the orbital angular momentum of 4f electron is

Options

A. `4(h/(2π))`
B. `sqrt12(h/(2π))`
C. `sqrt6π(h/(2π)`
D. `sqrt2×h/(2π)`

Answer

To find the orbital angular momentum of a 4f electron, we can follow these steps:

### Step 1: Identify the Angular Momentum Quantum Number (L)
The 4f electron belongs to the f subshell. The angular momentum quantum number (L) for the f subshell is defined as:
- L = 3

### Step 2: Use the Formula for Orbital Angular Momentum
The formula for calculating the orbital angular momentum (L) is given by:
\[ L = \sqrt{L(L + 1)} \cdot \frac{h}{2\pi} \]
where:
- \(L\) is the angular momentum quantum number,
- \(h\) is Planck's constant.

### Step 3: Substitute the Value of L into the Formula
Now, we substitute \(L = 3\) into the formula:
\[ L = \sqrt{3(3 + 1)} \cdot \frac{h}{2\pi} \]

### Step 4: Simplify the Expression
Calculating \(3 + 1\):
\[ 3 + 1 = 4 \]
Now, substituting this back into the equation:
\[ L = \sqrt{3 \cdot 4} \cdot \frac{h}{2\pi} \]

### Step 5: Calculate the Value Inside the Square Root
Now calculate \(3 \cdot 4\):
\[ 3 \cdot 4 = 12 \]
So we have:
\[ L = \sqrt{12} \cdot \frac{h}{2\pi} \]

### Step 6: Final Expression for Orbital Angular Momentum
Thus, the orbital angular momentum of a 4f electron is:
\[ L = \sqrt{12} \cdot \frac{h}{2\pi} \]

### Conclusion
The correct answer is:
\[ \sqrt{12} \cdot \frac{h}{2\pi} \]