Derive the formula for the binding energy of a nucleus by explaining the mass defect of the nucleus and write the formula for the binding energy per nucleon.

(N/A) The nucleus is made up of neutrons and protons. Therefore, it may be expected that the mass of the nucleus is equal to the total mass of its individual protons and neutrons.
However, the nuclear mass $M$ is found to be always less than the sum of the masses of its constituents (neutrons and protons) in the free state.
Suppose the mass of a nucleus $_{Z}^{A}X$ is $M$. If we indicate the masses of a proton and a neutron in the free state as $m_{p}$ and $m_{n}$ respectively, then $M < (Z m_{p} + N m_{n})$, where $N = A - Z$ is the neutron number. The difference between the total mass of the constituents of a nucleus and the actual mass of the nucleus is called the mass defect $(\Delta M)$.
$\therefore \Delta M = [Z m_{p} + (A - Z) m_{n}] - M$ is the formula for mass defect.
From Einstein's mass-energy equivalence, the energy equivalent to the mass defect is $E_{b} = \Delta M c^{2}$, where $c$ is the velocity of light in vacuum $(c \approx 3 \times 10^{8} \ m/s)$.
This energy $E_{b}$ is called the binding energy of the nucleus. It represents the energy that would be released when $Z$ protons and $N$ neutrons combine to form a nucleus, or the energy required to separate a nucleus into its individual nucleons.
By dividing the binding energy of the nucleus by the total number of nucleons $(A)$, the binding energy per nucleon $(E_{bn})$ is obtained.
$\therefore E_{bn} = \frac{E_{b}}{A}$
The binding energy per nucleon is a measure of the stability of a nucleus.