The radius of hydrogen shell is 0.53Å, then in first excited state radius of shell will be :

Options

A. 2.12 Å
B. 1.06 Å
C. 8.5 Å
D. 4.24 Å

Answer

To find the radius of the hydrogen atom in its first excited state, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The radius of the hydrogen atom in its ground state (n = 1) is given as 0.53 Å (angstroms).

2. **Use the Formula for Radius:**
- The formula for the radius of the nth shell in a hydrogen atom is given by:
\[ R_n = R_0 \cdot \frac{n^2}{Z} \]
- Where:
- \( R_n \) = radius of the nth shell
- \( R_0 \) = radius of the ground state (n = 1)
- \( n \) = principal quantum number (shell number)
- \( Z \) = atomic number (for hydrogen, \( Z = 1 \))

3. **Identify the Values:**
- For the ground state (n = 1):
- \( R_0 = 0.53 \) Å
- \( Z = 1 \)
- For the first excited state, we have:
- \( n = 2 \)

4. **Calculate the Radius for the First Excited State:**
- Substitute the values into the formula:
\[ R_2 = R_0 \cdot \frac{n^2}{Z} = 0.53 \cdot \frac{2^2}{1} \]
- Simplifying this gives:
\[ R_2 = 0.53 \cdot \frac{4}{1} = 0.53 \cdot 4 \]

5. **Perform the Calculation:**
- Calculate \( 0.53 \cdot 4 \):
\[ R_2 = 2.12 \text{ Å} \]

6. **Conclusion:**
- The radius of the hydrogen atom in its first excited state (n = 2) is **2.12 Å**.