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

Options

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

Answer

To determine which orbit has the same radius as the first Bohr's orbit of the hydrogen atom, we need to recall the formula for the radius of the nth orbit in a hydrogen atom according to Bohr's model.

### Step-by-Step Solution:

1. **Recall the Formula for the Radius of Bohr's Orbit:**
The radius of the nth orbit in a hydrogen atom is given by the formula:
\[ r_n = n^2 \cdot \frac{h^2}{4 \pi^2 k e^2 m} \]
where:
- \( n \) is the principal quantum number (1 for the first orbit),
- \( h \) is Planck's constant,
- \( k \) is Coulomb's constant,
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron.

For the first orbit (n=1), the radius simplifies to:
\[ r_1 = \frac{h^2}{4 \pi^2 k e^2 m} \]

2. **Calculate the Radius of the First Bohr's Orbit:**
The radius of the first Bohr orbit (n=1) for hydrogen is approximately:
\[ r_1 \approx 0.529 \text{ Å} \quad (\text{Angstroms}) \]

3. **Identify the Radius of Other Atoms:**
To find which orbit has the same radius, we need to compare this radius with the radii of the first orbit of other hydrogen-like atoms (atoms with only one electron, such as He\(^+\), Li\(^{2+}\), etc.).

The radius of the nth orbit for a hydrogen-like atom is given by:
\[ r_n = \frac{n^2}{Z} \cdot r_1 \]
where \( Z \) is the atomic number of the atom.

4. **Set Up the Equation:**
To find orbits with the same radius as the first Bohr orbit of hydrogen, we set:
\[ r_n = r_1 \]
This implies:
\[ \frac{n^2}{Z} \cdot r_1 = r_1 \]

5. **Solve for n and Z:**
Dividing both sides by \( r_1 \) (assuming \( r_1 \neq 0 \)):
\[ \frac{n^2}{Z} = 1 \]
This leads to:
\[ n^2 = Z \]

6. **Identify Possible Values of n and Z:**
- For \( n = 1 \), \( Z = 1 \) (Hydrogen)
- For \( n = 2 \), \( Z = 4 \) (Helium ion, He\(^+\))
- For \( n = 3 \), \( Z = 9 \) (Lithium ion, Li\(^{2+}\))

The only orbit that has the same radius as the first Bohr orbit of hydrogen is for \( n = 1 \) and \( Z = 1 \) (Hydrogen).

### Conclusion:
The radius of the first Bohr's orbit of hydrogen atom is the same as that of the first orbit of the hydrogen atom itself.