Nuclear Binding Energy Calculator
Work out the mass defect, the binding energy and the binding per nucleon of a nucleus from its atomic mass.
What it does
A nucleus is lighter than the pieces it was made from, and the difference is the energy that holds it together. Pull a heavy nucleus apart and that energy comes out, which is why a bomb works and why a star shines.
The atomic mass evaluation explains the arithmetic. The binding energy per nucleon is one over the mass number times the number of protons times the mass of the hydrogen atom plus the number of neutrons times the mass of the neutron, minus the mass of the nucleus. The National Nuclear Data Center lists the same quantity, the binding energy per nucleon in keV, as one of the ground state properties it publishes for every known nuclide.
That page also explains why the quantity is so useful. The National Nuclear Data Center describes the binding energy of a nucleus as the difference between the combined masses of the nucleons and the mass of the nucleus itself, and the history of the subject notes that the binding energy per nucleon varies as the mass number increases, which is the curve that decides whether a nucleus is stable and how much energy a reaction can release.
This page works that out for a list of common nuclides using the evaluated atomic masses. Give it a nuclide and it returns the mass defect, the total binding energy, the binding energy for each nucleon, and how that compares with iron 56, which is the most tightly bound of the common elements.
The limits are worth stating. The numbers come from evaluated atomic masses, which are measured rather than calculated, and the binding energy is only as good as the mass it comes from. The comparison with iron 56 says nothing about whether a nucleus is stable, which depends on the competition between the strong force that holds it together and the electrostatic repulsion inside it.
How to use
- Pick the nuclide from the list, which covers the light elements, the common heavy ones and the fuels used in reactors.
- Read the mass defect, which is the mass missing from the nucleus compared with its free parts.
- Read the total binding energy, which is what has to be supplied to take the nucleus apart.
- Read the binding energy per nucleon, which is the number that decides how much energy a reaction can release.
- Compare it with iron 56. A value below 8.790 MeV per nucleon means the nucleus has energy to give up by splitting, and one above means it would rather be part of something heavier.
- Remember the figure is a property of the nucleus, not a prediction of how it will behave in a reactor.
Formula
- Z
- Number of protons in the nucleus.
- N
- Number of neutrons in the nucleus.
- A
- Mass number, the total number of nucleons, Z plus N.
- M(atom)
- Measured atomic mass of the nuclide in atomic mass units.
The atomic mass is used rather than the nuclear mass because the electrons cancel out, since there are Z electrons in the atom and Z protons in the nucleus and the masses of the electron and the proton nearly cancel. The binding energy per nucleon is what the nuclear data centres tabulate, and the mass excess they quote is the same quantity seen from the other side.
Worked example
Example: the iron 56 nucleus
Defaults: iron 56, with 26 protons and 30 neutrons.
- The free protons and neutrons would weigh 26 x 1.00782503 + 30 x 1.008664916 = 56.463398 u.
- The measured atomic mass of iron 56 is 55.934942 u, so the mass defect is 0.528456 u.
- Turning that into energy with 931.494 MeV for each atomic mass unit gives 0.528456 x 931.494 = 492.254 MeV.
- Spread over the 56 nucleons that is 492.254 over 56 = 8.790 MeV for each one.
- That is the highest binding per nucleon of the common elements, which is why iron sits at the peak of the curve and why fusing anything lighter or splitting anything heavier both release energy.
- Uranium 235 by contrast gives 1783.869 MeV in total, which is 7.591 MeV per nucleon, or 86.36 percent of the iron figure.
- The tool shows a mass defect of 0.528456 u, a binding energy of 492.254 MeV, 8.790 MeV per nucleon, 56 nucleons and 100.00 percent of the binding of iron 56.
Reference table
| Nuclide | Mass defect | Binding energy | Binding per nucleon |
|---|---|---|---|
| Helium 4 | 0.030377 u | 28.296 MeV | 7.074 MeV |
| Carbon 12 | 0.09894 u | 92.162 MeV | 7.680 MeV |
| Oxygen 16 | 0.137005 u | 127.619 MeV | 7.976 MeV |
| Iron 56 | 0.528456 u | 492.254 MeV | 8.790 MeV |
| Uranium 235 | 1.915062 u | 1,783.869 MeV | 7.591 MeV |
Limits
- The numbers come from measured atomic masses, so the binding energy is only as good as the mass evaluation behind it.
- The comparison with iron 56 says nothing about whether a nucleus is stable, which depends on the balance between the strong force and the electrostatic repulsion.
- A binding energy per nucleon is an average, and for a light nucleus the individual nucleons are bound very differently from each other.
- Nothing here accounts for a nucleus that is excited, and an excited state has less binding than the ground state.
Common mistakes
- Using the mass number A as the mass of the nucleus, which loses the binding energy itself and makes the mass defect come out wrong.
- Forgetting the electron masses, which is why the atomic mass and the nuclear mass give slightly different answers for a light nucleus.
- Using the mass of a neutral atom with the number of protons and neutrons swapped around, which gives a negative mass defect.
- Expecting the binding per nucleon to rise with the mass number. It rises to iron and then falls.
- Reading a high total binding energy as a high binding per nucleon. Uranium has a huge total and a middling average.
Try it yourself
Quick checks. Answers sit tucked under each question.
How much energy is released when uranium 235 splits?
Each fission releases about 200 MeV, and the reason is the gap in binding per nucleon. Uranium 235 sits at 7.591 MeV per nucleon while the products of a split sit nearer iron at about 8.8, so the difference comes out as energy, and the total binding energy of 1783.869 MeV is what the whole nucleus is holding together.
Why is iron the end of the line for ordinary fusion?
Because iron 56 has the highest binding per nucleon of the common elements, at 8.790 MeV, so there is no energy left to win by fusing it into something heavier. Fusing light elements up towards iron releases energy, and splitting anything heavier than iron releases energy too, but iron itself is the peak of the curve.
How much heavier are the free parts than the iron nucleus?
The 26 protons and 30 neutrons would weigh 56.463398 u, while the iron atom weighs 55.934942 u, a difference of 0.528456 u. That missing mass is what holds the nucleus together, and at 931.494 MeV for each atomic mass unit it comes to 492.254 MeV.
Questions
What is the mass defect?
The difference between the mass of the free protons and neutrons and the mass of the nucleus they form. It is a small number, a fraction of an atomic mass unit, but it converts into a large amount of energy.
What is binding energy per nucleon?
The total binding energy divided by the number of nucleons. The atomic mass evaluation tabulates it, and the National Nuclear Data Center publishes it for every known nuclide as one of the ground state properties.
Why is iron 56 special?
It has the highest binding energy per nucleon of the common elements, at 8.790 MeV, so it is the point where the curve of binding against mass number turns over, and the point where fusion stops paying and fission stops paying.
How much energy does one atomic mass unit of mass defect give?
931.494 MeV, which is the conversion the atomic mass evaluation uses when it tabulates the binding energy per nucleon from the measured masses.
What is the mass excess?
The same information from the other side: the difference between the measured atomic mass and the mass number A, in keV. The nuclear data centres quote it alongside the binding energy, and the two carry the same information.
Does a high binding energy per nucleon mean a stable nucleus?
Not on its own. Stability also depends on the competition between the strong force, which holds the nucleons together, and the electrostatic repulsion of the protons, which pushes them apart, and that competition is what makes some light nuclei unstable.
Sources
- National Nuclear Data Center: NuDat 3 user guide, with the binding energy per nucleon as a ground state property
- International Atomic Energy Agency: the atomic mass evaluation, with the formula for the binding energy per nucleon
- National Nuclear Data Center: a short history of nuclear data and its evaluation, with the definition of the binding energy