18.2.3 Binding Energy

Chemistry students are familiar with the concept of bond energy which kind of measures how strong a chemical bond is. In nuclear physics, we talk about binding energy, which kind of measures how strongly all the nucleons in a nucleus are bound together.

The binding energy BE of a nucleus is related to its mass defect by the formula

\displaystyle BE=\Delta m.{{c}^{2}}

For the helium-4 nucleus (which has a mass defect of 0.03037u),

\displaystyle \begin{aligned}BE&=(0.03037)(1.66\times {{10}^{{-27}}}).{{(3.00\times {{10}^{8}})}^{2}}\\&=4.53\times {{10}^{{-12}}}\text{ J}\\&=28.3\text{ MeV}\end{aligned}

Let’s make some sense out of these numbers. When a helium-4 nucleus is formed by fusing 2 protons and 2 neutrons, 28.3 MeV of energy is released, as reflected by the mass defect of 0.03037u in the He-4 nucleus. Logically, to reverse the process completely, 28.3 MeV of energy must be supplied to the helium-4 nucleus to separate it back into the 2 protons and 2 neutrons.

Technically, binding energy is defined to be the energy required to separate a nucleus completely into its constituent neutrons and protons. But as you can see, 28.3 MeV is both the energy released when binding the He-4 nucleus as well as the energy required to unbind it.

Frankly, I think “binding energy” is a misnomer that confuses many students. Because BE is not energy that the nucleus possesses. It is the energy that must be returned to recover the energy that has already been lost. Personally I find it easier to think of BE as a kind of “energy debt”, or negative potential energy.

Anyway, since the BE of a nucleus depends on its nuclear composition, each nuclide will have its own BE. Tabulated below are the mass defects and binding energies of a few nuclides[1].

NuclideAZN(1) Mass of nucleons/u(2) Mass of nucleus/u(1)−(2) Mass Defect/uBE/MeV
Deuterium-22112.0159412.0135530.0023882.23
Helium-44224.0318824.0015850.03029728.29
Lithium-77347.0564887.0134890.04299940.15
Beryllium-99459.0724299.0101750.06225458.13
Iron-5656263056.44912655.9219610.527165492.24
Lead-20620682124207.671092205.9337211.7373711622.27
Polonium-21021084126211.702974209.9410891.7618851645.16
Uranium-23523592143236.908487234.9981261.9103611783.80
Uranium-23823892146239.934482238.0050261.9294561801.63

Note that:

  • BE is usually in the MeV range, which is many orders of magnitude larger than the energies we deal with in electronic transitions (eV) or even X-ray (keV).
  • By definition, a free proton or a free neutron has zero BE, since they are already the constituents.
  • Compilation of binding energies are of practical importance because they allow us to calculate the energies of nuclear reactions, as you shall see in the next section.

Concept Test

3611


[1] Actually, what’s published is usually the mass of a neutral atom (electrons included), instead of the mass of the nucleus (stripped of all electrons). So I had to work backwards to figure out the mass of the bare nucleus to compile this table.

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