Question

The number of spectral lines emitted by atomic hydrogen that is in the 4th energy level is:

• A. 6
• B. 3
• C. 4
• D. 1

Hint:

For atomic hydrogen, the number of spectral lines emitted from a particular energy level can be found using the formula for possible transitions between energy levels. The total number of transitions is the sum of transitions from higher to lower levels.

Answer 1

The Correct Option is A

The possible transitions for an electron in the 4th energy level are: \[ n = 4 \quad \text{to} \quad n = 3, 2, 1 \] \[ n = 3 \quad \text{to} \quad n = 2, 1 \] \[ n = 2 \quad \text{to} \quad n = 1 \] The total number of possible transitions is 6. These transitions lead to the emission of spectral lines. Thus, the correct answer is option (1).

Answer 2

Step 1: Recall the concept of spectral lines in hydrogen.
When an electron in a hydrogen atom transitions from a higher energy level \( n_2 \) to a lower energy level \( n_1 \), it emits radiation of a specific frequency. The number of different spectral lines corresponds to the number of possible transitions between these levels.

Step 2: Formula for number of spectral lines.
If an electron is in the \( n \)th energy level, the total number of possible spectral lines (transitions) is given by the formula:
\[ N = \frac{n(n - 1)}{2}. \] This formula accounts for all possible transitions between energy levels, as each pair of energy levels corresponds to one possible spectral line.

Step 3: Substitute \( n = 4 \).
For the 4th energy level:
\[ N = \frac{4(4 - 1)}{2} = \frac{4 \times 3}{2} = 6. \]

Step 4: Interpretation.
This means that when an electron in the 4th energy level of a hydrogen atom de-excites, it can emit a total of 6 different spectral lines corresponding to all possible transitions between the energy levels \( 4 \rightarrow 3 \), \( 4 \rightarrow 2 \), \( 4 \rightarrow 1 \), \( 3 \rightarrow 2 \), \( 3 \rightarrow 1 \), and \( 2 \rightarrow 1 \).

Final Answer:
\[ \boxed{6} \]