Radius of 2 nd shell of `He^(+) ` is
( where `a_(0)` is Bohr radius )

Options

A. `3a_(0)`
B. `a_(0)`
C. `(3)/(2) a_(0)`
D. `2a_(0)`

Answer

To find the radius of the second shell of the helium ion \( \text{He}^+ \), we can use the formula for the radius of the nth shell in a hydrogen-like atom:

\[ r_n = \frac{n^2 a_0}{Z} \]

where:
- \( r_n \) is the radius of the nth shell,
- \( n \) is the principal quantum number,
- \( a_0 \) is the Bohr radius,
- \( Z \) is the atomic number of the element.

### Step-by-step Solution:

1. **Identify the values for \( n \) and \( Z \)**:
- For helium ion \( \text{He}^+ \), the atomic number \( Z = 2 \).
- We are looking for the radius of the second shell, so \( n = 2 \).

2. **Substitute the values into the formula**:
\[ r_2 = \frac{n^2 a_0}{Z} \]
Plugging in the values:
\[ r_2 = \frac{(2)^2 a_0}{2} \]

3. **Calculate \( n^2 \)**:
\[ (2)^2 = 4 \]

4. **Substitute \( n^2 \) into the equation**:
\[ r_2 = \frac{4 a_0}{2} \]

5. **Simplify the equation**:
\[ r_2 = 2 a_0 \]

Thus, the radius of the second shell of \( \text{He}^+ \) is \( 2 a_0 \).

### Final Answer:
The radius of the second shell of \( \text{He}^+ \) is \( 2 a_0 \).

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