Question

The ratio of radii of first orbits of He\(^+ \) and Li\(^{2+}\) is:

• A. 2:3
• B. 3:2
• C. 1:2
• D. 4:3

Hint:

Key Fact: In hydrogen-like atoms, the radius of an orbit is inversely proportional to the atomic number \( Z \).

Answer 1

The Correct Option is B

Step 1: Use the Radius Formula for Hydrogen-like Atoms
The radius of the first orbit (\( n = 1 \)) is given by: \[ r \propto \frac{n^2}{Z} \] For \( n = 1 \): \[ r \propto \frac{1}{Z} \]

Step 2: Calculate for He\(^+ \)
For He\(^+ \), \( Z = 2 \), \( n = 1 \): \[ r_{\text{He}^+} \propto \frac{1}{2} = 0.5 \]

Step 3: Calculate for Li\(^{2+} \)
For Li\(^{2+} \), \( Z = 3 \), \( n = 1 \): \[ r_{\text{Li}^{2+}} \propto \frac{1}{3} \approx 0.333 \]

Step 4: Find the Ratio
\[ \text{Ratio} = \frac{r_{\text{He}^+}}{r_{\text{Li}^{2+}}} = \frac{0.5}{0.333} = \frac{3}{2} \]