Mass-energy and Nuclear Binding Energy

Mass–Energy Equivalence: The Deep Link

The story of nuclear binding energy begins with a profound idea from Einstein’s special relativity: mass and energy are not separate quantities. They are two faces of the same coin, connected by the most famous equation in physics:

Here, $E$ is energy, $m$ is mass, and $c$ is the speed of light in vacuum ($3 \times 10^8 \text{ m/s}$). This means that any object with mass possesses an intrinsic energy — its rest energy — simply by virtue of having mass. Conversely, if you add energy to a system, its mass increases; if you remove energy, its mass decreases. The conversion factor $c^2$ is enormous, so a tiny amount of mass corresponds to a huge amount of energy.

Mass Defect: Where Does the Mass Go?

Consider a nucleus of an atom. It is made of protons and neutrons (collectively called nucleons). If you add up the masses of all the individual protons and neutrons that make up a nucleus, you get a total mass that is larger than the actual mass of the nucleus itself. This difference is called the mass defect.

Let:

• $m_p$ = mass of a proton
• $m_n$ = mass of a neutron
• $M(A, Z)$ = actual mass of a nucleus with atomic number $Z$ and mass number $A$ (so it has $Z$ protons and $A-Z$ neutrons)

Then the mass defect $\Delta m$ is defined as:

The mass defect is always positive. It represents the mass that has "disappeared" when the nucleons came together to form the nucleus.

Nuclear Binding Energy: The Glue That Holds the Nucleus

The energy equivalent of the mass defect is the nuclear binding energy of the nucleus. It is the energy that must be supplied to the nucleus to separate it into its individual protons and neutrons. Equivalently, it is the energy released when the nucleus is formed from its constituent nucleons.

Using Einstein's relation, the binding energy $E_b$ is:

This is the total binding energy of the entire nucleus. It is a large positive number, typically in the range of hundreds to thousands of MeV.

Binding Energy per Nucleon: A Measure of Stability

A more useful quantity for comparing the stability of different nuclei is the binding energy per nucleon. It is simply the total binding energy divided by the total number of nucleons:

This quantity tells us, on average, how much energy is needed to remove a single nucleon from the nucleus. A higher binding energy per nucleon means the nucleus is more tightly bound and therefore more stable.

Properties of Binding Energy per Nucleon

The textbook lists several important properties of the binding energy per nucleon curve (a plot of binding energy per nucleon vs. mass number $A$). Here they are, with full derivations where the book provides them.

›Proof

Property 1: The binding energy per nucleon is practically constant for nuclei with mass number $A$ between 30 and 170.

This is an empirical observation from experimental data. For these medium-mass nuclei, the binding energy per nucleon is roughly 8.0 MeV per nucleon. The curve is nearly flat in this region. This constancy is a key feature of the nuclear force — it is short-ranged and saturates, meaning each nucleon only interacts with its nearest neighbours, not with all nucleons in the nucleus. For a large enough nucleus, the interior nucleons are fully surrounded, so the binding energy per nucleon becomes roughly the same.

›Proof

Property 2: The binding energy per nucleon is relatively small for both very light nuclei ($A < 30$) and very heavy nuclei ($A > 170$).

For light nuclei, the curve rises steeply from about 1.1 MeV for deuterium ($A=2$) to about 7.6 MeV for carbon ($A=12$), then continues to rise more slowly. For heavy nuclei, the curve gradually falls from about 8.5 MeV for iron ($A=56$) to about 7.6 MeV for uranium ($A=238$). This decrease is due to the increasing effect of the Coulomb repulsion between protons, which becomes more significant as the number of protons grows. The nuclear force is short-ranged and cannot overcome the long-range electrostatic repulsion in very large nuclei.

›Proof

Property 3: The binding energy per nucleon reaches a maximum at $A \approx 56$ (iron).

Iron-56 has the highest binding energy per nucleon of any known nucleus, at about 8.8 MeV. This means iron is the most stable nucleus. This is why iron is the endpoint of nuclear fusion in stars — fusing iron would require energy input rather than releasing it.

›Proof

Property 4: The binding energy per nucleon curve explains the release of energy in nuclear fission and fusion.

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