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Binding energy

Radiation and the atom · level 8, lesson 2 of 6

A question to start

Weigh two protons and two neutrons. Now weigh a helium nucleus, which is made of exactly those four. The helium nucleus is lighter. Where did the missing mass go?

The idea

When protons and neutrons join into a nucleus, energy is released, so the nucleus is lighter than its parts. The difference is the mass defect. The energy it stands for is the binding energy: the energy you would need to pull the nucleus apart again.

Using NIST's masses of whole atoms (so the electrons cancel), with a hydrogen atom at 1.007825 u and a neutron at 1.008665 u:

Nucleus Mass of the parts Mass of the atom Mass defect Binding energy Per nucleon
Helium-4 4.032980 u 4.002603 u 0.030377 u 28.3 MeV 7.07 MeV
Iron-56 56.463398 u 55.934936 u 0.528462 u 492 MeV 8.79 MeV
Uranium-238 239.984981 u 238.050788 u 1.934192 u 1,802 MeV 7.57 MeV

The binding energy per nucleon says how tightly each particle is held. It rises steeply through the light elements, peaks near iron, and falls slowly towards uranium. Nuclei near iron are the most tightly bound of all.

helium-4, 7.07iron-56, 8.79uranium-238, 7.57 8.8MeV per nucleon0mass number

Worked examples

1. Helium-4. Parts: \(2 \times 1.007825 + 2 \times 1.008665 = 4.032980\) u. Defect: \(4.032980 - 4.002603 = 0.030377\) u. Binding energy: \(0.030377 \times 931.5 = 28.3\) MeV, or \(28.3 / 4 = 7.07\) MeV per nucleon.

2. Iron-56. 26 protons and 30 neutrons: \(26 \times 1.007825 + 30 \times 1.008665 = 56.463398\) u. Defect 0.528462 u, \(\times 931.5 = 492\) MeV, \(/56 = 8.79\) MeV per nucleon.

3. As a share of the mass. For helium-4 the defect is \(0.030377 / 4.032980 = 0.75\%\) of the parts. For an everyday chemical bond it is tens of millions of times smaller.

Try it

  1. What is the binding energy of a nucleus?
  2. Uranium-235 has an atomic mass of 235.043930 u, with 92 protons and 143 neutrons. Find its binding energy per nucleon.
  3. Which nuclei are the most tightly bound?
  4. Why is the mass defect worked out with hydrogen atoms rather than bare protons?
Answers
  1. The energy needed to pull it apart into separate protons and neutrons.
  2. Parts: \(92 \times 1.007825 + 143 \times 1.008665 = 236.958986\) u. Defect 1.915056 u, \(\times 931.5 = 1{,}784\) MeV, \(/235 = 7.59\) MeV per nucleon.
  3. Those near iron.
  4. NIST lists the masses of whole atoms, with their electrons. Using hydrogen atoms for the protons adds the same number of electrons on both sides, so they cancel.

Common mistake

"Binding energy is energy stored inside the nucleus." It is the energy that was given out when the nucleus formed, and that you would have to put back to break it up. A more tightly bound nucleus has less energy, not more.

Where it leads