A-level Physics (Advancing Physics)/Binding Energy

It takes energy to pull nuclei apart. The amount of work (energy) which must be done in order to pull all of the neutrons and protons in a nucleus infinitely far apart from each other is known as the binding energy of the nucleus. Practically, pulling them all apart far enough to stop them interacting will do.

If energy must be put in to a nucleus to break it apart, where does this energy go? The answer lies in the fact that if you add up the masses of all the protons and neutrons in a nucleus individually, it is a little bit more than the actual mass of the nucleus. The binding energy put in to break the nucleus apart has 'become' mass in the individual baryons. So, the binding energy of a nucleus can be calculated using the following formula:

$E_{b}=(n_{N}m_{N}+n_{Z}m_{Z}-M)c^{2}$,

where n_N and n_Z are the numbers of neutrons and protons in the nucleus, m_N and m_Z are the masses of neutrons and protons, M is the mass of the nucleus and c is the speed of light (3 x 10⁸ ms⁻¹).

The Unified Atomic Mass Unit

The unified atomic mass unit, denoted u, is roughly equal to the mass of one proton or neutron. 1 u = 1.660538782 x 10⁻²⁷ kg. They are useful since 1 mole of atoms with a mass of 1 u each will weigh exactly 1 gram. However, when dealing with binding energy, you must never use atomic mass units in this way. The mass defect is so small that using atomic mass units will result in a completely wrong answer. If you want to use them with lots of decimal places, then you will save writing in standard form.

Data

The following table gives the masses in kg and u of the proton and the neutron:

Name	Mass (kg)	Mass (u)
Proton	1.67262164 x 10−27	1.00727647
Neutron	1.67492729 x 10−27	1.00866492

The Binding Energy Curve

Different nuclei have different binding energies. These are determined by the combination of protons and neutrons in the nucleus. These are shown in the following graph:

The position of Iron-56 at the top is important. If you take two nuclei completely apart, you do work. If you then put all the baryons back together again as one nucleus, you will get energy back out. Sometimes, the energy you get back out will be more than the work you had to do to take the nuclei apart. Overall, you release energy by fusing the nuclei together. This happens to nuclei which are smaller than Iron-56. Nuclei which are larger than Iron-56 will give out less energy when fused than you had to put in to take them apart into their constituent baryons in the first place. To the right of Iron-56, nuclear fusion, overall, requires energy.

If you take only one nucleus apart you still do work. If you stick its protons and neutrons back together, but this time in two lumps, you will get energy out. Again, sometimes this energy will be greater than the work you had to do to take them apart in the first place. Nuclear fission will be releasing energy. This occurs when the nucleus is larger than Iron-56. If the energy released is less than the initial work you put in, then nuclear fission, overall, requires energy. This happens when the nucleus is smaller than Iron-56.

This can be summarized in the following table:

Type of Nucleus	Nuclear Fusion	Nuclear Fission
smaller than Iron-56	releases energy	absorbs energy
Iron-56	≈ no energy change	≈ no energy change
Larger than Iron-56	absorbs energy	releases energy

Questions

1. Deuterium (an isotope of Hydrogen with an extra neutron) has a nuclear mass of 2.01355321270 u. What is its binding energy?

2. Uranium-235 has a nuclear mass of 235.0439299 u. It contains 92 protons. What is its binding energy?

3. How would you expect H-2 and U-235 to be used in nuclear reactors? Why?

Worked Solutions