1. Find oxidation state and d-electron count for each complex

The spin-only magnetic moment is $\mu = \sqrt{n(n+2)}\ \text{BM}$ where $n$ is the number of unpaired electrons. So we first determine $n$ for each complex.

2. Complex (I): $[Cr(H_2O)_6]Br_2$

Since $Br^-$ are counter ions, the complex cation is $[Cr(H_2O)_6]^{2+}$.

• Oxidation state of $Cr = +2$
• $Cr: [Ar]3d^5 4s^1$
• $Cr^{2+} \Rightarrow 3d^4$

$H_2O$ is a weak field ligand, so octahedral high-spin configuration: $d^4 \Rightarrow t_{2g}^3 e_g^1$

Number of unpaired electrons: $n=4$

Thus, $\mu_I = \sqrt{4(4+2)} = \sqrt{24}$

3. Complex (II): $Na_4[Fe(CN)_6]$

The complex ion is $[Fe(CN)_6]^{4-}$.

Let oxidation state of Fe be $x$: $x+6(-1)=-4 \Rightarrow x=+2$

So $Fe^{2+}$ is $3d^6$.

$CN^-$ is a strong field ligand, so octahedral low-spin: $d^6 \Rightarrow t_{2g}^6 e_g^0$

Number of unpaired electrons: $n=0$

Thus, $\mu_{II}=0$

4. Complex (III): $Na_3[Fe(C_2O_4)_3]$ with $\Delta_o>P$

The complex ion is $[Fe(C_2O_4)_3]^{3-}$.

Each oxalate ligand $C_2O_4^{2-}$ has charge $-2$. Let oxidation state of Fe be $x$: $x+3(-2)=-3 \Rightarrow x=+3$

So $Fe^{3+}$ is $3d^5$.

Given $\Delta_o>P$, it is low spin octahedral: $d^5 \Rightarrow t_{2g}^5 e_g^0$

Number of unpaired electrons: $n=1$

Thus, $\mu_{III} = \sqrt{1(1+2)}=\sqrt{3}$

5. Complex (IV): $(Et_4N)_2[CoCl_4]$

The complex ion is $[CoCl_4]^{2-}$.

Let oxidation state of Co be $x$: $x+4(-1)=-2 \Rightarrow x=+2$

So $Co^{2+}$ is $3d^7$.

$Cl^-$ is a weak field ligand, and tetrahedral complexes are generally high spin.

For tetrahedral $d^7$ high spin, number of unpaired electrons: $n=3$

Thus, $\mu_{IV} = \sqrt{3(3+2)} = \sqrt{15}$

6. Compare magnetic moments

• (I): $n=4$, $\mu=\sqrt{24}$
• (IV): $n=3$, $\mu=\sqrt{15}$
• (III): $n=1$, $\mu=\sqrt{3}$
• (II): $n=0$, $\mu=0$

Therefore, $\boxed{(I)>(IV)>(III)>(II)}$

7. Match with options

This corresponds to Option D.

8. Comparison with stored correct answer

Stored correct answer: D

Our derived answer: D

So the derived answer agrees with the stored answer.