) has an atomic number of 25. In the +3 oxidation state, its electron configuration is [Ar] 3d⁴.
- **Fe(CN)₆³⁻**: Iron (Fe) has an atomic number of 26. In the +3 oxidation state, its electron configuration is [Ar] 3d⁵.
- **Co(CN)₆³⁻**: Cobalt (Co) has an atomic number of 27. In the +3 oxidation state, its electron configuration is [Ar] 3d⁶.

2. **Determine the number of unpaired electrons for each complex:**
- **Cr(CN)₆³⁻**: CN⁻ is a strong field ligand, which causes pairing of electrons. The 3d³ configuration will fill the T2g orbitals:
- T2g: ↑↓ ↑↓ ↑ (3 electrons in T2g, 0 in Eg)
- **Unpaired electrons (n)** = 3
- Magnetic moment: \(\mu = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \text{ B.M.}\)

- **Mn(CN)₆³⁻**: With a 3d⁴ configuration, the strong field ligand causes pairing:
- T2g: ↑↓ ↑↓ ↑ (4 electrons in T2g, 1 in Eg)
- **Unpaired electrons (n)** = 2
- Magnetic moment: \(\mu = \sqrt{2(2 + 2)} = \sqrt{8} \approx 2.83 \text{ B.M.}\)

- **Fe(CN)₆³⁻**: With a 3d⁵ configuration:
- T2g: ↑↓ ↑↓ ↑ (5 electrons in T2g, 0 in Eg)
- **Unpaired electrons (n)** = 1
- Magnetic moment: \(\mu = \sqrt{1(1 + 2)} = \sqrt{3} \approx 1.73 \text{ B.M.}\)

- **Co(CN)₆³⁻**: With a 3d⁶ configuration:
- T2g: ↑↓ ↑↓ ↑↓ (6 electrons in T2g, 0 in Eg)
- **Unpaired electrons (n)** = 0
- Magnetic moment: \(\mu = \sqrt{0(0 + 2)} = 0 \text{ B.M.}\)

3. **Arrange the complexes in decreasing order of their magnetic moments:**
- Cr(CN)₆³⁻: 3.87 B.M.
- Mn(CN)₆³⁻: 2.83 B.M.
- Fe(CN)₆³⁻: 1.73 B.M.
- Co(CN)₆³⁻: 0 B.M.

### Final Order:
1. Cr(CN)₆³⁻ (3.87 B.M.)
2. Mn(CN)₆³⁻ (2.83 B.M.)
3. Fe(CN)₆³⁻ (1.73 B.M.)
4. Co(CN)₆³⁻ (0 B.M.)