Question

The ratio of the difference between the radii of \( 3^{rd} \) and \( 4^{th} \) orbits of the \( He^+ \) and those of \( Li^{2+} \) is:

• A. \( 2 : 3 \)
• B. \( 3 : 1 \)
• C. \( 1 : 3 \)
• D. \( 3 : 2 \)

Hint:

For hydrogen-like species, radius varies as \( \frac{n^2}{Z} \)Always compare orbit differences carefully using same formula.

Answer 1

The Correct Option is D

Step 1: Use Bohr radius formula.
For hydrogen-like species:
:contentReference[oaicite:0]{index=0} where \( n \) = orbit number, \( Z \) = atomic number, and \( a_0 \) = Bohr radius.

Step 2: Calculate difference for \( He^+ \) (Z = 2).
\[ r_3 = \frac{3^2}{2}a_0 = \frac{9}{2}a_0 \]
\[ r_4 = \frac{4^2}{2}a_0 = \frac{16}{2}a_0 = 8a_0 \]
Difference:
\[ \Delta r_{He^+} = r_4 - r_3 = 8a_0 - \frac{9}{2}a_0 = \frac{16 - 9}{2}a_0 = \frac{7}{2}a_0 \]

Step 3: Calculate difference for \( Li^{2+} \) (Z = 3).
\[ r_3 = \frac{9}{3}a_0 = 3a_0 \]
\[ r_4 = \frac{16}{3}a_0 \]
Difference:
\[ \Delta r_{Li^{2+}} = \frac{16}{3}a_0 - 3a_0 = \frac{16 - 9}{3}a_0 = \frac{7}{3}a_0 \]

Step 4: Take ratio.
\[ \frac{\Delta r_{He^+}}{\Delta r_{Li^{2+}}} = \frac{\frac{7}{2}a_0}{\frac{7}{3}a_0} = \frac{3}{2} \]

Step 5: Conclusion.
Thus, the required ratio is:
\[ \boxed{3 : 2} \]