Mass–Energy Equivalence, Mass Defect & Nuclear Binding Energy (Cambridge 9702)
Learning Objectives (mapped to Cambridge A‑Level Assessment Objectives)
- AO1 – Knowledge & understanding: Explain Einstein’s mass‑energy relation \(E=mc^{2}\); define mass defect, nuclear binding energy and the main modes of radioactive decay (α, β⁻, β⁺, γ).
- AO2 – Application: Calculate mass defects from atomic masses, convert them to binding energies, evaluate Q‑values for nuclear reactions and decays, and use the binding‑energy‑per‑nucleon curve to predict whether fusion or fission releases energy.
- AO3 – Practical/experimental skills: Design, analyse and evaluate an experiment (e.g. magnetic mass spectrometer) to determine a nucleus’s mass defect.
0. Prerequisite Recap (AS‑Level Foundations)
| Quantity | Symbol | SI unit | Key formula |
|---|
| Energy | E | J (or eV) | E=mc^{2},\;E=qV,\;E=\tfrac12mv^{2} |
| Mass | m | kg (or u) | m=E/c^{2} |
| Speed of light | c | m s⁻¹ | c=3.00 × 10⁸ m s⁻¹ |
| Atomic mass unit | u | kg | 1 u=1.660539 × 10⁻²⁷ kg |
| Energy–mass conversion | | | 1 u c²=931.5 MeV or 1 MeV c⁻²=1.073 × 10⁻³ u |
Key Conservation Laws (relevant to nuclear processes)
- Energy, linear momentum, electric charge and nucleon number (A) are conserved.
- Mass is not conserved; the “missing” mass appears as binding energy via \(E=mc^{2}\).
1. Einstein’s Mass–Energy Relation
For any system the total energy \(E\) and its invariant mass \(m\) are linked by
\(E = mc^{2}\)
In nuclear physics we normally use:
- Energy in mega‑electron‑volts (MeV) or giga‑electron‑volts (GeV).
- Mass in atomic mass units (u).
Exact conversion (to four significant figures):
1 u c² = 931.5 MeV or 1 MeV c⁻² = 1.073 × 10⁻³ u
Why the factor matters
When nucleons bind to form a nucleus the total mass of the bound system is less than the sum of the masses of the free nucleons. The difference – the mass defect – multiplied by 931.5 MeV u⁻¹ gives the binding energy released during formation.
2. Mass Defect
Definition: the amount of mass lost when free protons and neutrons combine to form a nucleus.
\(\Delta m = \bigl(Z\,m{p}+N\,m{n}\bigr)-M_{\text{nucl}}\)
Important points
- Masses in the syllabus are given to at least four significant figures.
- Always check whether you are using atomic or nuclear masses – the electron mass cancels when you compare like‑for‑like.
- \(\Delta m\) is always positive; the corresponding binding energy is the energy released when the nucleus forms.
3. Nuclear Binding Energy
Binding energy is the energy required to separate a nucleus into its constituent nucleons:
\(E_{b}= \Delta m\,c^{2}= \Delta m \times 931.5\;\text{MeV}\)
Binding energy per nucleon (a measure of stability):
\(\displaystyle \frac{E_{b}}{A}= \frac{\text{total binding energy}}{\text{mass number }A}\)
Stability trend – the binding‑energy‑per‑nucleon curve
- Rises steeply for light nuclei, peaks at \(A\approx 56\) (iron‑56), then falls slowly for heavier nuclei.
- Consequences:
- For \(A<56\) (light nuclei) fusion moves nuclei toward the peak → energy released.
- For \(A>56\) (heavy nuclei) fission** moves nuclei toward the peak → energy released.
Suggested sketch for the classroom (label the axes, indicate the peak at Fe‑56, and shade the regions where fusion or fission is exothermic).
4. Worked Examples
4.1 Alpha particle – \(^{4}\)He
| Particle | Mass (u) | Number in \(^{4}\)He |
|---|
| Proton (p) | 1.007276 | 2 |
| Neutron (n) | 1.008665 | 2 |
| Atomic mass of \(^{4}\)He (incl. 2 e⁻) | 4.002603 | – |
- Separate nucleon mass: \(M_{\text{sep}}=2(1.007276)+2(1.008665)=4.031882\;\text{u}\)
- Mass defect: \(\Delta m =4.031882-4.002603=0.029279\;\text{u}\)
- Binding energy: \(E_{b}=0.029279\times931.5=27.2\;\text{MeV}\)
- Binding energy per nucleon: \(E_{b}/A=27.2/4=6.8\;\text{MeV nucleon}^{-1}\)
4.2 Carbon‑12 – \(^{12}\)C (mid‑mass example)
Data: \(m{p}=1.007276\;\text{u},\; m{n}=1.008665\;\text{u},\; M(^{12}\text{C})=12.000000\;\text{u}\) (by definition).
- \(M{\text{sep}}=6m{p}+6m_{n}=12.099882\;\text{u}\)
- \(\Delta m=12.099882-12.000000=0.099882\;\text{u}\)
- \(E_{b}=0.099882\times931.5=93.1\;\text{MeV}\)
- \(E_{b}/A=93.1/12=7.8\;\text{MeV nucleon}^{-1}\)
4.3 Iron‑56 – the peak of the curve
Atomic mass \(M(^{56}\text{Fe})=55.934937\;\text{u}\).
- \(Z=26,\; N=30\)
- \(M_{\text{sep}}=26(1.007276)+30(1.008665)=56.10844\;\text{u}\)
- \(\Delta m=56.10844-55.934937=0.173503\;\text{u}\)
- \(E_{b}=0.173503\times931.5=161.5\;\text{MeV}\)
- \(E_{b}/A=161.5/56=2.89\;\text{MeV nucleon}^{-1}\) (maximum per‑nucleon value in the chart).
5. Radioactive Decay (Syllabus 23.2)
Four principal types of decay are required for the Cambridge syllabus.
| Decay type | Change in (Z, A) | Particle emitted | Typical Q‑value (MeV) |
|---|
| α‑decay | \((Z,A)\rightarrow(Z-2,A-4)\) | \(^{4}\text{He}\) nucleus (mass 4, charge +2) | 4–9 |
| β⁻‑decay | \((Z,A)\rightarrow(Z+1,A)\) | Electron + antineutrino | 0.1–5 |
| β⁺‑decay (positron emission) | \((Z,A)\rightarrow(Z-1,A)\) | Positron + neutrino | 1.02–5 |
| γ‑decay | No change in (Z, A) | High‑energy photon | 0.01–10 (usually follows α or β) |
General Q‑value expression
\(Q = \bigl[M{\text{parent}}-M{\text{products}}\bigr]c^{2}\)
Use atomic masses; the electron masses cancel automatically for β⁻ and β⁺ (provided the correct atomic species are used).
Example – α‑decay of \(^{238}\)U
- \(^{238}\text{U}\rightarrow^{234}\text{Th}+^{4}\text{He}\)
- Atomic masses: \(M(^{238}\text{U})=238.050788\;\text{u},\; M(^{234}\text{Th})=234.043601\;\text{u},\; M(^{4}\text{He})=4.002603\;\text{u}\)
- \(Q=[238.050788-(234.043601+4.002603)]\times931.5=4.27\;\text{MeV}\)
6. Practical Skills – Determining a Mass Defect (AO3)
Design an Experiment to Measure the Mass Defect of \(^{4}\)He
- Apparatus: Magnetic sector mass spectrometer, calibrated with a reference ion (e.g., \(^{12}\)C⁺), ion source for helium, vacuum system, detector (Faraday cup or channeltron), power supply, safety shield.
- Procedure
- Ionise a sample of helium to produce He⁺ ions.
- Accelerate ions through a known potential difference \(V\).
- Pass the ion beam through a uniform magnetic field \(B\); record the radius of curvature \(r\) on a phosphor screen or via a position‑sensitive detector.
- Calculate the mass‑to‑charge ratio using \(\displaystyle \frac{m}{q}= \frac{(B r)^{2}}{2V}\).
- Since He⁺ carries a single positive charge, \(q=e\); convert the result to atomic mass units.
- Subtract the electron mass (\(0.00054858\;\text{u}\)) to obtain the pure nuclear mass of \(^{4}\)He.
- Data analysis
- Compute the mass defect \(\Delta m\) using the formula in Section 2.
- Convert \(\Delta m\) to binding energy with the factor 931.5 MeV u⁻¹.
- Propagate uncertainties from measurements of \(B\), \(r\) and \(V\) to obtain the final error on \(E_{b}\).
- Evaluation
- Systematic errors: magnetic field inhomogeneity, fringe fields, voltage stability, ion‑charge‑state contamination.
- Random errors: detector resolution, counting statistics.
- Safety: helium is inert, but the high‑voltage and radiation shielding must be addressed.
- Improvements: use a time‑of‑flight spectrometer for higher precision or a double‑focusing magnetic analyser.
7. Common Pitfalls & How to Avoid Them
- Electron mass omission: When using atomic masses, remember to subtract \(Z\times0.00054858\;\text{u}\) only if a pure nuclear mass is required.
- Sign of \(\Delta m\): \(\Delta m\) = (mass of separate nucleons) − (mass of nucleus). It must be positive.
- Units: Keep the conversion factor 931.5 MeV u⁻¹ separate; do not insert an extra \(c^{2}\) after multiplication.
- Significant figures: Use four‑significant‑figure masses and the conversion factor; round the final answer to the same number of figures as the least‑precise input.
- Energy per nucleon vs total energy: \(E_{b}/A\) is a stability indicator; the Q‑value of a reaction is the total energy change.
8. Practice Questions (with marks allocation)
- Mass defect & binding energy of \(^{12}\)C (6 marks)
Given: \(m{p}=1.007276\;\text{u},\; m{n}=1.008665\;\text{u},\; M(^{12}\text{C})=12.000000\;\text{u}\).
Calculate \(\Delta m\), \(E{b}\) (MeV) and \(E{b}/A\) (MeV nucleon⁻¹).
[Show all steps, use 931.5 MeV u⁻¹, give answers to three significant figures.]
- Binding‑energy per nucleon to total energy (5 marks)
An isotope has \(E_{b}/A=8.5\;\text{MeV}\) and \(A=56\). Find the total binding energy and the corresponding mass defect (in u).
[Use the conversion factor, express \(\Delta m\) to four significant figures.]
- Q‑value of a nuclear reaction (7 marks)
Calculate the energy released when \(^{235}\)U undergoes fission into \(^{141}\)Ba, \(^{92}\)Kr and three neutrons.
Atomic masses:
- \(^{235}\)U = 235.0439 u
- \(^{141}\)Ba = 140.9144 u
- \(^{92}\)Kr = 91.9262 u
- n = 1.008665 u
[Explain each step, comment on the magnitude of the result.]
- Conceptual – Why does the binding‑energy curve peak at iron? (4 marks)
Explain in terms of the short‑range attractive nuclear force and the long‑range Coulomb repulsion, and state the implication for fusion and fission as energy‑producing processes.
- Practical – Design task (8 marks)
Outline a method to measure the mass defect of \(^{4}\)He using a magnetic mass spectrometer. Include apparatus, procedure, data analysis, sources of error and safety considerations.
- Decay calculation – α‑decay of \(^{238}\)U (5 marks)
Using the masses given in Section 5, calculate the Q‑value and state whether the decay is energetically allowed.
9. Summary
Einstein’s equation \(E=mc^{2}\) tells us that mass and energy are interchangeable. In the nucleus the binding of protons and neutrons reduces the total mass; the “missing” mass – the mass defect – appears as binding energy. By:
- Measuring atomic masses,
- Applying the conversion factor \(1\;\text{u}c^{2}=931.5\;\text{MeV}\),
- Calculating \(\Delta m\,c^{2}\),
we obtain the binding energy, a key quantity for:
- Predicting the energy yield of nuclear reactions (fusion, fission, decay).
- Understanding nuclear stability via the binding‑energy‑per‑nucleon curve.
- Linking nuclear physics to applications in medicine, energy production and astrophysics.
Mastery of these concepts satisfies the Cambridge A‑Level objectives AO1–AO3 and provides a solid foundation for the rest of the nuclear physics syllabus.