The complex showing a spin -magnetic momnet of `2.82 BM` is .

Options

A. `Ni(CO)_(4)`
B. `[NiCI_(4)]^(2-)`
C. `Ni(PPh_(3))_(4)`
D. `[Ni(CN)_(4)]^(2-)`

Explanation

To solve the problem of identifying the complex that shows a spin magnetic moment of 2.82 Bohr magnetons (BM), we can follow these steps:

### Step 1: Understand Magnetic Moment
The magnetic moment (μ) of a complex can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Determine the Value of n
Given that the magnetic moment is 2.82 BM, we can set up the equation:
\[ 2.82 = \sqrt{n(n + 2)} \]
Squaring both sides gives:
\[ (2.82)^2 = n(n + 2) \]
Calculating \( (2.82)^2 \):
\[ 7.9524 = n(n + 2) \]

### Step 3: Solve the Quadratic Equation
Rearranging the equation:
\[ n^2 + 2n - 7.9524 = 0 \]
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, c = -7.9524 \):
\[ n = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-7.9524)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ n = \frac{-2 \pm \sqrt{4 + 31.8096}}{2} \]
\[ n = \frac{-2 \pm \sqrt{35.8096}}{2} \]
Calculating \( \sqrt{35.8096} \):
\[ n = \frac{-2 \pm 5.983}{2} \]
This gives two possible values for \( n \):
1. \( n = \frac{3.983}{2} \approx 1.9915 \) (approximately 2)
2. \( n = \frac{-7.983}{2} \) (not a valid solution since n cannot be negative)

Thus, \( n \) is approximately 2.

### Step 4: Identify the Complex
Now that we know there are 2 unpaired electrons, we can analyze the types of ligands:
- Weak field ligands (like Cl⁻) do not cause pairing of electrons, allowing for unpaired electrons, while strong field ligands (like CN⁻, CO) tend to cause pairing.

Given that we have 2 unpaired electrons, we can conclude that the complex is likely to contain a weak field ligand.

### Step 5: Conclusion
The complex that shows a spin magnetic moment of 2.82 BM is likely one that contains nickel in the +2 oxidation state with weak field ligands like chloride (Cl⁻).

### Final Answer
The complex showing a spin magnetic moment of 2.82 BM is likely a nickel(II) complex with chloride ligands.

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