The ratio of the radii of the first three Bohr orbit in H atom is

Options

A. `1:0.5 :0:0.33`
B. `1:2:3`
C. `1:4:9`
D. `1:8:27`

Answer

To find the ratio of the radii of the first three Bohr orbits in a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the Orbits**:
We need to find the radii for the first three orbits, which correspond to \( n = 1, 2, \) and \( 3 \).

3. **Calculate the Radii**:
- For \( n = 1 \):
\[ r_1 = \frac{0.529 \times 1^2}{1} = 0.529 \, \text{Å} \]
- For \( n = 2 \):
\[ r_2 = \frac{0.529 \times 2^2}{1} = 0.529 \times 4 = 2.116 \, \text{Å} \]
- For \( n = 3 \):
\[ r_3 = \frac{0.529 \times 3^2}{1} = 0.529 \times 9 = 4.761 \, \text{Å} \]

4. **Establish the Ratio of the Radii**:
The ratio of the radii \( r_1 : r_2 : r_3 \) can be expressed as:
\[ r_1 : r_2 : r_3 = 1^2 : 2^2 : 3^2 = 1 : 4 : 9 \]

5. **Final Answer**:
Therefore, the ratio of the radii of the first three Bohr orbits in a hydrogen atom is:
\[ 1 : 4 : 9 \]