In a certain electronic transition from the quantum level, 'n' to the ground state in atomic hydrogen in one or more steps, no line belonging to the Bracket series is observed. What wave number may be observed in the Balmer series ? (R = Rydberg Constant)

Options

A. `(8R)/(9), (5R)/(36)`
B. `(3R)/(16), (8R)/(9)`
C. `(5R)/(36), (3R)/(16)`
D. `(3R)/(4), (3R)/(16)`

Answer

To solve the problem, we need to determine the wave numbers that may be observed in the Balmer series during electronic transitions in atomic hydrogen, given that no lines belonging to the Brackett series are observed.

### Step-by-Step Solution:

1. **Understanding the Brackett Series**: The Brackett series corresponds to electronic transitions that end at the n=4 energy level. If no lines from the Brackett series are observed, it implies that the initial quantum level (n) must be greater than 4.

2. **Determining Possible Transitions**: If n > 4, the possible transitions can be from n=5, n=6, etc., down to the ground state (n=1). The transitions that will contribute to the Balmer series are those that end at n=2.

3. **Identifying the Balmer Series**: The Balmer series consists of transitions that end at the n=2 level. The relevant transitions from higher levels to n=2 are:
- From n=3 to n=2
- From n=4 to n=2
- From n=5 to n=2
- And so on...

4. **Calculating Wave Numbers for Transitions**: The wave number (ν) can be calculated using the Rydberg formula:
\[ \nu = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, \( n_1 \) is the final energy level (2 for Balmer series), and \( n_2 \) is the initial energy level (which can be 3, 4, 5, etc.).

5. **Calculating for n=4 to n=2**:
\[ \nu_1 = R \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{4} - \frac{1}{16} \right) = R \left( \frac{4 - 1}{16} \right) = \frac{3R}{16} \]

6. **Calculating for n=3 to n=2**:
\[ \nu_2 = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = \frac{5R}{36} \]

7. **Conclusion**: The wave numbers that may be observed in the Balmer series from the transitions calculated are:
- From n=4 to n=2: \( \frac{3R}{16} \)
- From n=3 to n=2: \( \frac{5R}{36} \)

### Final Answer:
The wave numbers that may be observed in the Balmer series are \( \frac{3R}{16} \) and \( \frac{5R}{36} \).