# Binding Energy

When the different components of an atom’s nucleus are glued together, a force opposed to the electromagnetic force acts. The electrostatic force will try to separate the protons in the nucleus, but the nuclear strong force will hold them together.

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The energy produced by this force keeping the particles together is known as binding energy. This is also the minimum amount of energy needed to separate a particle.

## Origin of binding energy

Binding energy exists as part of the interaction between the electromagnetic force and the strong nuclear force. The nuclear force decays very rapidly. There is a distance at which the attraction caused by the strong nuclear force can win over the electromagnetic force, repelling protons. The strong force works against the electromagnetic force, repelling the other positive particles. This is the origin of binding energy.

The left-hand side shows that the balance of the forces in the nucleus is responsible for binding energy. On the right, the forces have an area where they are stronger. When the strong nuclear force is stronger, the binding energy increases.

The work produced by the strong nuclear force against the electromagnetic force is equal to the binding energy.

## How to calculate binding energy

Binding energy is a form of mass-energy equivalence, the principle expressed as a formula by Albert Einstein that allows us to calculate the energy stored by the strong nuclear force in the atomic nucleus. Here is the binding energy equation:

$c^2(m_f-m_i) = E$

Here mf and mi are the final and initial masses in kilograms, E is the energy released in joules, and c is the light velocity in a vacuum. For particles, m is more easily measured in atomic mass units, which is the mass of 1/12th of a neutral, unbonded carbon-12 atom. It is equal to 1.6605 ⋅ 10-27kg.

### The mass defect

The binding energy of an atom is linked to a phenomenon called the mass defect. The components of an atom are heavier than the atom itself when formed, and this mass difference is known as the mass defect. The mass is not lost but converted into energy.

### Binding energy atomic mass dependence

As the binding energy depends on the strong nuclear force, the amount of energy available will depend on the total nuclear force in the nucleus. Heavier elements such as uranium or plutonium, which contain many particles in their nucleus, will have a larger binding energy.

As an example, we can calculate the difference between the binding energy of uranium-235 and helium.

Helium

A helium nucleus comprises two neutrons and two protons, having four particles in total. To calculate the mass difference, we need to multiply the mass of protons and neutrons in amu (atomic mass units) by its total quantity.

The proton has a mass of 1.0073 amu, and the neutron has a mass of 1.0087 amu.

$$\text{Mass of particles}(He) = 2 \cdot (1.0073) amu + 2 \cdot (1.0087) amu = 2.0146 amu + 2.0174 amu = 4.032 amu$$

The mass of helium in amu is 4.0026 amu. The mass defect is equal to the difference between the calculated mass and the actual mass of the element.

$$\text{Mass defect}(He) = 4.032 amu - 4.0026 amu = 0.0006 amu$$

Uranium-235

Uranium-235 has a mass in amu of 235.0439. It is composed of 92 protons and 143 neutrons. If we add the total mass of the particles, we get the calculated mass below.

$$\text{Mass defect}(U235) = 92 \cdot (1.00727 amu) + 143 \cdot (1.00866 amu) = 92.6688 amu + 1442383 amu = 236.9072 amu$$

The mass defect is equal to the difference between the calculated mass and the actual mass of the element.

$$\text{Mass defect}(U235) = 236.9072 amu - 235.0439 amu = 1.8633 amu$$

Conversions

You can see how the difference between both quantities is large, and the energy available in uranium is greater due to its large quantity of particles. You can convert this to joules by dividing by the square value of the light velocity in a vacuum. In this case, we can convert one amu to kg.

$1 amu = 1.6605 \cdot 10^{-27} kg$

$$E(U235) = (3.0940 \cdot 10^{-27} kg) \cdot (299792 \cdot 10^3 m/s)^2 = 2.78074 \cdot 10^{-10} J$$

$$E(He) = (9.963 \cdot 10^{-31} kg) \cdot (299792 \cdot 10^3 m/s)^2 = 8.95427 \cdot 10^{-14} J$$

We can see that the energy available in uranium is four orders of magnitude larger than in helium.

## The binding energy per nucleon graph

In physics, it is helpful to plot the relationship between the binding energy and the atomic mass. This graph is known as the binding energy per nucleon for stable nuclei and gives us the following relevant information:

1. The amount of energy per nuclei.
2. The force that is dominant in the nucleus.
3. Which nuclear process is more likely to occur (fusion or fission).

The graph representing the binding energy per nucleon gives us the amount of binding energy in each isotope

In the plot, we can see that higher binding energy is usual for heavier nuclei. Isotopes with higher binding energy are more stable. However, at very large masses, stability decays as the nuclear binding energy decays. For lighter elements, fusion is more likely to occur, and electrostatic forces dominate the element. For heavier elements, fission is likely to occur, and the strong nuclear force dominates the nucleus. Lets look at fission and fusion in a bit more detail.

### Fission

The breakaway of heavy nuclei such as uranium will release energy in a process known as fission. Elements heavier than iron-56 are prone to fission.

During fission, heavy elements (left) disintegrate and create lighter elements and high-energy photons (right)

### Fusion

Contrary to heavy elements, lighter elements are prone to fusion. In this case, the principle of mass-energy equivalence intervenes. Because the mass of the particle that is produced is less than its original components, the lost mass should be transformed into energy.

During fusion, lighter elements (like tritium and deuterium) create heavier elements (alpha nucleus) and high-energy photons

The processes of fission and fusion are two of the most important processes in the universe. Fusion is how the Sun creates energy in the form of electromagnetic radiation. Stars are mostly made of hydrogen and helium, and the enormous gravity of the stars creates large amounts of pressure. The pressure then creates heat and ignites the fusion process, allowing the lighter elements to create heavier ones.

Star fusion works by using deuterium and tritium, two isotopes of hydrogen. The fusion process converts these two hydrogen atoms into helium. The residual energy made of photons of different wavelengths escapes into space. The process by which these atoms escape into space can take thousands of years or more as they collide with trillions of atoms on their way to the Suns surface.

When the radiation finally escapes and reaches Earth, it provides energy where it powers many physical, chemical, and biological processes that make life possible.

## Binding Energy - Key takeaways

• Binding energy is the product of the work done by the strong nuclear force against the electromagnetic force repelling the protons in the atomic nucleus.
• We can calculate the binding energy by multiplying the mass lost when the atom or particle is formed by the square of the light in the vacuum.
• The binding energy per nucleon graph gives us information on the amount of energy per nuclei, the force that is dominant in the nucleus, and which nuclear process is more likely to occur (fusion or fission).
• For lighter elements, fusion is more likely to occur, and electrostatic forces dominate the element.
• For heavier elements, fission is likely to occur, and the strong nuclear force dominates the nucleus.

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What is binding energy?

Binding energy is the product of the work done by the strong nuclear force against the electromagnetic force repelling the protons in the atomic nucleus.

How do you calculate binding energy?

The binding energy can be calculated by multiplying the mass lost when the atom or particle is formed by the square of the light in a vacuum. The equation is E=mc2.

What is the binding energy curve?

The binding energy per nucleon graph gives us information on the amount of energy per nuclei, the force that is dominant in the nucleus, and which nuclear process is more likely to occur (fusion or fission).

Are isotopes with higher binding energy more stable?

Yes, isotopes with higher binding energy are more stable. However, at very large masses, stability decays as the nuclear binding energy decays.

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