When an electron is excited to `n^(th)` energy state in hydrogen , the posssible 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 1: Understanding Energy Levels
In a hydrogen atom, the electron can occupy different energy levels, denoted by the principal quantum number \( n \). When an electron is excited to a higher energy level, it can transition back to lower energy levels, emitting photons in the process.

### Step 2: Calculating the Number of Transitions
When an electron is in the \( n^{th} \) energy state, it can transition to any of the lower energy states, which are \( n-1, n-2, \ldots, 1 \). The number of possible transitions from the \( n^{th} \) level to the lower levels can be calculated using the formula for combinations.

### Step 3: Using the Combination Formula
The number of ways to choose 2 levels from \( n \) levels (the initial and final states) is given by the combination formula:
\[ \text{Number of spectral lines} = \binom{n}{2} = \frac{n(n-1)}{2} \]
This formula counts the number of unique pairs of energy levels that the electron can transition between.

### Step 4: Final Result
Thus, the total number of spectral lines emitted when an electron is excited to the \( n^{th} \) energy state is:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

### Conclusion
This formula provides the answer to the question regarding the number of spectral lines emitted when an electron is excited to the \( n^{th} \) energy state in hydrogen.