The correct increasing o rd e r of spin-only magnetic moment values of the complex ions $$[MnBr_{4}]^{2-}(A),[Cu(H_{2}O)_{6}]^{2+}(B),[Ni(CN)_{4}]^{2-}(C)\text{ and }[Ni(H_{2}O)_{6}]^{2+}(D)$$ is:

The spin-only magnetic moment ($$\mu$$) is calculated using the formula:

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

where $$n$$ is the number of unpaired electrons in the central metal ion of the complex. We need to determine $$n$$ for each complex ion.

The central metal ion is Mn²⁺. Manganese has an atomic number of 25, so Mn²⁺ has the electron configuration [Ar] 3d⁵. Br⁻ is a weak field ligand, and the complex is tetrahedral. In tetrahedral complexes, the crystal field splitting is small, resulting in a high spin configuration. For d⁵ in a tetrahedral field, all five electrons are unpaired. Thus, $$n = 5$$.

Magnetic moment: $$\mu_A = \sqrt{5 \times 7} = \sqrt{35} \approx 5.92 \text{ BM}$$.

The central metal ion is Cu²⁺. Copper has an atomic number of 29, so Cu²⁺ has the configuration [Ar] 3d⁹. H₂O is a weak field ligand, and the complex is octahedral. For d⁹ in an octahedral field, the configuration is t₂g⁶ eg³. The eg orbitals consist of two orbitals: one orbital has two electrons (paired), and the other has one electron (unpaired). Thus, $$n = 1$$.

Magnetic moment: $$\mu_B = \sqrt{1 \times 3} = \sqrt{3} \approx 1.73 \text{ BM}$$.

The central metal ion is Ni²⁺. Nickel has an atomic number of 28, so Ni²⁺ has the configuration [Ar] 3d⁸. CN⁻ is a strong field ligand, and the complex is square planar. In a square planar field for d⁸, all electrons are paired. Thus, $$n = 0$$.

Magnetic moment: $$\mu_C = \sqrt{0 \times 2} = 0 \text{ BM}$$.

The central metal ion is Ni²⁺ (d⁸). H₂O is a weak field ligand, and the complex is octahedral. This results in a high spin configuration. The configuration is t₂g⁶ eg², with two unpaired electrons in the eg orbitals. Thus, $$n = 2$$.

Magnetic moment: $$\mu_D = \sqrt{2 \times 4} = \sqrt{8} \approx 2.83 \text{ BM}$$.

Summary of magnetic moments:

• Complex A: 5.92 BM
• Complex B: 1.73 BM
• Complex C: 0 BM
• Complex D: 2.83 BM

Increasing order: Complex C (0 BM) < Complex B (1.73 BM) < Complex D (2.83 BM) < Complex A (5.92 BM), i.e., C < B < D < A.

Comparing with the options:

• A. A = B < C < D → Incorrect (A and B are not equal, and C is not greater than B)
• B. A = B < D < C → Incorrect (A and B are not equal, and D is not less than C)
• C. C = D < B < A → Incorrect (C and D are not equal)
• D. C < B < D < A → Correct

The correct option is D.