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Binding Energy

The key concept behind the release of energy in fusion (and fission) reactions is binding energy. Binding energy is the energy that is lost when a nucleus is created from protons and neutrons. If you added up the total mass of the nucleons (protons and neutrons) that compose an atom, you would notice that this sum is less than the actual mass of the atom. This missing mass, called the mass defect, is a measure of the atom's binding energy. It is released during the formation of a nucleus from the composing nucleons. This energy would have to be put back into the nucleus in order to decompose it into its individual nucleons. The greater the binding energy per nucleon in the atom, the greater the atom's stability. To calculate the binding energy of a nucleus, all you have to do is sum the mass of the individual nucleons, and then subtract the mass of the atom itself. The mass leftover is then converted into its energy equivalent. The relation between mass and energy is shown in Einstein's famous equation E = mc2. However, we will just multiply the mass by a conversion factor to have the units of energy in millions of electron volts (MeV), a standard unit of energy in nuclear physics. Therefore, the equation for binding energy that you can use later is:

Eb = (Z × mH + N × mn - misotope) × 931.5 MeV/amu

Eb = binding energy, in MeV
Z = number of protons
mH = mass of a hydrogen atom (1.007825 atomic mass units, or amu)
N = number of neutrons
mn = mass of a neutron (1.008664904 amu)
misotope = actual mass of the isotope
931.5 Mev/amu = the conversion factor to convert mass into energy, in units of MeV

Remember how I said that the greater the binding energy per nucleon of an atom, the greater it's stability? Well, above is a graph of the relative binding energy per nucleon vs. mass number (total number of nucleons composing an atom). Notice that the nuclei of the light elements are generally less stable than the heavier nuclei up to those with a mass number around 56. The nuclei of the heaviest elements are less stable than the nuclei that have a mass number of around 56. From this, you can see that the nuclei around iron are the most stable. This information implies two methods towards the converting of mass into useful amounts of energy: fusion and fission.

Fusing two nuclei of very small mass, such as hydrogen, will create a more massive nucleus and release a small amount of mass which appears as energy. Meanwhile, fissioning elements of great mass, like uranium, will create two lower-mass and more stable nuclei while losing mass in the form of kinetic and/or radiant energy. The calculation to find the energy released in these reactions is similar to calculating, and related to, binding energy. If the reactants (the things that went into the reaction) are bound more weakly than the products (the stuff that comes out of the reaction), then the reaction releases energy. Just sum the masses of the reactants and subtract the sum of the masses of the products. As an example, lets take a look at a step in the proton-proton reaction:

(2.) 2H + 1H 3H + gamma ray (y)
The fusion of the deuteron (2H) and another proton (a hydrogen nucleus) resulted in the formation of 3He and a gamma ray. If you summed the masses 2H (2.0140 amu) and the proton (1.007825 amu), and subtracted the isotopic mass of 3He (3.01603 amu), you would end up with 0.005795 amu of missing mass. This is equivalent to 5.398MeV of released energy (not including any kinetic energy the reactants had), in this case taking the form of a gamma ray and any additional kinetic energy of the products.

Let's Review:

Knowing that stars are created mostly from hydrogen and helium, tell which type of process, fusion or fission, generates energy within stars and explain why.

The questions below will use information found in this table.
 Mass of... Mass (atomic mass units) hydrogen atom 1.007825 neutron 1.008665 4He 4.002603 56Fe 55.934939 931.5 MeV per amu Periodic Table

Calculate the binding energy per nucleon of 56Fe in MeV, given the information in the table.

Let's take a look at the proton-proton reaction again. Simply put, this reaction converts four protons (hydrogen nuclei) into a helium nucleus, while releasing a significant amount of energy. Calculate how much energy is released in MeV, given the information in the table.