The radius of the first Bohr orbit of hydrogen atom is r. The radius of the 3rd orbit would be

Options

A. 3r
B. 9r
C. 27r
D. none of these

Answer

To find the radius of the third Bohr orbit of a hydrogen atom, we can use the formula for the radius of the nth Bohr orbit:

### Step-by-Step Solution:

1. **Understand the Formula**: The radius of the nth Bohr orbit is given by the formula:
\[ r_n = r_0 \cdot \frac{n^2}{Z} \]
where:
- \( r_n \) is the radius of the nth orbit,
- \( r_0 \) is the radius of the first orbit,
- \( n \) is the principal quantum number (1 for the first orbit, 2 for the second, etc.),
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

2. **Identify Known Values**: From the question, we know:
- The radius of the first Bohr orbit (\( r_1 \)) is \( r \).
- For hydrogen, \( Z = 1 \).

3. **Set Up the Equation for the First Orbit**: For the first orbit (\( n = 1 \)):
\[ r_1 = r_0 \cdot \frac{1^2}{1} = r_0 \]
Since \( r_1 = r \), we can conclude that:
\[ r_0 = r \]

4. **Calculate the Radius for the Third Orbit**: Now, we need to find the radius for the third orbit (\( n = 3 \)):
\[ r_3 = r_0 \cdot \frac{3^2}{1} = r_0 \cdot 9 \]
Substituting \( r_0 = r \):
\[ r_3 = r \cdot 9 = 9r \]

5. **Final Answer**: Therefore, the radius of the third Bohr orbit of the hydrogen atom is:
\[ r_3 = 9r \]