Mass, energy and binding energyAQA A-Level Physics: Revision notes
Section 1
Mass–energy equivalence
Einstein's equation E = mc² links mass and energy. When the energy of a system changes by ΔE, its mass changes by Δm, where
ΔE = c²Δm
This applies to all energy changes, not just nuclear ones. Burning 1.0 kg of coal releases about 3 × 10⁷ J, so the mass falls by 3 × 10⁷ ÷ (3.00 × 10⁸)² ≈ 3 × 10⁻¹⁰ kg. That is far too small to measure. Nuclear reactions release about a million times more energy per kilogram, so their mass changes can be measured.
Energy released means the mass of the system decreases; energy supplied means it increases.
Do not say that mass is 'converted into' energy in nuclear reactions only. ΔE = c²Δm applies to chemical reactions too, but the mass change is far too small to notice.
Section 2
The atomic mass unit and unit conversions
Nuclear masses are tiny in kilograms, so they are given in the atomic mass unit, u. 1 u is one twelfth of the mass of a carbon-12 atom:
1 u = 1.661 × 10⁻²⁷ kg
Using ΔE = c²Δm, the energy equivalent of 1 u is
1 u = 931.5 MeV
To convert between energy units, 1 eV = 1.60 × 10⁻¹⁹ J, so 1 MeV = 1.60 × 10⁻¹³ J.
- mass in u × 931.5 gives energy in MeV
- mass in kg × c² gives energy in J
Use 931.5 MeV per u when masses are in u. Only use c² when masses are in kg. Mixing the two is a classic source of wrong powers of ten.
Section 3
Mass defect and binding energy
The mass of a nucleus is always less than the total mass of its separate protons and neutrons. The difference is the mass defect:
Δm = [Z m(proton) + (A − Z) m(neutron)] − m(nucleus)
The binding energy is the energy equivalent of the mass defect: the minimum energy needed to separate a nucleus into its constituent nucleons. Binding energy = c²Δm, or Δm (in u) × 931.5 MeV.
Worked example (helium-4). Δm = 2(1.00728) + 2(1.00867) − 4.00151 = 0.03039 u. Binding energy = 0.03039 × 931.5 = 28.3 MeV. Per nucleon: 28.3 ÷ 4 = 7.08 MeV.
Binding energy is not energy stored 'inside' the nucleus as extra mass. The bound nucleus has less mass than its parts; the binding energy is what you must supply to separate them.
Section 4
Average binding energy per nucleon against nucleon number
Plotting average binding energy per nucleon against nucleon number A gives a characteristic curve:
- it rises steeply for the lightest nuclei, with helium-4 unusually high (about 7 MeV)
- it reaches a maximum of about 8.8 MeV at iron-56, the most stable nucleus
- it then falls slowly for heavier nuclei, to about 7.6 MeV for uranium-235
The higher a nucleus is on the curve, the more tightly bound it is and the lower its mass per nucleon.
Section 5
Where fusion and fission release energy
A nuclear change releases energy when the products have a higher average binding energy per nucleon than the reactants, because the total mass then decreases.
- Fusion of light nuclei (A below about 56) moves them up the steep part of the curve, so energy is released.
- Fission of heavy nuclei (A above about 56) gives fragments nearer the peak, so energy is released.
- Fusing nuclei heavier than iron-56, or splitting nuclei lighter than iron-56, would not release energy.
Energy released = total binding energy of products − total binding energy of reactants.
Worked example. U-235 (7.6 MeV per nucleon) splits into fragments of average 8.5 MeV per nucleon. Energy released ≈ 235 × (8.5 − 7.6) = 212 MeV.
Must Know
- ΔE = c²Δm applies to all energy changes
- 1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV; 1 MeV = 1.60 × 10⁻¹³ J
- Mass defect = mass of separate nucleons − mass of nucleus
- Binding energy = mass defect × 931.5 MeV per u
- Average binding energy per nucleon peaks at iron-56 (about 8.8 MeV)
- Fusion of light nuclei and fission of heavy nuclei both increase binding energy per nucleon and release energy
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Mass, energy and binding energy
- A student is analysing the helium-4 nucleus using these data: nuclear mass of helium-4 = 4.00151 u; mass of a proton = 1.00728 u; mass of a neutron = 1.00867 u; 1 u = 931.5 MeV.Explain why the mass of a helium-4 nucleus is less than the total mass of its separate nucleons.2 marks
- A power station burns coal. Burning 1.0 kg of coal releases 3.0 × 10⁷ J of energy. The speed of light in a vacuum is c = 3.00 × 10⁸ m s⁻¹.Suggest why the mass change in a chemical reaction is never noticed, but the mass change in a nuclear reaction can be measured.2 marks
- Average binding energy per nucleon: hydrogen-2 (deuterium) 1.1 MeV; helium-4 7.1 MeV; iron-56 8.8 MeV; uranium-235 7.6 MeV. When a uranium-235 nucleus undergoes fission, the fragments formed have an average binding energy per nucleon of 8.5 MeV. 1 MeV = 1.60 × 10⁻¹³ J.Explain, in terms of binding energy per nucleon, why energy is released both when light nuclei such as hydrogen-2 fuse and when heavy nuclei such as uranium-235 undergo fission.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).