The difference between the radii of 3rd and 4th orbits of Li2+ is R1. The difference between the radii of 3rd and 4th…

🧠 Measuring the Gap We are calculating the spatial distance between the 3rd and 4th orbits ($\Delta R = r_4 - r_3$) for two different ions: $\ce{Li^2+}$ and $\ce{He^+}$. The "gap width" depends inversely on $Z$.

🗺️ The Algebraic Ratio

1. Orbit Gap general formula: $\Delta R = r_4 - r_3 = \frac{(4^2 - 3^2) a_0}{Z} = \frac{7 a_0}{Z}$
2. For $\ce{Li^2+}$ ($Z=3$): $\Delta R_1 = 7a_0/3$.
3. For $\ce{He^+}$ ($Z=2$): $\Delta R_2 = 7a_0/2$.
4. Ratio Generation: $\frac{\Delta R_1}{\Delta R_2} = \frac{7a_0/3}{7a_0/2} = \frac{2}{3}$

⚡ The Inverse Z Law Since the shells (3 and 4) are the same for both cases, the gap size is strictly inversely proportional to $Z$. $\text{Ratio} = Z(\ce{He^+}) : Z(\ce{Li^2+}) = 2:3$. You can leap to the answer without calculating "7" or using "a0" at all!

⚠️ Common Traps The common $Z_1/Z_2$ vs $Z_2/Z_1$ flip. Always remember: a LARGER $Z$ (stronger pull) makes the atom SMALLER and the gaps TIGHTER. So Li ($Z=3$) must have the smaller gap. Ratio must be $2:3$, not $3:2$.

$\boxed{\text{Answer: (c)}}$