The radius of hydrogen atom in the ground state `0.53 Å`. The radius of `Li^(2+)` ion (Atomic number = 3) in a similar state is

Options

A. (a) 1.06 Å
B. (b) 0.265 Å
C. (c) 0.17 Å
D. (d) 0.53 Å

Explanation

To find the radius of the `Li^(2+)` ion in the ground state, we can use the formula derived from Bohr's theory of atomic structure. The radius of an atom in a particular state is given by the formula:

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

where:
- \( r \) is the radius of the atom,
- \( n \) is the principal quantum number (which is 1 for the ground state),
- \( Z \) is the atomic number,
- \( r_H \) is the radius of the hydrogen atom in the ground state (0.53 Å).

### Step-by-Step Solution:

1. **Identify the parameters:**
- For hydrogen (H), the radius in the ground state \( r_H = 0.53 \, \text{Å} \).
- The atomic number \( Z \) for lithium ion \( Li^{2+} \) is 3.
- The principal quantum number \( n \) for the ground state is 1.

2. **Use the formula for the radius:**
\[ r = \frac{n^2}{Z} \times r_H \]

3. **Substitute the values into the formula:**
\[ r = \frac{1^2}{3} \times 0.53 \, \text{Å} \]

4. **Calculate the radius:**
\[ r = \frac{1}{3} \times 0.53 \, \text{Å} = 0.1767 \, \text{Å} \]

5. **Round the answer:**
\[ r \approx 0.17 \, \text{Å} \]

### Final Answer:
The radius of the `Li^(2+)` ion in the ground state is approximately \( 0.17 \, \text{Å} \).

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