If the radius of first orbit of hydrogen atom is `a_(0)`, which of the following cannot be the radius of any other orbits of hydrogen?

Options

A. `4a_(0)`
B. `6a_(0)`
C. `9a_(0)`
D. `16a_(0)`

Explanation

To solve the problem of determining which radius cannot be the radius of any other orbit of the hydrogen atom, we will use the formula derived from Bohr's theory of the hydrogen atom.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The radius of the nth orbit of a hydrogen atom is given by the formula:
\[ R_n = a_0 \cdot n^2 \]
where \( a_0 \) is the radius of the first orbit (approximately 0.529 Å) and \( n \) is the principal quantum number (1, 2, 3, ...).

2. **Calculating the Radii for Different Orbits**:
- For \( n = 1 \):
\[ R_1 = a_0 \cdot 1^2 = a_0 \]
- For \( n = 2 \):
\[ R_2 = a_0 \cdot 2^2 = 4a_0 \]
- For \( n = 3 \):
\[ R_3 = a_0 \cdot 3^2 = 9a_0 \]
- For \( n = 4 \):
\[ R_4 = a_0 \cdot 4^2 = 16a_0 \]

3. **Identifying Possible Radii**:
The possible radii for the hydrogen atom orbits based on the calculations above are:
- \( R_1 = a_0 \)
- \( R_2 = 4a_0 \)
- \( R_3 = 9a_0 \)
- \( R_4 = 16a_0 \)

4. **Analyzing the Options**:
Now, we need to analyze the options provided in the question to see which one does not fit the form \( R_n = a_0 \cdot n^2 \) for any integer \( n \).

5. **Conclusion**:
If an option is given that does not match any of the calculated values (i.e., \( a_0, 4a_0, 9a_0, 16a_0 \)), it cannot be the radius of any orbit of the hydrogen atom.

### Final Answer:
The radius that cannot be the radius of any other orbit of hydrogen is option B (assuming option B is a value that does not match \( a_0, 4a_0, 9a_0, \) or \( 16a_0 \)).