JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Modern Physics

The volume energy counts the bonds of the interior nucleons, each surrounded by the same number of neighbours, so it is proportional to the number of nucleons, $b_v=a_{3}A$; statement C is correct. Nucleons at the surface have fewer neighbours, so a correction is subtracted that is proportional to the surface area of the nucleus, $\propto R^2\propto A^{2/3}$; this decrease in binding energy with surface area makes statement D correct. The total surface energy scales as $A^{2/3}$, so the surface energy per nucleon scales as $A^{2/3}/A=A^{-1/3}$, not $A^{2/3}$, so statement A is wrong. The Coulomb energy of the protons scales as $\frac{Z(Z-1)}{A^{1/3}}$, not $A^{4/3}$, so statement B is wrong. The assumption of twelve neighbours applies to the bulk nucleons used for the volume energy, whereas surface nucleons have fewer, so statement E is wrong. Hence only statements C and D are correct.