The radius of which of the following orbit is same as that of the first Bohr's orbit of Hydrogen atom?

Options

A. `He^(+) (n=2)`
B. `Li^(2+)(n=2)`
C. `Li^(2+)(n=3)`
D. `Be^(3+)(n=2)`

Explanation

To determine which of the given options has the same radius as the first Bohr's orbit of the hydrogen atom, we will use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ R_n = \frac{0.529 \, n^2}{Z} \, \text{angstroms} \]

where:
- \( R_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (energy level),
- \( Z \) is the atomic number of the atom.

### Step-by-Step Solution:

1. **Calculate the radius of the first Bohr's orbit of Hydrogen:**
- For Hydrogen, \( Z = 1 \) and \( n = 1 \).
- Using the formula:
\[ R_1 = \frac{0.529 \times 1^2}{1} = 0.529 \, \text{angstroms} \]

2. **Check the radius of the second orbit of Helium ion (He\(^+\)):**
- For He\(^+\), \( Z = 2 \) and \( n = 2 \).
- Calculate:
\[ R_2 = \frac{0.529 \times 2^2}{2} = \frac{0.529 \times 4}{2} = 1.058 \, \text{angstroms} \]
- This is not equal to 0.529 angstroms.

3. **Check the radius of the second orbit of Lithium ion (Li\(^{2+}\)):**
- For Li\(^{2+}\), \( Z = 3 \) and \( n = 2 \).
- Calculate:
\[ R_2 = \frac{0.529 \times 2^2}{3} = \frac{0.529 \times 4}{3} \approx 0.705 \, \text{angstroms} \]
- This is not equal to 0.529 angstroms.

4. **Check the radius of the third orbit of Lithium ion (Li\(^{2+}\)):**
- For Li\(^{2+}\), \( Z = 3 \) and \( n = 3 \).
- Calculate:
\[ R_3 = \frac{0.529 \times 3^2}{3} = \frac{0.529 \times 9}{3} = 1.587 \, \text{angstroms} \]
- This is not equal to 0.529 angstroms.

5. **Check the radius of the second orbit of Beryllium ion (Be\(^{3+}\)):**
- For Be\(^{3+}\), \( Z = 4 \) and \( n = 2 \).
- Calculate:
\[ R_2 = \frac{0.529 \times 2^2}{4} = \frac{0.529 \times 4}{4} = 0.529 \, \text{angstroms} \]
- This is equal to 0.529 angstroms.

### Conclusion:
The radius of the second orbit of Beryllium ion (Be\(^{3+}\)) is the same as that of the first Bohr's orbit of the Hydrogen atom. Therefore, the correct answer is option D.