# What is the relationship between binding energy, energy released, and mass defect? [duplicate]

Im learning about fission/fusion and cant get mass defect, binding energy, and energy released to fit together in my head.

The three are all equal in magnitude if I understand it correctly?

When, for example, $$4$$ hydrogen fuse into $$\mathrm{He}$$, mass is lost - this mass is converted into energy, via $$E = mc^2$$, which is released.

So then how can the binding energy also increase? there seems to be "too much energy" on one side of the equation for me.

If the mass lost is converted into energy, which is released, where is the extra energy coming from to also increase the total binding energy?

Can't make it work in my brain, and have to teach high school physics this year so would like to understand it properly!

• There is a plethora of questions and answers about binding energy. This is only one. You should use the search box at the top left and simply type in binding energy and read all the questions and answers. Jan 21 at 20:46

Let's take a nucleus $$Z$$, composed of $$x$$ neutrons and $$y$$ protons (that is $$x$$ and $$y$$ nucleons). If we create such a nucleus from its constituent nucleons and assuming the process is exothermic then the energy released in the creation process is the Binding Energy $$\Delta E_{binding}$$.

This energy is released by converting some of the nucleonic mass to energy as per Einstein's, where $$\Delta m$$ is the Mass Defect:

$$\Delta E_{binding}=\Delta m c^2$$

So the mass defect can also be calculated from:

$$\Delta m=\frac{\Delta E_{binding}}{c^2}$$

I'm not sure what is meant here by 'energy released' but it may be the binding energy but expressed per $$\mathrm{mol}$$ or $$\mathrm{kg}$$ of $$Z$$, i.e. some macroscopic unit of mass.

• thanks for the reply. by "energy released" I mean the energy that is given off during a fusion or fission reaction - say in a fission power plant. Basically I cant understand how both that energy can be released and the binding energy can increase. I understand that mass is lost, and that mass is converted to energy. I can understand mass being converted to binding energy. and I can understand mass being released as energy via mc^2 and released. But there isnt enough mass lost to do both those things? There seems to be twice as much "energy out" as "mass in". Jan 21 at 18:21
• I see your confusion. In an exothermic nuclear reaction energy is released because mass in converted to energy (via Einstein). If that reaction is a creation reaction starting from the constituent nucleons we call it the 'binding energy'. I.o.w. the energy released IS the binding energy.
– Gert
Jan 21 at 19:28

In an exothermic reaction (nuclear or chemical), the mass (rest mass) of the reactants is greater than the mass of the products, and this decrease in mass from the reaction results in an increase in kinetic energy of the products compared to the reactants. This is sometimes called "releasing energy". In a nuclear reaction the change in mass is about a million times greater than the change mass of a chemical reaction. (In an endothermic reaction the mass of the reactants is less than the mass of the products.)

Binding energy is a measure of how tightly the nucleons (electrons for chemical reaction) are bound in a nucleus (atom/molecule). Specifically, it is the sum of the masses of the individual nucleons minus the mass of the nucleus consisting of those nucleons. For an exothermic reaction, the binding energy of the reactants is less than the binding energy of the products; that is the nucleons for the products are more tightly bound (more binding energy) than those for the reactants. For example, in a nuclear fission or fusion reaction, we "moveup" the curve of binding energy, as indicated in the following curve.

Binding energy is just another way to address the change of mass (rest mass) to kinetic energy.