If radius of second stationary orbit (in Bohr's atom) is R then radius of third orbit will be :

Options

A. `R//3`
B. `9R`
C. `R//9`
D. `2.25R`

Answer

To find the radius of the third stationary orbit in Bohr's atom given that the radius of the second stationary orbit is \( R \), we can follow these steps:

### Step-by-step Solution:

1. **Understand the Formula for Bohr's Radius**:
The radius of the \( n^{th} \) stationary orbit in Bohr's model is given by the formula:
\[ R_n = \frac{a_0 n^2}{Z} \]
where:
- \( R_n \) is the radius of the \( n^{th} \) orbit,
- \( a_0 \) is the Bohr radius,
- \( n \) is the principal quantum number (the orbit number),
- \( Z \) is the atomic number.

2. **Calculate the Radius of the Second Orbit**:
For the second stationary orbit (\( n = 2 \)):
\[ R_2 = \frac{a_0 (2^2)}{Z} = \frac{4a_0}{Z} \]
According to the problem, this radius is given as \( R \):
\[ R = \frac{4a_0}{Z} \]

3. **Calculate the Radius of the Third Orbit**:
For the third stationary orbit (\( n = 3 \)):
\[ R_3 = \frac{a_0 (3^2)}{Z} = \frac{9a_0}{Z} \]

4. **Relate the Radii**:
We can express \( R_3 \) in terms of \( R \):
\[ R_3 = \frac{9a_0}{Z} \]
From the equation for \( R \):
\[ R = \frac{4a_0}{Z} \]
Now, we can divide \( R_3 \) by \( R \):
\[ \frac{R_3}{R} = \frac{\frac{9a_0}{Z}}{\frac{4a_0}{Z}} = \frac{9}{4} \]

5. **Final Calculation**:
Thus, we can express \( R_3 \) as:
\[ R_3 = \frac{9}{4} R \]
This means that the radius of the third stationary orbit is \( 2.25 R \).

### Conclusion:
The radius of the third stationary orbit is \( 2.25 R \).