An element belonging to `3d`series of modern periodic table has spin magnetic moment `= 5.92 B.M in +3 oxidation state. Determine the atomic number of element.

Answer

To determine the atomic number of the element belonging to the 3d series of the modern periodic table with a spin magnetic moment of 5.92 Bohr magnetons (B.M) in the +3 oxidation state, we can follow these steps:

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

### Step 2: Set up the equation
Given that the magnetic moment is 5.92 B.M, we can set up the equation:
\[ 5.92 = \sqrt{n(n + 2)} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (5.92)^2 = n(n + 2) \]
Calculating \( (5.92)^2 \):
\[ 35.0464 = n(n + 2) \]

### Step 4: Rearranging the equation
This can be rearranged to form a quadratic equation:
\[ n^2 + 2n - 35.0464 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 2, c = -35.0464 \).

Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-35.0464) = 4 + 140.1856 = 144.1856 \]

Now, substituting into the quadratic formula:
\[ n = \frac{-2 \pm \sqrt{144.1856}}{2} \]
Calculating \( \sqrt{144.1856} \approx 12.01 \):
\[ n = \frac{-2 \pm 12.01}{2} \]

Calculating the two possible values for \( n \):
1. \( n = \frac{10.01}{2} \approx 5.005 \) (taking the positive root)
2. \( n = \frac{-14.01}{2} \) (not applicable since \( n \) cannot be negative)

Thus, we find:
\[ n \approx 5 \]

### Step 6: Determine the element in the +3 oxidation state
In the +3 oxidation state, the element has lost 3 electrons. To find the neutral state, we need to add these 3 electrons back.

### Step 7: Identify the electronic configuration
The 3d series elements have the general electronic configuration of:
\[ \text{[Ar]} \, 3d^{x} \, 4s^{y} \]
where \( x + y \) is the total number of electrons.

Since we have 5 unpaired electrons in the +3 oxidation state, we can deduce that:
- In the neutral state, the element must have \( 5 + 3 = 8 \) unpaired electrons.

### Step 8: Find the atomic number
The only element in the 3d series that fits this configuration is Iron (Fe), which has an atomic number of 26:
- Neutral Iron: \([Ar] \, 3d^6 \, 4s^2\)
- In +3 state: \([Ar] \, 3d^5\) (5 unpaired electrons)

### Conclusion
The atomic number of the element is **26**.

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