The electron in hydrogen atom makes a transition `n_(1)ton_(2)` where `n_1` and `n_2` are the principal quantum number of two states. Assuming 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 value 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=2`

Explanation

To solve the problem, we need to analyze the relationship between the time periods of the electron in different energy states of a hydrogen atom according to the Bohr model.

### Step-by-Step Solution:

1. **Understanding the Time Period in Bohr Model**:
The time period \( T \) of the 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 proportional to the cube of the principal quantum number.

2. **Setting Up the Relationship**:
According to the problem, we have two states: the initial state with principal quantum number \( n_1 \) and the final state with principal quantum number \( n_2 \). The problem states that the time period of the electron in the initial state is eight times that in the final state:
\[ T_1 = 8 T_2 \]

3. **Using the Proportional Relationship**:
From the proportionality, we can express the time periods in terms of the principal 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. **Substituting the Given Condition**:
Substituting the condition \( T_1 = 8 T_2 \) into the equation gives us:
\[ 8 = \frac{n_1^3}{n_2^3} \]

5. **Simplifying the Equation**:
Rearranging the equation, we get:
\[ \frac{n_1^3}{n_2^3} = 8 \implies \left(\frac{n_1}{n_2}\right)^3 = 8 \]
Taking the cube root of both sides, we find:
\[ \frac{n_1}{n_2} = 2 \]

6. **Finding Possible Values of \( n_1 \) and \( n_2 \)**:
This means that \( n_1 \) is twice \( n_2 \):
\[ n_1 = 2n_2 \]
Now 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 \( n_1 = 2 \times 4 = 8 \)

7. **Conclusion**:
The possible pairs \( (n_1, n_2) \) that satisfy the condition \( n_1 = 2n_2 \) are:
- \( (2, 1) \)
- \( (4, 2) \)
- \( (6, 3) \)
- \( (8, 4) \)

### Final Answer:
The possible values of \( n_1 \) and \( n_2 \) are \( (2, 1) \) and \( (4, 2) \).