How many spectral lines are emitted by atomic hydrogen excited to the `n-th` energy level?

Options

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

Answer

To determine how many spectral lines are emitted by atomic hydrogen when it is excited to the `n-th` energy level, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Energy Levels**:
- Atomic hydrogen has discrete energy levels denoted by quantum numbers (n = 1, 2, 3, ...). When an electron transitions from a higher energy level to a lower one, it emits a photon, which corresponds to a spectral line.

2. **Identifying Possible Transitions**:
- An electron in the `n-th` energy level can transition to any of the lower energy levels (n = 1, 2, ..., n-1). Therefore, the possible transitions from the `n-th` level are to the (n-1)-th, (n-2)-th, ..., down to the 1st level.

3. **Counting the Transitions**:
- The number of ways an electron can transition from the `n-th` level to the lower levels is given by the formula:
\[ \text{Number of transitions} = n - 1 + n - 2 + ... + 1 = \frac{(n-1) \cdot n}{2} \]
- This is because for each level below `n`, there is a corresponding transition.

4. **Calculating the Spectral Lines**:
- Each transition corresponds to a unique spectral line. Therefore, the total number of spectral lines emitted when an electron transitions from the `n-th` level to all lower levels is:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

5. **Final Result**:
- Thus, the number of spectral lines emitted by atomic hydrogen excited to the `n-th` energy level is:
\[ \frac{n(n-1)}{2} \]

### Conclusion:
The answer to the question is:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]