Spin angular momentum for unpaired electron in sodium ( Atomic No. `=11)` is

Options

A. `(sqrt(3))/(2)`
B. `0.866h//2pi`
C. `-(sqrt(3))/(2)(h)/(2pi)`
D. None of these

Explanation

To calculate the spin angular momentum for the unpaired electron in sodium (atomic number 11), we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Electronic Configuration of Sodium:**
- Sodium has an atomic number of 11, which means it has 11 electrons.
- The electronic configuration can be written as:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^1 \]
- Here, the 3s orbital contains one unpaired electron.

2. **Identify the Spin Quantum Number (s):**
- For an unpaired electron, the spin quantum number \( s \) is \( +\frac{1}{2} \).

3. **Use the Formula for Spin Angular Momentum:**
- The formula for calculating the magnetic moment (spin angular momentum) is given by:
\[ \mu = \sqrt{s(s + 1)} \cdot \frac{h}{2\pi} \]
- Substituting the value of \( s \):
\[ \mu = \sqrt{\frac{1}{2} \left( \frac{1}{2} + 1 \right)} \cdot \frac{h}{2\pi} \]

4. **Calculate the Value Inside the Square Root:**
- Calculate \( s + 1 \):
\[ s + 1 = \frac{1}{2} + 1 = \frac{3}{2} \]
- Now, substituting back into the equation:
\[ \mu = \sqrt{\frac{1}{2} \cdot \frac{3}{2}} \cdot \frac{h}{2\pi} \]
- This simplifies to:
\[ \mu = \sqrt{\frac{3}{4}} \cdot \frac{h}{2\pi} = \frac{\sqrt{3}}{2} \cdot \frac{h}{2\pi} \]

5. **Final Expression for the Spin Angular Momentum:**
- Thus, the spin angular momentum for the unpaired electron in sodium is:
\[ \mu = \frac{\sqrt{3}}{2} \cdot \frac{h}{2\pi} \]

6. **Choose the Correct Option:**
- From the options provided, we can identify the correct answer based on the calculated value.