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 follow these steps:

### Step-by-Step Solution:

1. **Understanding Energy Levels**: In a hydrogen atom, electrons can occupy different energy levels, denoted by the principal quantum number \( n \). When an electron is excited to the \( n^{th} \) energy level, it can transition back to lower energy levels, emitting spectral lines in the process.

2. **Calculating the Number of Transitions**: The number of possible transitions from the \( n^{th} \) energy level to lower energy levels can be calculated using the formula for combinations. Specifically, the number of ways to choose 2 energy levels from \( n \) levels is given by \( \binom{n}{2} \).

3. **Using the Combinatorial Formula**: The formula for combinations is:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
This formula counts the number of unique pairs of energy levels that the electron can transition between.

4. **Final Formula for Spectral Lines**: Therefore, the total number of spectral lines emitted when the electron is excited to the \( n^{th} \) energy state is:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

5. **Conclusion**: The final result indicates that the number of spectral lines emitted when an electron is excited to the \( n^{th} \) energy state in hydrogen is given by the formula \( \frac{n(n-1)}{2} \).