The ratio of the first three Bohr orbit radii is

Options

A. 1:4:9
B. `1:2:3`
C. `3:4:5`
D. `1:8:27`

Explanation

To find the ratio of the first three Bohr orbit radii, we can follow these steps:

### Step 1: Understand the formula for Bohr orbit radius
According to the Bohr model, the radius of the nth orbit (Rn) for hydrogen-like atoms is given by the formula:
\[ R_n = R_0 \cdot \frac{n^2}{Z} \]
where:
- \( R_0 \) is the radius of the first orbit (approximately 0.53 Å),
- \( n \) is the principal quantum number (1 for the first orbit, 2 for the second, etc.),
- \( Z \) is the atomic number of the atom.

### Step 2: Identify the first three orbits
We need to find the radii for the first three orbits, which correspond to \( n = 1, 2, 3 \):
- For \( n = 1 \): \( R_1 = R_0 \cdot \frac{1^2}{Z} = \frac{R_0}{Z} \)
- For \( n = 2 \): \( R_2 = R_0 \cdot \frac{2^2}{Z} = \frac{4R_0}{Z} \)
- For \( n = 3 \): \( R_3 = R_0 \cdot \frac{3^2}{Z} = \frac{9R_0}{Z} \)

### Step 3: Write the ratios of the radii
Now we can write the ratio of the first three radii:
\[ R_1 : R_2 : R_3 = \frac{R_0}{Z} : \frac{4R_0}{Z} : \frac{9R_0}{Z} \]

### Step 4: Simplify the ratio
Since \( R_0 \) and \( Z \) are constants, they can be canceled out from the ratio:
\[ R_1 : R_2 : R_3 = 1 : 4 : 9 \]

### Conclusion
Thus, the ratio of the first three Bohr orbit radii is:
\[ \boxed{1 : 4 : 9} \]

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