The Bohr radius of the fifth Valence electron of phosphorous atom (atomic number = 15) acting as a dopant in silicon (relative dielectric constant = 12) is …….. Å

Explanation

To find the Bohr radius of the fifth valence electron of a phosphorus atom acting as a dopant in silicon, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Atomic number of phosphorus (Z) = 15
- Relative dielectric constant of silicon (ε_r) = 12
- The Bohr radius (r₀) = 0.53 Å (angstroms)

2. **Determine the Principal Quantum Number (n):**
- The electronic configuration of phosphorus is 1s² 2s² 2p⁶ 3s² 3p³.
- The valence electrons are in the outermost shell, which is n = 3.

3. **Use the Formula for Bohr Radius in a Medium:**
The formula for the modified Bohr radius (r_n) in a medium with dielectric constant is given by:
\[ r_n = \frac{\epsilon_r \cdot n^2}{Z} \cdot r_0 \]
where:
- \( r_n \) = Bohr radius in the medium
- \( \epsilon_r \) = relative dielectric constant
- \( n \) = principal quantum number
- \( Z \) = atomic number
- \( r_0 \) = Bohr radius in vacuum (0.53 Å)

4. **Substitute the Values into the Formula:**
- Substitute \( \epsilon_r = 12 \), \( n = 3 \), \( Z = 15 \), and \( r_0 = 0.53 \) Å into the formula:
\[ r_n = \frac{12 \cdot 3^2}{15} \cdot 0.53 \]

5. **Calculate \( n^2 \):**
- \( n^2 = 3^2 = 9 \)

6. **Calculate the Radius:**
\[ r_n = \frac{12 \cdot 9}{15} \cdot 0.53 \]
\[ r_n = \frac{108}{15} \cdot 0.53 \]
\[ r_n = 7.2 \cdot 0.53 \]
\[ r_n = 3.816 \text{ Å} \]

7. **Final Result:**
- The Bohr radius of the fifth valence electron of the phosphorus atom in silicon is approximately **3.81 Å**.