Step 1: Understanding the Concept:
Bohr's model explains that for hydrogen-like species (ions with only one electron), the orbit radius is dictated by the balance of electrostatics and centripetal force. Increasing the number of protons in the nucleus increases the attraction, pulling the electron closer.
Key Formula or Approach:
The radius \( r_n \) for the \( n \)-th orbit is given by:
\[ r_n = 0.529 \frac{n^2}{Z} \, \text{\AA} \]
where \( n \) is the principal quantum number and \( Z \) is the atomic number.
Step 2: Detailed Explanation:
For Hydrogen (H): \( Z = 1 \). Ground state means \( n = 1 \).
\( r_H = 0.529 \frac{1^2}{1} = 0.529 \text{\AA} \approx 0.53 \text{\AA} \).
For Lithium ion (\( \text{Li}^{2+} \)): \( Z = 3 \). "Similar state" implies ground state, so \( n = 1 \).
Using the ratio:
\[ \frac{r_{Li^{2+}}}{r_H} = \frac{Z_H}{Z_{Li^{2+}}} \]
\[ r_{Li^{2+}} = r_H \times \frac{1}{3} \]
\[ r_{Li^{2+}} = \frac{0.53}{3} \approx 0.1766 \text{\AA} \]
Looking at the options, \( 0.17 \text{\AA} \) is the closest approximation.
Step 3: Final Answer:
The radius of the \( \text{Li}^{2+} \) ion is \( 0.17 \text{\AA} \).