I am a high school student and am struggling with understanding the significance of the mass per nucleon graph.

What does this graph show with regard to nuclear reactions? (y-axis: M/A, x-axis:A) I do know that in fusion and fission, there is more than one particle involved. As an example in the fusion reaction in which two hydrogen atoms are fused and the product is helium, we can calculate the mass difference but the graph doesn't really help considering the fact that we can't compare the mass of two hydrogen atoms with one Helium atom and a neutron.

To summarise: what is the significance of this graph? how can it help to have a better understanding of nuclear reactions? Why is it based on the mass per nucleon and not mass alone?


3 Answers 3


That is not a graph of mass, it is a graph of potential energy (vertical axis) vs. atomic number (horizontal axis). In order for a reaction, chemical or nuclear, to be exothermic (release more energy than was put into it) the products need to have a lower potential energy than the reactants. For example, multiple hydrogens H can fuse into helium He and release energy because the protons being bound together as He is a state with lower potential energy than the protons being free hydrogen. Conversely, Uranium has higher potential energy than lighter nuclei such as lead, so it can decay into lead and release energy.

Iron Fe has the lowest potential energy of any nucleon, and so either fusing it or fissioning it requires energy input. There is no lower energy state available, so there is nowhere to go from Iron. This is why stars go supernova. Once they start creating iron, they no longer produce fusion energy to support their own weight, and collapse – just as a building collapses when you knock out its support columns during a demolition.

  • $\begingroup$ I see but the the graph that we have received is a graph with M/A on the vertical axis and A on the horizontal axis. $\endgroup$ Jan 9, 2023 at 18:51
  • 2
    $\begingroup$ I see. It appears they are showing "average mass per nucleon" $M/A$ on the vertical axis. But you can think of it as a stand-in for potential energy. Hydrogen has the most energy per nucleon because it is fully unbound. It will really help you in this stage of your education if you start thinking in terms of energy and not mass when it comes to nuclear and particle physics. Mass is simply a property that energy has when it is confined somewhere like in a particle. Chemistry also deals with changes in energy, but they are so small that the mass of reactants and products stays roughly constant $\endgroup$
    – RC_23
    Jan 9, 2023 at 21:20
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    $\begingroup$ High school physics definitely can't make up its mind properly about what exactly it thinks mass should be, and whether it's distinct from but can be transformed into energy, or is a form of energy, or is a consequence of the presence of energy. It gets real messy when it tries to teach the poor students about relativistic mass, but nuclear mass is bad too. Skirting around any form of rigor in a misguided attempt at simplifying. $\endgroup$
    – Arthur
    Jan 10, 2023 at 9:29

You may be familiar with the Einstein relation

$$ E=mc^2$$

Free protons and free neutrons have particular masses, which you can measure by seeing e.g. how they accelerate in response to electromagnetic forces. You can also measure the masses of other nuclei. What you find is that stable nuclei are always less massive than their protons and neutrons would be if you measured them separately. Furthermore, if you measure the energy released when two nuclei fuse into a heavier nucleus, the total energy released is equal to the mass difference between the starting components — with the unit conversion factor $c^2$ from the Einstein equation.

The mass differences are small, typically less than 0.1% of the nucleon masses. If you are doing chemistry, it’s fine say that “the proton and neutron each have mass 1” and “helium has mass 4.”

But a helium nucleus is less massive than two protons and two neutrons. That difference, or quantities related to it, are variously called the “mass deficit,” flipped around into a related quantity called the “mass excess,” or converted to energy units and called the “binding energy.”

In the sun, the “proton-proton process” turns four protons into a helium nucleus. (Two protons change “flavor” to neutrons in the process.) The “binding energy,” released as the protons are bound into the helium nucleus, explains why the sun is hot.

Your plot shows these mass differences. But it’s not helpful to directly compare the mass of hydrogen, $A≈1$, directly to the mass of iron, $A≈50$-ish. This graph tells you that, if you started with fifty-ish hydrogens and fused them to iron, you would release some of their mass as energy. But if you tried to fuse two irons into tin, $A≈100$-ish, that would cost energy. The energy difference gets big enough for very heavy nuclei that they sometimes fission on their own. A big pile of wet uranium will spontaneously release a bunch of mid-mass elements like cesium, barium, iodine, and xenon, as well as a substantial amount of heat.


If you study physics further you will become familiar with special relativity and quantum mechanics that allow the possibility of some of the mass to be turned into binding energy by the fusion of two or more nucleons into a nucleus.

The plot is showing a complement of the usual plot of binding energy per nucleon , by giving how much smaller the average mass will be due to the binding of the nucleons in the nucleus. In my opinion it is a confusing way to look at the binding energy curve data:

bind energ

  • $\begingroup$ Would it be possible for you to provide me with an explanation of this complement way of showing the binding energy per nucleon graph? $\endgroup$ Jan 9, 2023 at 18:56
  • $\begingroup$ In you plot, they have taken the mass of hydrogen as 1, If there were no binding energy, the mass would be a multiple of hydrogen, N the number of nucleons. They have subtracted Nx(binding energy per nucleon shown in the plot in my answer) to get at your figure. $\endgroup$
    – anna v
    Jan 9, 2023 at 19:06
  • $\begingroup$ What is Nx? And what are they subtracting from? $\endgroup$ Jan 9, 2023 at 19:10
  • $\begingroup$ the multiplication sign, I should say calculated, the mass/-per nucleon becomes smaller because mass is lost as binding energy $\endgroup$
    – anna v
    Jan 9, 2023 at 19:13
  • $\begingroup$ But what are they subtracting from? $\endgroup$ Jan 9, 2023 at 19:15

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