1. Find oxidation state and electronic configuration of the metal in each complex

We need the spin-only magnetic moment, which depends on the number of unpaired electrons:

$\mu = \sqrt{n(n+2)}\;\text{BM}$

where $n$ = number of unpaired electrons.

2. Complex (A): $Ni(CO)_4$

• $CO$ is a neutral ligand.
• So oxidation state of $Ni$ is $0$.
• Atomic number of $Ni = 28$

$Ni: [Ar]3d^8 4s^2$

For $Ni^0$, electrons become:

$3d^{10}$

In $Ni(CO)_4$, nickel is tetrahedral but with $d^{10}$ configuration.

• All electrons are paired.
• Number of unpaired electrons $n=0$

So,

$\mu = 0$

3. Complex (B): $[Ni(H_2O)_6]Cl_2$

This is actually $[Ni(H_2O)_6]^{2+}$ with two $Cl^-$ outside.

• $H_2O$ is neutral.
• Oxidation state of $Ni$ is $+2$.

Thus,

$Ni^{2+}: 3d^8$

$H_2O$ is a weak-field ligand, so octahedral complex is high spin.

For octahedral $d^8$:

$t_{2g}^6 e_g^2$

This gives 2 unpaired electrons.

Hence,

$\mu = \sqrt{2(2+2)} = \sqrt{8}$

So (B) is paramagnetic.

4. Complex (C): $Na_2[Ni(CN)_4]$

The complex ion is $[Ni(CN)_4]^{2-}$.

• Let oxidation state of $Ni$ be $x$:

$x + 4(-1) = -2 \Rightarrow x=+2$

So metal is $Ni^{2+} = 3d^8$.

$CN^-$ is a strong-field ligand, so electrons pair up.

For $d^8$ with strong field in 4-coordinate complex, geometry is square planar.

• Square planar $d^8$ complexes are diamagnetic.
• Number of unpaired electrons $n=0$

Thus,

$\mu = 0$

5. Complex (D): $PdCl_2(PPh_3)_2$

• $PPh_3$ is neutral.
• $Cl^-$ is $-1$ each.
• Hence oxidation state of $Pd$ is $+2$.

So,

$Pd^{2+}: 4d^8$

For $4d$ metals, pairing is stronger, and $d^8$ complexes commonly form square planar structures. Thus all electrons are paired.

• Number of unpaired electrons $n=0$
• Therefore,

$\mu = 0$

6. Compare magnetic moments

• (A): $0$
• (B): $\sqrt{8}$
• (C): $0$
• (D): $0$

Therefore,

$(A) \approx (C) \approx (D) < (B)$

7. Match with options

This corresponds to:

Option C: $(A) \approx (C) \approx (D) < (B)$

(Note: the printed option order in the image-style statement appears slightly unusual compared to increasing order, but among the given options, C is the one expressing that $A, C, D$ have equal lower magnetic moments and $B$ is higher.)