Mass–energy and binding energy

Mass is energy

Narrated video. Transcript:

Before Einstein, we treated mass and energy as two separate things, each conserved on its own in any reaction.

Einstein’s special theory of relativity changed this. Mass is another form of energy. Mass-energy can turn into other forms, like kinetic energy, and back again.

The link is E equals m c squared. Here c is the speed of light in vacuum, about 3 × 10⁸ metre per second.

c squared is huge, 9 × 10¹⁶. So a tiny mass stands for an enormous energy.

Take just one gram, 10⁻³ kilogram. E equals 10⁻³ times 9 × 10¹⁶, which is 9 × 10¹³ joule.

That is about 2.5 crore kilowatt-hours of electricity, from a single gram, if all of it could be turned into energy.

In nuclear physics we measure mass in u, and energy in MeV, mega-electron-volts. One MeV is 1.6 × 10⁻¹³ joule. So what energy is one u worth?

1 u is 1.6605 × 10⁻²⁷ kilogram. Multiply by c squared, 2.9979 × 10⁸ squared, and we get 1.4924 × 10⁻¹⁰ joule.

Divide by 1.602 × 10⁻¹⁹ to change joules into electron volts: 0.9315 × 10⁹ eV. That is 931.5 MeV.

So 1 u of mass is equivalent to 931.5 MeV of energy. We often write 1 u equals 931.5 MeV over c squared. Remember this number: we will use it all through the chapter.

The missing mass

Narrated video. Transcript:

An oxygen-16 nucleus has 8 protons and 8 neutrons. You might expect its mass to be the mass of 8 protons plus 8 neutrons. Let us check.

8 neutrons: 8 × 1.00866 u. 8 protons: 8 × 1.00727 u. Together, 8 × 2.01593, which is 16.12744 u.

The mass spectrometer gives the oxygen-16 atom as 15.99493 u. Take away 8 electrons, 8 × 0.00055 u, and the nucleus alone is 15.99053 u.

The nucleus is lighter than its parts! The difference is 0.13691 u. This is called the mass defect, ΔM.

In general, ΔM equals Z m p plus A minus Z times m n, minus M, where M is the mass of the nucleus. It is always positive: a nucleus always weighs less than its separate protons and neutrons.

Where did the mass go? When the nucleons came together, they gave out energy. Less energy means less mass.

To pull the nucleus apart again into 8 free protons and 8 free neutrons, we must put this energy back in. It is called the binding energy, E b, and E b equals ΔM c squared.

For oxygen-16: 0.13691 u times 931.5 MeV per u gives 127.5 MeV. That is the energy needed to break oxygen-16 into its 16 nucleons.

Put the other way round: if 8 protons and 8 neutrons came together to form oxygen-16, 127.5 MeV would be released.

We cannot really tear a nucleus apart nucleon by nucleon, but the binding energy is still a good measure of how tightly a nucleus is held together.

Binding energy per nucleon

Narrated video. Transcript:

Big nuclei have more nucleons, so naturally they have bigger binding energies. A fairer measure is the binding energy per nucleon, E b n, equal to E b divided by A.

It is the average energy needed to pull one nucleon out of the nucleus. For oxygen-16 it is 127.5 over 16, about 8.0 MeV.

Now plot E b n against A for many nuclei. This curve is one of the most important graphs in physics.

In the middle, from A about 30 to about 170, the curve is nearly flat, close to 8 MeV per nucleon.

It peaks at about 8.75 MeV for A equals 56, iron. Iron-56 is among the most tightly bound nuclei in nature.

On the right, the heavy nuclei slope down: uranium-238 is only about 7.6 MeV. On the left, light nuclei are much lower: deuterium is just 1.1 MeV.

What does the curve tell us? First, the nuclear force is attractive and strong enough to bind each nucleon by a few MeV, nearly a million times more than an electron is bound in a hydrogen atom.

Second, the flat middle. Each nucleon feels only the few neighbours within the short range of the nuclear force. Add more nucleons far away, and its binding does not change. This is called saturation.

Third, a nucleus of A equals 240 is less tightly bound than two nuclei of A equals 120. Split it, and the nucleons become more tightly bound, so energy is released. This is fission.

Fourth, two very light nuclei that join into a heavier one also become more tightly bound, and again energy is released. This is fusion, the energy source of the Sun.

Try it: the binding energy curve

Interactive simulation. Things to try:

• Hover over (or tap) any point to see its mass defect, binding energy and binding energy per nucleon.
• The app uses atomic masses: Δm = Z m_H + N m_n − M. The Z electrons in Z hydrogen atoms cancel the Z electrons in the atom.
• Find the top of the curve. Then compare ²³⁸U with ¹²⁰Sn: which is more tightly bound per nucleon?
• Look for ⁴He, ¹²C and ¹⁶O: they sit above their neighbours, so they are especially stable.
• Hollow points are radioactive fission fragments like ¹⁴⁴Ba. They lie a little below the curve.

Binding energy of nitrogen

Find the binding energy of ¹⁴₇N. Atomic masses: m(¹⁴N) = 14.00307 u, m_H = 1.007825 u, m_n = 1.008665 u; 1 u = 931.5 MeV/c².

1. 7.48 MeV
2. 101.0 MeV
3. 104.7 MeV
4. 1.30 × 10⁴ MeV

More is not always tighter

The binding energy of ²³⁸U is about 1802 MeV and that of ⁵⁶Fe about 492 MeV. Which nucleus is more tightly bound?

1. ²³⁸U, because its binding energy is larger
2. ⁵⁶Fe, because its binding energy per nucleon is larger
3. Both equally, because both are stable
4. ²³⁸U, because it has more nucleons pulling on each other

Summary

• Mass–energy: E = mc²; 1 u = 931.5 MeV/c²
• Mass defect: ΔM = [Z m_p + (A − Z) m_n] − M
• Binding energy: E_b = ΔM c²
• Per nucleon: E_bn = E_b / A

• Mass is a form of energy. 1 g of matter is equivalent to 9 × 10¹³ J.
• A nucleus is always lighter than its separate nucleons. The missing mass ΔM is the binding energy E_b = ΔMc², e.g. 127.5 MeV for ¹⁶O.
• E_bn is nearly constant, about 8 MeV, for 30 < A < 170 (saturation, because the nuclear force is short-ranged). It peaks at about 8.75 MeV near A = 56 and falls to 7.6 MeV at A = 238.
• Energy is released when nuclei become more tightly bound: when a heavy nucleus splits (fission) or light nuclei join (fusion).