Number of spectral lines in hydrogen atom is

Options

A. 8
B. 6
C. 15
D. `oo`

Explanation

To find the number of spectral lines in a hydrogen atom, we can use the formula for the number of spectral lines produced when an electron transitions between energy levels. The formula is:

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

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

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n \) can take any positive integer value starting from 1 (i.e., \( n = 1, 2, 3, \ldots \)).
- For the hydrogen atom, there is no upper limit to the value of \( n \) since it can theoretically go to infinity.

2. **Apply the Formula**:
- The formula for the number of spectral lines is applicable for transitions between energy levels. For any given \( n \), the number of possible transitions (or spectral lines) is calculated using the formula:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]

3. **Calculate for Different Values of n**:
- If we take \( n = 1 \), there are no transitions possible (0 lines).
- For \( n = 2 \):
\[ \frac{2(2-1)}{2} = 1 \text{ line} \]
- For \( n = 3 \):
\[ \frac{3(3-1)}{2} = 3 \text{ lines} \]
- For \( n = 4 \):
\[ \frac{4(4-1)}{2} = 6 \text{ lines} \]
- Continuing this way, as \( n \) increases, the number of spectral lines increases.

4. **Consider Infinite Transitions**:
- Since \( n \) can theoretically go to infinity, the number of spectral lines is also infinite. This means that the electron can transition from \( n = \infty \) to any lower energy level, resulting in an infinite number of spectral lines.

5. **Conclusion**:
- Therefore, the total number of spectral lines in a hydrogen atom is infinite due to the unlimited possible transitions between energy levels.

### Final Answer:
The number of spectral lines in a hydrogen atom is infinite.