182 Binding Energy
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Learning Objectives
- Define and discuss binding energy.
- Calculate the binding energy per nucleon of a particle.
The more tightly bound a system is, the stronger the forces that hold it together and the greater the energy required to pull it apart. We can therefore learn about nuclear forces by examining how tightly bound the nuclei are. We define the binding energy (BE) of a nucleus to be the energy required to completely disassemble it into separate protons and neutrons. We can determine the BE of a nucleus from its rest mass. The two are connected through Einstein’s famous relationship \(E=\left(\Delta m\right){c}^{2}\). A bound system has a smaller mass than its separate constituents; the more tightly the nucleons are bound together, the smaller the mass of the nucleus.
Imagine pulling a nuclide apart as illustrated in (Figure). Work done to overcome the nuclear forces holding the nucleus together puts energy into the system. By definition, the energy input equals the binding energy BE. The pieces are at rest when separated, and so the energy put into them increases their total rest mass compared with what it was when they were glued together as a nucleus. That mass increase is thus \(\text{Δ}m=\text{BE}/{c}^{2}\). This difference in mass is known as mass defect. It implies that the mass of the nucleus is less than the sum of the masses of its constituent protons and neutrons. A nuclide \({}^{A}\text{X}\) has \(Z\) protons and \(N\) neutrons, so that the difference in mass is
Thus,
where \({m}_{\text{tot}}\) is the mass of the nuclide \({}^{A}\text{X}\), \({m}_{p}\) is the mass of a proton, and \({m}_{n}\) is the mass of a neutron. Traditionally, we deal with the masses of neutral atoms. To get atomic masses into the last equation, we first add \(Z\) electrons to \({m}_{\text{tot}}\), which gives \(m\left({}^{A}\text{X}\right)\), the atomic mass of the nuclide. We then add \(Z\) electrons to the \(Z\) protons, which gives \(\text{Zm}\left({}^{1}\text{H}\right)\), or \(Z\) times the mass of a hydrogen atom. Thus the binding energy of a nuclide \({}^{A}\text{X}\) is
The atomic masses can be found in Appendix A, most conveniently expressed in unified atomic mass units u (\(1\phantom{\rule{0.25em}{0ex}}\text{u}=\text{931}\text{.}5\phantom{\rule{0.25em}{0ex}}\text{MeV}/{c}^{2}\)). BE is thus calculated from known atomic masses.
Nuclear Decay Helps Explain Earth’s Hot Interior
A puzzle created by radioactive dating of rocks is resolved by radioactive heating of Earth’s interior. This intriguing story is another example of how small-scale physics can explain large-scale phenomena.
Radioactive dating plays a role in determining the approximate age of the Earth. The oldest rocks on Earth solidified about \(3\text{.}5×{\text{10}}^{9}\) years ago—a number determined by uranium-238 dating. These rocks could only have solidified once the surface of the Earth had cooled sufficiently. The temperature of the Earth at formatio