A weak field complex of `Ni^(2+)` has magnetic moment of 2.82 BM. The number of electron in the `t_(2g)` level of `Ni^(2+)` will be

Answer

To solve the problem, we need to determine the number of electrons in the \( t_{2g} \) level of \( Ni^{2+} \) based on its magnetic moment. Let's break it down step by step.

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

### Step 2: Substitute the Given Magnetic Moment
We are given that the magnetic moment of the complex is 2.82 BM. We can set up the equation:
\[ 2.82 = \sqrt{n(n + 2)} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides:
\[ (2.82)^2 = n(n + 2) \]
Calculating \( (2.82)^2 \):
\[ 7.9524 = n(n + 2) \]

### Step 4: Rearrange the Equation
This gives us a quadratic equation:
\[ n^2 + 2n - 7.9524 = 0 \]

### Step 5: Solve the Quadratic Equation
We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 2, c = -7.9524 \):
\[ n = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-7.9524)}}{2(1)} \]
Calculating the discriminant:
\[ n = \frac{-2 \pm \sqrt{4 + 31.8096}}{2} \]
\[ n = \frac{-2 \pm \sqrt{35.8096}}{2} \]
Calculating \( \sqrt{35.8096} \):
\[ n = \frac{-2 \pm 5.983}{2} \]
Calculating the two possible values for \( n \):
1. \( n = \frac{3.983}{2} \approx 1.9915 \) (not a valid solution since \( n \) must be an integer)
2. \( n = \frac{-7.983}{2} \) (not valid since \( n \) cannot be negative)

Since we are looking for integer values, we can round \( n \) to 2.

### Step 6: Determine the Electron Configuration of \( Ni^{2+} \)
Nickel (Ni) has an atomic number of 28. The electron configuration of neutral nickel is:
\[ [Ar] 3d^8 4s^2 \]
For \( Ni^{2+} \), we remove two electrons, typically from the 4s orbital first:
\[ Ni^{2+}: [Ar] 3d^8 \]

### Step 7: Distribution of Electrons in \( t_{2g} \) and \( e_g \)
In an octahedral field, the \( 3d \) orbitals split into \( t_{2g} \) and \( e_g \) levels. For \( d^8 \):
- The \( t_{2g} \) level can hold a maximum of 6 electrons.
- The \( e_g \) level can hold a maximum of 4 electrons.

Since we found that there are 2 unpaired electrons, and we know that weak field ligands do not cause pairing, the distribution of the 8 electrons in \( Ni^{2+} \) will be:
- \( t_{2g} \) level: 6 electrons (fully filled)
- \( e_g \) level: 2 electrons (unpaired)

### Conclusion
Thus, the number of electrons in the \( t_{2g} \) level of \( Ni^{2+} \) is **6**.

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