The radius of electron's second stationary orbit in Bohr's atom is `R`. The radius of the third orbit will be

Options

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

Explanation

To find the radius of the third stationary orbit in a Bohr atom when the radius of the second stationary orbit is given as \( R \), we can use the formula for the radius of the \( n \)-th orbit in a hydrogen-like atom:

\[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi m z e^2} \]

Where:
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant,
- \( \epsilon_0 \) is the permittivity of free space,
- \( m \) is the mass of the electron,
- \( z \) is the atomic number,
- \( e \) is the charge of the electron.

### Step-by-step Solution:

1. **Identify the relationship between radius and principal quantum number**:
The radius of the orbit is proportional to the square of the principal quantum number:
\[ r_n \propto n^2 \]

2. **Write the ratio of the radii for different orbits**:
For the second orbit (\( n = 2 \)):
\[ r_2 \propto 2^2 = 4 \]
For the third orbit (\( n = 3 \)):
\[ r_3 \propto 3^2 = 9 \]

3. **Set up the ratio of the radii**:
The ratio of the radii for the second and third orbits can be expressed as:
\[ \frac{r_2}{r_3} = \frac{4}{9} \]

4. **Express \( r_3 \) in terms of \( r_2 \)**:
Rearranging the above ratio gives:
\[ r_3 = \frac{9}{4} r_2 \]

5. **Substituting the value of \( r_2 \)**:
Since we know that \( r_2 = R \):
\[ r_3 = \frac{9}{4} R \]

6. **Calculate the numerical value**:
To express \( r_3 \) in terms of \( R \):
\[ r_3 = 2.25 R \]

### Final Answer:
The radius of the third orbit is \( 2.25 R \).