# Binding Energy and its Calculation

The binding energy of a nucleus is the amount of energy required to completely disassemble it i.e., separate its protons and neutrons. The binding energy per nucleon is the total binding energy of the nucleus divided by the number of nucleons (protons and neutrons) in the nucleus. It is typically expressed in units of MeV.

The binding energy per nucleon generally increases as the atomic number of the nucleus increases up to a certain point, after which it begins to decrease. This is because the nuclear force is strongest for nuclei with an intermediate number of protons and neutrons, and weaker for nuclei with either very low or very high atomic numbers. This behavior can be explained by the balance between the attractive nuclear force and the repulsive electromagnetic force between the protons in the nucleus.

Consider a nucleus of atmic number $Z$ and mass number $A$. Its has $Z$ protons each of mass $m_p$ and $(A-Z)$ neutrons each of rest mass $m_n$. The mass of the nucleus is $M$.

The mass defect of a nucleus is the difference between total rest masses of its protons and neutrons and the mass of the nucleus i.e., \begin{align} \Delta m=\left[Zm_p+(A-Z)m_n\right]-M. \end{align}

The binding energy of the nucleus is given by \begin{align} B=\left[Zm_p+(A-Z)m_n-M\right]c^2. \end{align}

The $Q$-value of a nuclear reaction is given by \begin{align} Q=U_i-U_f. \end{align}

The energy released in a nuclear reaction is given by \begin{align} \Delta E=\Delta m c^2, \end{align} where \begin{align} \Delta m=m_\text{reactants}-m_\text{products} \end{align}

## Problems from IIT JEE

Problem (IIT JEE 2008): Assume that the nuclear binding energy per nucleon $(B/A)$ versus mass number $(A)$ is as shown in the figure. Use this plot to choose the correct choice(s) given below,

1. Fusion of two nuclei with mass number lying in the range of $1 < A < 50$ will release energy.
2. Fusion of two nuclei with mass number lying in the range of $51 < A < 100$ will release energy.
3. Fission of a nucleus lying in the mass range of $100 < A < 200$ will release energy when broken into two equal fragments.
4. Fission of a nucleus lying in the mass range of $200 < A < 260$ will release energy when broken into two equal fragments.

Solution: If $B/A$ of the products is more than that of the reactants than energy is released in the reaction. Fusion of two nuclei with mass number in $1 < A < 50$ produces a nuclei with mass number in $1 < A < 100$. The $B/A$ of product is equal to that of reactants, hence no energy is released.

Fusion of two nuclei with mass number in $51 < A < 100$ produces a nuclei with mass number in $100 < A < 200$. The $B/A$ of product is more than that of reactants, hence energy is released.

The fission of nucleus with mass number $100 < A < 200$ produces products with mass number lying in $1 < A < 200$. The $B/A$ of products is less than or equal to that of reactants hence no energy is released. The fission of nucleus with mass number $200 < A < 260$ produces products with mass number lying in $100 < A < 200$. The $B/A$ of products is more than that of the reactants, hence energy is released.