The radius of which of the following orbit is the 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 orbit has the same radius as the first Bohr orbit of the hydrogen atom, we will follow these steps:

### Step 1: Understand the formula for the radius of Bohr orbits
The radius of the nth Bohr orbit for a hydrogen-like atom is given by the formula:
\[ r_n = 0.529 \times \frac{n^2}{Z} \text{ angstroms} \]
where \( n \) is the principal quantum number and \( Z \) is the atomic number.

### Step 2: Identify the parameters for the first Bohr orbit of hydrogen
For hydrogen:
- The atomic number \( Z = 1 \)
- The principal quantum number for the first orbit \( n = 1 \)

### Step 3: Calculate the radius of the first Bohr orbit of hydrogen
Substituting the values into the formula:
\[ r_1 = 0.529 \times \frac{1^2}{1} = 0.529 \text{ angstroms} \]

### Step 4: Set up the condition for other orbits
We need to find another orbit where the radius is also \( 0.529 \) angstroms. This means we need to find a condition where:
\[ \frac{n^2}{Z} = 1 \]
This simplifies to:
\[ n^2 = Z \]

### Step 5: Analyze the options
We will check each option to see if \( n^2 = Z \).

1. **Option A: Helium ion (He\(^+\))**
- \( Z = 2 \), \( n = 2 \)
- \( n^2 = 2^2 = 4 \) which is not equal to \( Z \) (4 ≠ 2).

2. **Option B: Lithium ion (Li\(^{2+}\))**
- \( Z = 3 \), \( n = 2 \)
- \( n^2 = 2^2 = 4 \) which is not equal to \( Z \) (4 ≠ 3).

3. **Option C: Lithium ion (Li\(^{2+}\))**
- \( Z = 3 \), \( n = 3 \)
- \( n^2 = 3^2 = 9 \) which is not equal to \( Z \) (9 ≠ 3).

4. **Option D: Beryllium ion (Be\(^{3+}\))**
- \( Z = 4 \), \( n = 2 \)
- \( n^2 = 2^2 = 4 \) which is equal to \( Z \) (4 = 4).

### Step 6: Conclusion
The only option that satisfies the condition \( n^2 = Z \) is **Option D: Beryllium ion (Be\(^{3+}\))**. Therefore, the radius of the second orbit of the beryllium ion is the same as that of the first Bohr orbit of the hydrogen atom.

### Final Answer
**The radius of the second orbit of the beryllium ion (Be\(^{3+}\)) is the same as that of the first Bohr orbit of the hydrogen atom.**

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