The magnetic moment of a transition metal of 3 d series 6. 92 BM. Its electronic configuration would be

Options

A. `3d^4 4s^2`
B. `3d^45 4s^1`
C. `3d^B 4s^1`
D. `3d^5 4s^0`

Answer

To determine the electronic configuration of a transition metal with a magnetic moment of 6.92 Bohr Magneton (BM), we can follow these steps:

### Step 1: Understand the relationship between magnetic moment and unpaired electrons

The magnetic moment (μ) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \text{ BM} \]
where \( n \) is the number of unpaired electrons.

### Step 2: Set up the equation with the given magnetic moment

Given that the magnetic moment is 6.92 BM, we can set up the equation:
\[ 6.92 = \sqrt{n(n + 2)} \]

### Step 3: Square both sides to eliminate the square root

Squaring both sides gives:
\[ (6.92)^2 = n(n + 2) \]
Calculating \( (6.92)^2 \):
\[ 6.92^2 = 47.9364 \]
So, we have:
\[ n(n + 2) = 47.9364 \]

### Step 4: Rearrange the equation

Rearranging gives us a quadratic equation:
\[ n^2 + 2n - 47.9364 = 0 \]

### Step 5: Solve the quadratic equation

Using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 2, c = -47.9364 \):
\[ n = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-47.9364)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 4 + 191.7456 = 195.7456 \]
Taking the square root:
\[ \sqrt{195.7456} \approx 14.0 \]
Now substituting back:
\[ n = \frac{-2 \pm 14}{2} \]
Calculating the two possible values for \( n \):
1. \( n = \frac{12}{2} = 6 \)
2. \( n = \frac{-16}{2} = -8 \) (not a valid solution)

Thus, \( n = 6 \).

### Step 6: Determine the electronic configuration

Transition metals in the 3d series can have a maximum of 6 unpaired electrons. The electronic configuration for a transition metal with 6 unpaired electrons is typically:
\[ [Ar] 3d^6 4s^2 \]
However, since we have 6 unpaired electrons, we can conclude that the metal is likely in a high-spin state, which is common for \( Fe^{3+} \ or \ Co^{2+} \).

### Final Answer

The electronic configuration of the transition metal with a magnetic moment of 6.92 BM is:
\[ [Ar] 3d^6 \]

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