The ratio of the radii of the three Bohr orbits for a given atom is:

Options

A. `1:1//2 : 1//3`
B. `1:2:3`
C. `1:4:9`
D. `1:8:27`

Explanation

To find the ratio of the radii of the three Bohr orbits for a given atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Bohr Radius**:
The radius of the nth orbit in a hydrogen-like atom is given by the formula:
\[ r_n = \frac{0.529 \, n^2}{Z} \, \text{angstroms} \]
where \( n \) is the principal quantum number (1, 2, 3, ...) and \( Z \) is the atomic number of the atom.

2. **Calculate the Radius for Each Orbit**:
- For the **1st orbit** (\( n = 1 \)):
\[ r_1 = \frac{0.529 \cdot 1^2}{Z} = \frac{0.529}{Z} \, \text{angstroms} \]
- For the **2nd orbit** (\( n = 2 \)):
\[ r_2 = \frac{0.529 \cdot 2^2}{Z} = \frac{0.529 \cdot 4}{Z} = \frac{2.116}{Z} \, \text{angstroms} \]
- For the **3rd orbit** (\( n = 3 \)):
\[ r_3 = \frac{0.529 \cdot 3^2}{Z} = \frac{0.529 \cdot 9}{Z} = \frac{4.761}{Z} \, \text{angstroms} \]

3. **Express Each Radius in Terms of a Common Variable**:
We can express the radii in terms of \( r_0 = \frac{0.529}{Z} \):
- \( r_1 = r_0 \)
- \( r_2 = 4r_0 \)
- \( r_3 = 9r_0 \)

4. **Form the Ratio of the Radii**:
Now, we can write the ratio of the radii of the three orbits:
\[ r_1 : r_2 : r_3 = r_0 : 4r_0 : 9r_0 \]
Simplifying this gives:
\[ 1 : 4 : 9 \]

5. **Conclusion**:
Therefore, the ratio of the radii of the first, second, and third Bohr orbits for a given atom is:
\[ \boxed{1 : 4 : 9} \]