The magnetic moment of a complex ion is `2.83BM` The complex ion is
(a)`[Cr(H_(2)O)_(6)]^(3+)`
(b) `[Cu(CN)_(6)]^(2-)`
(c ) `[V(H_(2)O)_(6)]^(3+)`
(d) `[MnCI_(4)]^(2-)` .

Options

A. `[V(H_(2)O)_(6)]^(3+)`
B. `[Cr(H_(2)O)_(6)]^(3+)`
C. `[Cu(CN)_(4)]^(2-)`
D. `[MnCl_(4)]^(2-)`

Explanation

To determine which complex ion corresponds to a magnetic moment of `2.83 BM`, we need to analyze the oxidation states and electron configurations of the given complexes. The magnetic moment is related to the number of unpaired electrons in the d-orbitals of the metal ion. The formula for calculating the magnetic moment (μ) in Bohr Magnetons (BM) is:

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

where \( n \) is the number of unpaired electrons. For a magnetic moment of `2.83 BM`, we can set up the equation:

\[ 2.83 = \sqrt{n(n + 2)} \]

### Step 1: Calculate the number of unpaired electrons
Squaring both sides gives:

\[ (2.83)^2 = n(n + 2) \]

Calculating \( (2.83)^2 \):

\[ 8.0089 = n(n + 2) \]

This simplifies to the quadratic equation:

\[ n^2 + 2n - 8.0089 = 0 \]

### Step 2: Solve the quadratic equation
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 2, c = -8.0089 \):

\[ n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8.0089)}}{2 \cdot 1} \]
\[ n = \frac{-2 \pm \sqrt{4 + 32.0356}}{2} \]
\[ n = \frac{-2 \pm \sqrt{36.0356}}{2} \]
\[ n = \frac{-2 \pm 6}{2} \]

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

1. \( n = \frac{4}{2} = 2 \)
2. \( n = \frac{-8}{2} = -4 \) (not possible)

Thus, \( n = 2 \). This means there are 2 unpaired electrons.

### Step 3: Analyze each complex
Now we will analyze the given complexes to see which one has 2 unpaired electrons.

**(a) \([Cr(H_2O)_6]^{3+}\)**

- Oxidation state of Cr: +3
- Electron configuration: \( [Ar] 4s^0 3d^3 \) (3 unpaired electrons)

**(b) \([Cu(CN)_6]^{2-}\)**

- Oxidation state of Cu: +2
- Electron configuration: \( [Ar] 4s^0 3d^9 \) (1 unpaired electron)

**(c) \([V(H_2O)_6]^{3+}\)**

- Oxidation state of V: +3
- Electron configuration: \( [Ar] 4s^0 3d^2 \) (2 unpaired electrons)

**(d) \([MnCl_4]^{2-}\)**

- Oxidation state of Mn: +2
- Electron configuration: \( [Ar] 4s^2 3d^5 \) (5 unpaired electrons)

### Conclusion
The complex ion with a magnetic moment of `2.83 BM`, which corresponds to 2 unpaired electrons, is:

**Answer: (c) \([V(H_2O)_6]^{3+}\)**