The time period for revolutioin of Bohr electron in an orbit of ground state `(n_(1))` is `T_(1)` and time period for revolution of electron in higher orbit `(n_(2))` is `T_(2)`. Which values of `n_(1)` and `n_(2)` are not correct if `((T_(1))/(T_(2)))=1/8`?

Options

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

Explanation

To solve the problem, we need to analyze the relationship between the time periods \( T_1 \) and \( T_2 \) for the electron in different orbits according to Bohr's model of the atom.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) for an electron in a given orbit \( n \) can be expressed as:
\[ T \propto n^3 \]
This means that the time period is proportional to the cube of the principal quantum number \( n \).

2. **Setting Up the Ratio**:
Given that:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]
We can express this in terms of \( n_1 \) and \( n_2 \):
\[ \frac{T_1}{T_2} = \frac{n_1^3}{n_2^3} \]
Therefore, we can write:
\[ \frac{n_1^3}{n_2^3} = \frac{1}{8} \]

3. **Simplifying the Ratio**:
From the above equation, we can deduce:
\[ n_1^3 = \frac{1}{8} n_2^3 \]
This implies:
\[ n_1 = \frac{1}{2} n_2 \]

4. **Testing Possible Values**:
Now we will test the given options for \( n_1 \) and \( n_2 \):

- **Option 1**: \( n_1 = 1 \), \( n_2 = 2 \)
\[ \frac{1^3}{2^3} = \frac{1}{8} \quad \text{(Correct)} \]

- **Option 2**: \( n_1 = 2 \), \( n_2 = 4 \)
\[ \frac{2^3}{4^3} = \frac{8}{64} = \frac{1}{8} \quad \text{(Correct)} \]

- **Option 3**: \( n_1 = 1 \), \( n_2 = 3 \)
\[ \frac{1^3}{3^3} = \frac{1}{27} \quad \text{(Not Correct)} \]

- **Option 4**: \( n_1 = 3 \), \( n_2 = 6 \)
\[ \frac{3^3}{6^3} = \frac{27}{216} = \frac{1}{8} \quad \text{(Correct)} \]

5. **Conclusion**:
The values of \( n_1 \) and \( n_2 \) that are not correct are:
- \( n_1 = 1 \) and \( n_2 = 3 \)