The electron in a hydrogen atom makes a transition `n_(1) rarr n_(2)`, where `n_(1)` and `n_(2)` are the principle quantum numbers of the two states. Assume the Bohr model to be valid. The time period of the electron in the initial state is eight times that in the final state. the possible values of `n_(1)` and `n_(2)` are

Options

A. `n_(1)=4,n_(2)=2`
B. `n_(1)=8,n_(2)=2`
C. `n_(1)=8,n_(2)=1`
D. `n_(1)=6,n_(2)=3`

Answer

To solve the problem, we will use the Bohr model of the hydrogen atom to relate the time periods of the electron in different energy states.

### Step-by-Step Solution:

1. **Understanding the Time Period in Bohr Model**:
The time period \( T \) of an electron in a particular orbit is given by the formula:
\[ T \propto n^3 \]
where \( n \) is the principal quantum number. This means that the time period is proportional to the cube of the principal quantum number.

2. **Setting Up the Ratio of Time Periods**:
Given that the time period of the electron in the initial state \( T_1 \) is 8 times that in the final state \( T_2 \), we can write:
\[ T_1 = 8 T_2 \]

3. **Expressing Time Periods in Terms of Quantum Numbers**:
From the proportionality, we can express the time periods in terms of their respective quantum numbers:
\[ T_1 \propto n_1^3 \quad \text{and} \quad T_2 \propto n_2^3 \]
Therefore, we can write:
\[ \frac{T_1}{T_2} = \frac{n_1^3}{n_2^3} \]

4. **Setting Up the Equation**:
Substituting the relationship of the time periods into the equation gives:
\[ \frac{n_1^3}{n_2^3} = 8 \]
This implies:
\[ n_1^3 = 8 n_2^3 \]

5. **Taking the Cube Root**:
Taking the cube root of both sides, we find:
\[ n_1 = 2 n_2 \]

6. **Finding Possible Values of \( n_1 \) and \( n_2 \)**:
Since \( n_1 \) and \( n_2 \) must be positive integers, we can assign values to \( n_2 \) and calculate \( n_1 \):
- If \( n_2 = 1 \), then \( n_1 = 2 \times 1 = 2 \)
- If \( n_2 = 2 \), then \( n_1 = 2 \times 2 = 4 \)
- If \( n_2 = 3 \), then \( n_1 = 2 \times 3 = 6 \)
- If \( n_2 = 4 \), then \( n_1 = 2 \times 4 = 8 \)
- And so on...

Thus, the pairs \( (n_1, n_2) \) can be:
- \( (2, 1) \)
- \( (4, 2) \)
- \( (6, 3) \)
- \( (8, 4) \)
- etc.

### Conclusion:
The possible values of \( n_1 \) and \( n_2 \) are pairs where \( n_1 = 2n_2 \) for any positive integer \( n_2 \).