What is the radius of `2^(nd)` Bohr orbit, given the radius of the first Bohr orbit is r?

Options

A. 8r
B. 2r
C. 4r
D. `2sqrt(2r)`

Explanation

To find the radius of the \(2^{nd}\) Bohr orbit given that the radius of the first Bohr orbit is \(r\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Bohr Radii**:
The radius of the \(n^{th}\) Bohr orbit is given by the formula:
\[ R_n = \frac{0.529 \, n^2}{Z} \, \text{angstroms} \]
where \(n\) is the principal quantum number (orbit number) and \(Z\) is the atomic number.

2. **Identify the Radius of the First Orbit**:
For the first Bohr orbit (\(n = 1\)):
\[ R_1 = \frac{0.529 \, (1)^2}{Z} = \frac{0.529}{Z} \, \text{angstroms} \]
We are given that this radius is \(r\):
\[ r = \frac{0.529}{Z} \, \text{angstroms} \]

3. **Calculate the Radius of the Second Orbit**:
For the second Bohr orbit (\(n = 2\)):
\[ R_2 = \frac{0.529 \, (2)^2}{Z} = \frac{0.529 \cdot 4}{Z} = \frac{2.116}{Z} \, \text{angstroms} \]

4. **Relate \(R_2\) to \(r\)**:
Since we know \(r = \frac{0.529}{Z}\), we can express \(R_2\) in terms of \(r\):
\[ R_2 = 4 \cdot \frac{0.529}{Z} = 4r \]

5. **Conclusion**:
Therefore, the radius of the \(2^{nd}\) Bohr orbit is:
\[ R_2 = 4r \]

### Final Answer:
The radius of the \(2^{nd}\) Bohr orbit is \(4r\).

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