Q.Explain the terms mass defect and binding energy. How are they related ?

Concept: Nuclear binding energy and mass-energy equivalence ($E = mc^2$).

Mass defect is the difference between the calculated mass of the individual nucleons (protons + neutrons) and the actual measured mass of the nucleus. This "missing" mass ($\Delta m$) is not lost — it is converted into the energy that holds the nucleus together. …

The Core Idea

When you add up the masses of the protons and neutrons that make up a nucleus, you get a number that is larger than the mass of the actual nucleus itself. That missing mass is the mass defect. It isn't really "lost" — it has been converted into the energy that holds the nucleus together, called the binding energy. This is a direct consequence of Einstein's famous equation $E = mc^2$, which tells us that mass and energy are two sides of the same coin.

Think of it this way: to assemble a nucleus from its individual nucleons, you would need to supply energy to overcome the electrostatic repulsion between protons. But nature does the opposite — when nucleons come together, they release energy (the binding energy), and that released energy carries away some mass. So the assembled nucleus is lighter than its parts.

Step-by-Step Explanation

1. Define mass defect ($\Delta m$) For a nucleus with $Z$ protons, $N$ neutrons, and mass $M_{\text{nucleus}}$, the mass defect is:

where $m_p$ is the mass of a proton and $m_n$ is the mass of a neutron.

2. Define binding energy ($E_b$) The binding energy is the energy released when a nucleus forms from its constituent nucleons. It is also the energy you would need to supply to break the nucleus apart. Using Einstein's mass-energy equivalence:

where $c = 3 \times 10^8 \text{ m/s}$ is the speed of light.

3. Understand the relationship

   The mass defect and binding energy are directly proportional. A larger mass defect means more mass has been converted into energy, so the nucleus is more tightly bound and has a larger binding energy. Conversely, a smaller mass defect means a less stable nucleus.

   $E_b = \Delta m \, c^2$

4. Why this matters for stability

   A more useful quantity for comparing nuclei is the binding energy per nucleon:

where $A = Z + N$ is the total number of nucleons.

• Higher binding energy per nucleon → more stable nucleus.
• Iron-56 has the highest binding energy per nucleon (~8.8 MeV), making it the most stable nucleus. …