When hydrogen atom is in its first excited level, its radius is

Options

A. Twice
B. 4 times
C. Same
D. Half

Answer

To find the radius of a hydrogen atom when it is in its first excited state, we can use the formula for the radius of an electron in a hydrogen atom, which is given by:

\[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi e^2 m} \]

Where:
- \( r_n \) is the radius of the nth energy level,
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant,
- \( \epsilon_0 \) is the permittivity of free space,
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number**:
- For the ground state of hydrogen, the principal quantum number \( n = 1 \).
- For the first excited state, \( n = 2 \).

2. **Calculate the Radius for the Ground State (n=1)**:
- Using the formula, the radius for \( n = 1 \) is:
\[ r_1 = \frac{1^2 h^2 \epsilon_0}{\pi e^2 m} = r_1 \]

3. **Calculate the Radius for the First Excited State (n=2)**:
- Using the formula for \( n = 2 \):
\[ r_2 = \frac{2^2 h^2 \epsilon_0}{\pi e^2 m} = \frac{4 h^2 \epsilon_0}{\pi e^2 m} = 4 r_1 \]

4. **Conclusion**:
- Therefore, when the hydrogen atom is in its first excited state, its radius is \( 4 \) times the radius of the ground state.
- Thus, the correct answer is option 2: four times.

### Final Answer:
The radius of the hydrogen atom in its first excited level is **four times** the radius in its ground state.

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