The orbit having Bohr radius equal to 1st Bohr orbit of H-atom is :

Options

A. `n=2` of `He^(+)`
B. `n=2` of `B^(+4)`
C. `n=3` of `Li^(2+)`
D. `n=2` of `Be^(+3)`

Explanation

To find the orbit that has a Bohr radius equal to the first Bohr orbit of the hydrogen atom, we can follow these steps:

### Step 1: Calculate the Bohr radius for the first orbit of the hydrogen atom.
The formula for the radius of the nth Bohr orbit is given by:

\[ r_n = 0.529 \times \frac{n^2}{Z} \]

For the hydrogen atom (Z = 1) and the first orbit (n = 1):

\[ r_1 = 0.529 \times \frac{1^2}{1} = 0.529 \text{ Å} \]

### Step 2: Set up the condition for other orbits to have the same radius.
We want to find orbits where the radius \( r \) is equal to \( 0.529 \text{ Å} \). This means we need to find conditions where:

\[ 0.529 = 0.529 \times \frac{n^2}{Z} \]

This simplifies to:

\[ \frac{n^2}{Z} = 1 \]

### Step 3: Analyze the options provided.
We will evaluate each option to see which satisfies the condition \( \frac{n^2}{Z} = 1 \).

1. **Option 1: Helium (n = 2, Z = 2)**
\[ \frac{2^2}{2} = \frac{4}{2} = 2 \quad (\text{Not equal to } 1) \]

2. **Option 2: Boron (n = 2, Z = 5)**
\[ \frac{2^2}{5} = \frac{4}{5} = 0.8 \quad (\text{Not equal to } 1) \]

3. **Option 3: Lithium ion (n = 3, Z = 3)**
\[ \frac{3^2}{3} = \frac{9}{3} = 3 \quad (\text{Not equal to } 1) \]

4. **Option 4: Beryllium ion (n = 2, Z = 4)**
\[ \frac{2^2}{4} = \frac{4}{4} = 1 \quad (\text{Equal to } 1) \]

### Step 4: Conclusion
The orbit that has a Bohr radius equal to the first Bohr orbit of the hydrogen atom is when \( n = 2 \) for the beryllium ion (Z = 4).

### Final Answer:
The orbit having a Bohr radius equal to the first Bohr orbit of the hydrogen atom is **n = 2 for the beryllium ion (Be\(^+\))**.

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