An electron in hydrogen atom makes a transition `n_i to n_2` where `n_1` and `n_2` are principal quantum numbers of the two states. Assuming Bohr's 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 and n_2 =2`
B. `n_1 = 6 and n_2 =2`
C. `n_1 =8 and n_2 =1`
D. `n_1 =8 and n_2 =2`

Explanation

To solve the problem, we will use the relationships derived from Bohr's model of the hydrogen atom regarding the time period of an electron in different energy states.

### Step-by-Step Solution:

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

2. **Setting Up the Equation**:
From the problem, we know that:
\[ T_1 = 8 T_2 \]
where \( T_1 \) is the time period for the initial state \( n_1 \) and \( T_2 \) is for the final state \( n_2 \).

3. **Expressing the Time Periods**:
Using the proportionality, we can write:
\[ T_1 = k n_1^3 \quad \text{and} \quad T_2 = k n_2^3 \]
where \( k \) is a constant. Therefore, we can express the ratio of the time periods as:
\[ \frac{T_1}{T_2} = \frac{n_1^3}{n_2^3} \]

4. **Substituting the Known Ratio**:
From the relation \( T_1 = 8 T_2 \), we can substitute:
\[ \frac{n_1^3}{n_2^3} = 8 \]

5. **Taking the Cube Root**:
Taking the cube root of both sides gives:
\[ \frac{n_1}{n_2} = 2 \]
This implies that:
\[ n_1 = 2 n_2 \]

6. **Finding Possible Values**:
Now we need to find integer values for \( n_1 \) and \( n_2 \) that satisfy this relation. We can express \( n_1 \) in terms of \( n_2 \):
- 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 \( n_1 = 2 \times 4 = 8 \)

7. **Validating the Values**:
We need to ensure that both \( n_1 \) and \( n_2 \) are valid principal quantum numbers (positive integers). The pairs we have are:
- \( (n_1, n_2) = (2, 1) \)
- \( (n_1, n_2) = (4, 2) \)
- \( (n_1, n_2) = (6, 3) \)
- \( (n_1, n_2) = (8, 4) \)

8. **Conclusion**:
The possible values of \( n_1 \) and \( n_2 \) that satisfy the condition \( n_1 = 2 n_2 \) are:
- \( n_1 = 4 \) and \( n_2 = 2 \) (which is a valid pair).

### Final Answer:
The possible values of \( n_1 \) and \( n_2 \) are \( n_1 = 4 \) and \( n_2 = 2 \).