Velence bond theroy describes the bonding in complexs in terms of coordinate -covalent bond resulting from overlap filled ligand orbitals with vacant metal hybrid orbitals This theory explains magnetic behaviour and geometrical shape of coordination compounds Magnetic moment of a complex compound can be determined experimentally and theoretically by using spin only formula
Magnetic moment `sqrtn (n+2)BM` (where n = No. unpaired electrons) .
The value of of spin only magnetic moment for octahedral complex of the following configuration is `2.84BM` The correct statement is
(a) `d^(4)` (in weak field ligand)
(b) `d^(2)` (in weak field and in strong field ligand)
(c) `d^(3)` (in weak field and in strong field ligand)
(d) `d^(5)` (in strong field ligand) .

Explanation

To solve the problem, we need to determine which electronic configuration corresponds to the given magnetic moment of 2.84 BM for an octahedral complex. We'll use the spin-only magnetic moment formula:

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

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

### Step 1: Calculate the number of unpaired electrons for the given magnetic moment

Given that the magnetic moment \( \mu \) is 2.84 BM, we can set up the equation:

\[ \sqrt{n(n + 2)} = 2.84 \]

Squaring both sides gives:

\[ n(n + 2) = (2.84)^2 \]

Calculating \( (2.84)^2 \):

\[ (2.84)^2 = 8.0656 \]

So, we have:

\[ n(n + 2) = 8.0656 \]

### Step 2: Solve the quadratic equation

This can be rearranged into a standard quadratic equation:

\[ n^2 + 2n - 8.0656 = 0 \]

Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = 2, c = -8.0656 \).

Calculating the discriminant:

\[ b^2 - 4ac = 2^2 - 4 \times 1 \times (-8.0656) = 4 + 32.2624 = 36.2624 \]

Now, applying the quadratic formula:

\[ n = \frac{-2 \pm \sqrt{36.2624}}{2} \]

Calculating \( \sqrt{36.2624} \):

\[ \sqrt{36.2624} \approx 6.02 \]

Now substituting back:

\[ n = \frac{-2 \pm 6.02}{2} \]

Calculating the two possible values for \( n \):

1. \( n = \frac{4.02}{2} = 2.01 \) (not possible since \( n \) must be an integer)
2. \( n = \frac{-8.02}{2} = -4.01 \) (not possible)

Since \( n \) must be a whole number, we can conclude that \( n = 2 \) is the only feasible solution.

### Step 3: Identify the correct electronic configuration

Now we need to check the options provided:

- **(a)** \( d^4 \) (in weak field ligand): This would have 4 unpaired electrons in a weak field, giving \( \mu = \sqrt{4(4 + 2)} = \sqrt{24} \approx 4.9 \) BM.

- **(b)** \( d^2 \) (in weak field and in strong field ligand): This would have 2 unpaired electrons in both cases, giving \( \mu = \sqrt{2(2 + 2)} = \sqrt{8} \approx 2.84 \) BM.

- **(c)** \( d^3 \) (in weak field and in strong field ligand): This would have 3 unpaired electrons, giving \( \mu = \sqrt{3(3 + 2)} = \sqrt{15} \approx 3.87 \) BM.

- **(d)** \( d^5 \) (in strong field ligand): This would have 1 unpaired electron, giving \( \mu = \sqrt{1(1 + 2)} = \sqrt{3} \approx 1.73 \) BM.

### Conclusion

The only configuration that gives a magnetic moment of approximately 2.84 BM is option **(b)** \( d^2 \) in both weak and strong field ligands.

Thus, the correct answer is **(b)** \( d^2 \) (in weak field and in strong field ligand).

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