Question

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nucleus. Which of the following statement(s) is(are) correct?

• A. $E_b^p-E_b^n$ is proportional to $Z(Z-1)$ where $Z$ is the atomic number of the nucleus.
• B. $E_b^p-E_b^n$ is proportional to $A^{-\frac{1}{3}}$ where $A$ is the mass number of the nucleus.
• C. $E_b^p-E_b^n$ is positive.
• D. $E_b^p$ increases if the nucleus undergoes a beta decay emitting a positron.

Answer 1

Answer: (A), (B), (D)

Binding energy of proton \& neutron due to nuclear force is same. So difference in binding energy is only due to electrostatic P.E. and it is positive
$E _0^{ P }- E _0^{ n }=\text { electrostatic P.E. }$
$= Z \times P . E$. of one proton
$=Z \times \frac{1}{4 \pi \varepsilon_0} \frac{(Z-1) e ^2}{ R }$
Where $R = R _0 A ^{1 / 3}$
$=\frac{1}{4 \pi \varepsilon_0} \frac{Z(Z-1) e ^2}{ R _0 A ^{\frac{1}{3}}}$