The radius of Bohr 's first orbit in hydrogen atom is 0.53 Å the radius of second orbit in He+ will be

Options

A. 0.0265 nm
B. 0.0530 nm
C. 0.1060 nm
D. 0.2120 nm

Answer

To find the radius of the second orbit in the He+ ion, we can use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ R_n = R_0 \frac{n^2}{Z} \]

Where:
- \( R_n \) is the radius of the nth orbit,
- \( R_0 \) is the Bohr radius for hydrogen (0.53 Å),
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number of the element.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- The radius of the first orbit in hydrogen (\( R_1 \)) = 0.53 Å = \( 0.53 \times 10^{-10} \) m.
- For He+, the atomic number \( Z = 2 \).
- The principal quantum number for the second orbit \( n = 2 \).

2. **Use the Formula for the Radius of the nth Orbit:**
\[ R_n = R_0 \frac{n^2}{Z} \]

3. **Substituting the Values:**
\[ R_2 = 0.53 \times 10^{-10} \, \text{m} \cdot \frac{2^2}{2} \]

4. **Calculate \( n^2 \):**
- \( n^2 = 2^2 = 4 \).

5. **Substituting \( n^2 \) into the Equation:**
\[ R_2 = 0.53 \times 10^{-10} \, \text{m} \cdot \frac{4}{2} \]
\[ R_2 = 0.53 \times 10^{-10} \, \text{m} \cdot 2 \]

6. **Perform the Calculation:**
\[ R_2 = 1.06 \times 10^{-10} \, \text{m} \]

7. **Convert to Nanometers:**
- Since \( 1 \, \text{nm} = 10^{-9} \, \text{m} \),
\[ R_2 = 1.06 \times 10^{-10} \, \text{m} = 0.106 \, \text{nm} \]

8. **Final Answer:**
- The radius of the second orbit in He+ is **0.1060 nm**.