Question

Show that the minimum energy needed to separate a proton from a nucleus with Z protons and N neutrons is
$∆E=(M_{Z-1,N}+M_B-M_{Z,N})c^2$
where M_Z,N = mass of an atom with Z protons and N neutrons in the nucleus and M_B = mass of a hydrogen atom. This energy is known as proton-separation energy.

Answer 1

Given:
Mass of an atom with Z protons and N neutrons = M_Z,_N
Mass of hydrogen atom = M_H
As hydrogen contains only protons, the reaction will be given by
$$\begin{gathered} E_{Z\mathit{,}N}\to E_{\mathrm{Z}-1,N}+p_1 \\ \Rightarrow E_{Z\mathit{,}N}\to E_{z-1,N}{+}^1{\mathrm{H}}_1 \end{gathered}$$
∴ Minimum energy needed to separate a proton, $∆E=(M_{\mathrm{Z}-1,\mathrm{N}}+M_{\mathrm{H}}-M_{\mathrm{Z},\mathrm{N}})c^2$