The correct order of the spin - only magnetic moments of the following complexes is
(I) `[Cr(H_(2)O)_(5)]Br_(2)`
(II) `Na_(4)[Fe(CN)_(6)]`
(III) `Na_(3)[Fe(C_(2)O_(4))_(3)](Delta_(0) gtP)`
(IV) `(Et_(4)N)_(4)[CoCl_(4)]`

Options

A. (I) gt (IV) gt (III) gt (II)
B. (II) `~~` (I) gt (IV) gt (III)
C. (III) gt (I) gt (IV) gt (III)
D. (III) gt (I) gt (II) gt (IV)

Answer

To determine the correct order of the spin-only magnetic moments for the given complexes, we will follow these steps:

### Step 1: Identify the oxidation states and electronic configurations of the central metal ions.

1. **For `[Cr(H₂O)₅]Br₂`:**
- The oxidation state of Cr can be calculated as follows:
\[ x + 0 + 2(-1) = 0 \implies x = +2 \]
- The electronic configuration of Cr in +2 state (3d⁴) is:
\[ [Ar] 3d^4 \]
- Since H₂O is a weak field ligand, there will be no pairing of electrons. Thus, the number of unpaired electrons (n) = 4.

2. **For `Na₄[Fe(CN)₆]`:**
- The oxidation state of Fe can be calculated as follows:
\[ +4 + x + 6(-1) = 0 \implies x = +2 \]
- The electronic configuration of Fe in +2 state (3d⁶) is:
\[ [Ar] 3d^6 \]
- CN⁻ is a strong field ligand, leading to pairing of electrons. Therefore, the number of unpaired electrons (n) = 0.

3. **For `Na₃[Fe(C₂O₄)₃]`:**
- The oxidation state of Fe can be calculated as follows:
\[ 3 + x + 3(-2) = 0 \implies x = +3 \]
- The electronic configuration of Fe in +3 state (3d⁵) is:
\[ [Ar] 3d^5 \]
- Given that Δ₀ > P (pairing energy), pairing will occur. Thus, the number of unpaired electrons (n) = 1.

4. **For `(Et₄N)₄[CoCl₄]`:**
- The oxidation state of Co can be calculated as follows:
\[ 0 + x + 4(-1) = 0 \implies x = +4 \]
- The electronic configuration of Co in +4 state (3d⁶) is:
\[ [Ar] 3d^6 \]
- Cl⁻ is a weak field ligand, so there will be no pairing. Therefore, the number of unpaired electrons (n) = 4.

### Step 2: Calculate the spin-only magnetic moments using the formula:
\[ \mu_{\text{spin only}} = \sqrt{n(n + 2)} \]

1. **For `[Cr(H₂O)₅]Br₂`:**
- \( n = 4 \)
\[ \mu = \sqrt{4(4 + 2)} = \sqrt{24} \approx 4.9 \, \mu_B \]

2. **For `Na₄[Fe(CN)₆]`:**
- \( n = 0 \)
\[ \mu = \sqrt{0(0 + 2)} = 0 \, \mu_B \]

3. **For `Na₃[Fe(C₂O₄)₃]`:**
- \( n = 1 \)
\[ \mu = \sqrt{1(1 + 2)} = \sqrt{3} \approx 1.73 \, \mu_B \]

4. **For `(Et₄N)₄[CoCl₄]`:**
- \( n = 3 \)
\[ \mu = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \, \mu_B \]

### Step 3: Order the complexes based on their magnetic moments:
- `[Cr(H₂O)₅]Br₂`: \( \approx 4.9 \, \mu_B \) (4 unpaired electrons)
- `(Et₄N)₄[CoCl₄]`: \( \approx 3.87 \, \mu_B \) (3 unpaired electrons)
- `Na₃[Fe(C₂O₄)₃]`: \( \approx 1.73 \, \mu_B \) (1 unpaired electron)
- `Na₄[Fe(CN)₆]`: \( 0 \, \mu_B \) (0 unpaired electrons)

### Final Order:
Thus, the correct order of the spin-only magnetic moments is:
**(I) > (IV) > (III) > (II)**