When an electron is excited to `n^(th)` energy state in hydrogen, the possible number of spectral lines emitted are

Options

A. `n`
B. `2n`
C. `(n^(2)-n)/2`
D. `(n^(2)+n)/2`

Answer

To determine the possible number of spectral lines emitted when an electron is excited to the \( n^{th} \) energy state in hydrogen, we can use the formula:

\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

where \( n \) is the principal quantum number of the excited state.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number \( n \)**:
- When an electron is excited to a certain energy level, we denote that level by the principal quantum number \( n \).

2. **Apply the Formula**:
- The formula to calculate the number of spectral lines emitted when the electron returns to lower energy states is given by:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

3. **Example Calculation**:
- Let's say the electron is excited to the \( n = 3 \) state.
- Substitute \( n = 3 \) into the formula:
\[ \text{Number of spectral lines} = \frac{3(3-1)}{2} = \frac{3 \times 2}{2} = 3 \]
- This means that when the electron is excited to the \( n = 3 \) state, it can emit 3 spectral lines as it transitions back to lower energy states.

4. **Generalization**:
- The formula can be generalized for any \( n \):
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]
- This indicates that the number of spectral lines increases as \( n \) increases.

### Conclusion:
Thus, the possible number of spectral lines emitted when an electron is excited to the \( n^{th} \) energy state in hydrogen is given by \( \frac{n(n-1)}{2} \).