How many different wavelengths may be observed in the spectrum form a hydrogen sample, if the atoms are excited to third excited state?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To determine how many different wavelengths may be observed in the spectrum from a hydrogen sample when the atoms are excited to the third excited state, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Excited State**:
The third excited state corresponds to the principal quantum number \( n = 4 \) (since the ground state is \( n = 1 \), the first excited state is \( n = 2 \), the second excited state is \( n = 3 \), and the third excited state is \( n = 4 \)).

2. **Determine the Maximum Number of Spectral Lines**:
The formula to calculate the maximum number of spectral lines (or wavelengths) that can be observed when electrons transition from the nth energy level to lower energy levels is given by:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
where \( n \) is the principal quantum number of the excited state.

3. **Substitute the Value of n**:
Here, since we are considering the third excited state, we have \( n = 4 \). Plugging this value into the formula:
\[ \text{Number of lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 \]

4. **Conclusion**:
Therefore, the number of different wavelengths that may be observed in the spectrum from a hydrogen sample when the atoms are excited to the third excited state is **6**.

### Final Answer:
The number of different wavelengths observed is **6**.

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