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 first three Bohr orbits for a given atom, we can use the formula for the radius of the nth orbit in the Bohr model of the atom:

### Step-by-Step Solution:

1. **Write the Formula for the Radius of Bohr Orbits**:
The radius of the nth orbit (r_n) is given by the formula:
\[ r_n = \frac{r_0 n^2}{Z} \]
where:
- \( r_0 \) is a constant (0.529 Å),
- \( n \) is the principal quantum number (the orbit number),
- \( Z \) is the atomic number of the atom.

2. **Calculate the Radii for n = 1, 2, and 3**:
- For \( n = 1 \):
\[ r_1 = \frac{r_0 \cdot 1^2}{Z} = \frac{r_0}{Z} \]
- For \( n = 2 \):
\[ r_2 = \frac{r_0 \cdot 2^2}{Z} = \frac{4r_0}{Z} \]
- For \( n = 3 \):
\[ r_3 = \frac{r_0 \cdot 3^2}{Z} = \frac{9r_0}{Z} \]

3. **Establish the Ratio of the Radii**:
Now we can write the ratio of the radii \( r_1 : r_2 : r_3 \):
\[ r_1 : r_2 : r_3 = \frac{r_0}{Z} : \frac{4r_0}{Z} : \frac{9r_0}{Z} \]
Since \( \frac{r_0}{Z} \) is common in all terms, we can cancel it out:
\[ r_1 : r_2 : r_3 = 1 : 4 : 9 \]

4. **Conclusion**:
Therefore, the ratio of the radii of the first three Bohr orbits for a given atom is:
\[ \text{Ratio} = 1 : 4 : 9 \]