Mass and energy (A-level only) — AQA A-Level Physics

This topic explores the profound relationship between mass and energy as described by Einstein's famous equation E = mc^2. You will learn how mass can be converted into energy in nuclear processes, and how the tiny mass defects in nuclei account for the enormous energy released in fission and fusion. It builds on your understanding of atomic structure, the nucleus, and conservation laws from earlier in the course.

Understanding mass-energy equivalence is crucial for explaining nuclear stability, the energy output of stars, and the operation of nuclear power stations. It also provides a foundation for later topics such as particle physics and cosmology. In AQA A-Level Physics, this topic is assessed in both calculation-heavy questions and extended explanations, so you must be confident with both the maths and the physical principles.

• →Mass-energy equivalence: E = mc^2, where E is energy in joules, m is mass in kg, and c is the speed of light in vacuum (3.00 x 10^8 m s^-1).
• →Mass defect: the difference between the mass of a nucleus and the sum of the masses of its individual protons and neutrons.
• →Binding energy: the energy equivalent of the mass defect, representing the energy required to separate a nucleus into its constituent nucleons.
• →Binding energy per nucleon: total binding energy divided by nucleon number; peaks at iron-56, indicating maximum nuclear stability.
• →Nuclear fission and fusion: both release energy because the products have higher binding energy per nucleon than the reactants, resulting in a mass defect that is converted into energy.

• State that the atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom.
• Recall that 1 u ≈ 1.661 × 10⁻²⁷ kg.
• Use u as a unit of mass in nuclear calculations, such as mass defect.
• Convert mass in u to energy using the equivalence 1 u = 931.5 MeV.
• Explain why u is used instead of kg for atomic and nuclear masses.
• State the conversion factor 1 u = 931.5 MeV.
• Convert mass defect in u to energy in MeV by multiplying by 931.5.
• Convert energy in MeV to mass in u by dividing by 931.5.
• Use the conversion in calculations involving nuclear binding energy or reaction energy.
• Explain the origin of the conversion factor from E = mc².
• Fission: a heavy nucleus absorbs a neutron and splits into two smaller nuclei, releasing two or three neutrons and energy.
• Fusion: two light nuclei combine to form a heavier nucleus, releasing energy.
• In both processes the total mass after the reaction is less than the total mass before, and the mass difference Δm corresponds to energy released by ΔE = Δmc².
• The energy release is linked to an increase in binding energy per nucleon of the products compared with the reactants, placing them higher on the curve.
• Write a balanced nuclear equation for the fission or fusion reaction, ensuring nucleon number and proton number are conserved.
• Calculate the total mass of reactants and the total mass of products using the nuclear masses supplied.
• Find the mass difference Δm = total mass of reactants − total mass of products.
• Convert Δm to kilograms if using SI units, or use the energy equivalent of 1 u, before applying ΔE = Δmc².
• State the energy released with an appropriate unit, such as J or MeV, and link it to the kinetic energy of the products.
• Average binding energy per nucleon is total binding energy divided by nucleon number A, with units of MeV per nucleon.
• The curve rises sharply for light nuclei, reaches a broad maximum around A = 56 near iron, then decreases gradually for heavy nuclei.
• A larger value of average binding energy per nucleon indicates a more stable nucleus because more energy is needed per nucleon to separate it.
• Binding energy arises from the mass defect, calculated using E = mc², where m is the difference between the mass of separate nucleons and the nuclear mass.
• The graph is a smooth curve through nuclide data, not a straight line, and the peak region is broad rather than a single sharp point.
• Energy is released when the average binding energy per nucleon increases from reactants to products, because the total mass decreases.
• Fusion releases energy for light nuclei on the rising left-hand region of the curve, where products lie closer to the peak.
• Fission releases energy for heavy nuclei on the falling right-hand region, where mid-mass fragments lie closer to the peak.
• The change in binding energy per nucleon multiplied by the number of nucleons gives the total energy released, consistent with E = mc².
• Nuclei near the maximum around iron-56 are the most stable, so reactions producing them release little further energy.
• State that mass–energy equivalence, ΔE = Δmc², explains the energy released when a nucleus undergoes fission or fusion.
• Explain that the total mass of the products of fission is slightly less than the mass of the original nucleus plus neutron, and this mass difference appears as energy.
• Use quantitative data, such as energy released per kilogram of fuel, to compare nuclear energy with fossil fuels in a decision-making context.
• Describe how scientific evidence about energy release, waste and safety feeds into public and political decisions alongside economic, ethical and environmental considerations.
• Recognise that physics can quantify risks and benefits but cannot by itself settle questions of value or policy.
• Describes the method for speed: electrons are accelerated through a known potential difference and their flight time over a measured distance gives speed.
• Explains that kinetic energy is determined independently using calorimetry, by measuring the temperature rise of a target struck by the electrons.
• States the key result: measured speed approaches but never reaches c, while kinetic energy continues to rise without limit.
• Uses Eₖ = (γ − 1)mc² to show that the classical ½mv² result fails when v is a significant fraction of c.
• Explains that the experiment supports special relativity rather than classical mechanics because the classical prediction diverges from the data at high speed.
• Recognises that the result implies a limiting speed c for massive particles and links this to the relativistic increase of energy.

• 💡Learn the definition of u precisely, including the carbon-12 reference.
• 💡In calculations, keep track of units and convert to kg if the equation requires SI units.
• 💡Use the conversion 1 u = 931.5 MeV to simplify energy calculations in nuclear reactions.
• 💡Memorise the conversion factor 1 u = 931.5 MeV for quick calculations.
• 💡When using E = mc², ensure mass is in kg and energy in J, or use the direct u to MeV conversion.
• 💡Check that your final answer has the correct unit and magnitude.
• 💡When asked to describe fission or fusion, name the nuclei involved and state what happens to them before discussing energy.
• 💡Use the binding-energy-per-nucleon curve to justify why energy is released: products must have higher binding energy per nucleon than reactants.
• 💡Lay out the calculation in clear steps: balanced equation, total reactant mass, total product mass, mass difference, then energy.
• 💡Keep track of units throughout; if masses are in u, either convert to kg or use the u-to-energy equivalence consistently.
• 💡Check that the sign of the energy released is positive and that the magnitude is reasonable for a nuclear reaction.
• 💡Label both axes with quantity and unit, for example average binding energy per nucleon / MeV and nucleon number A.
• 💡Quote the peak position approximately, for example around A = 56, and its value near 8.8 MeV per nucleon, rather than claiming an exact universal figure.
• 💡When explaining stability, refer to energy required per nucleon to separate the nucleus, not to the total energy alone.
• 💡Mark arrows on the graph showing light nuclei moving up towards the peak for fusion and heavy nuclei moving towards the peak for fission.
• 💡State the direction of change in average binding energy per nucleon explicitly, then link it to the mass decrease and energy release.
• 💡Use the phrase per nucleon when comparing values, and avoid quoting a single fixed energy release for all reactions.
• 💡Link every decision-making point back to a specific physics quantity, such as Δm, ΔE or energy per kilogram, rather than writing general opinions.
• 💡When asked to evaluate, give both a benefit and a limitation of using nuclear energy, and state which evidence supports each.
• 💡Use correct units and powers of ten, for example writing energy in J or MJ kg⁻¹, and keep the mass difference in kg when using ΔE = Δmc².
• 💡Describe the independent measurements of speed (time of flight) and kinetic energy (calorimetry) separately, because the strength of the evidence depends on them being measured by different means.
• 💡Sketch or interpret the graph shape: classical ½mv² rising as a parabola against the relativistic curve that steepens near c.
• 💡Use the phrase direct evidence and explain what would be expected classically, so the contrast with the data is explicit.
• 💡Quote the limiting behaviour of speed near c rather than saying electrons were stopped or slowed.
• 💡Always show your unit conversions clearly. In calculations, convert atomic mass units (u) to kilograms (kg) before using E = mc^2. Remember 1 u = 1.66 x 10^-27 kg.
• 💡When explaining binding energy, use precise language: 'the energy required to separate a nucleus into its individual protons and neutrons' or 'the energy released when a nucleus forms from its constituent nucleons'.
• 💡For graph questions on binding energy per nucleon, label axes correctly (binding energy per nucleon / MeV on y-axis, nucleon number A on x-axis), mark the peak at A=56, and annotate regions where fusion and fission occur.

• Confusing the atomic mass unit with the relative atomic mass; correction: u is a unit of mass, while relative atomic mass is a dimensionless ratio.
• Using 1 u = 1.66 × 10⁻²⁷ kg without recognising it is an approximation; correction: use the more precise value 1.661 × 10⁻²⁷ kg when needed.
• Forgetting to convert mass from u to kg before using SI equations; correction: either convert to kg or use the energy equivalent 1 u = 931.5 MeV directly.
• Using 1 u = 931.5 MeV as a mass conversion without recognising it is an energy equivalent; correction: it converts mass in u to energy in MeV.
• Forgetting to multiply by the number of particles when calculating total energy release; correction: ensure the mass defect used corresponds to the total number of nuclei involved.
• Mixing units, such as using MeV with kg without conversion; correction: consistently use either u and MeV or kg and J.
• Thinking that mass is destroyed in fission and fusion; the correct idea is that mass is converted into energy, so total mass-energy is conserved.
• Believing that fission releases energy because the nucleus is unstable, without linking the release to the increase in binding energy per nucleon of the products.
• Confusing the two processes, for example describing fusion as the splitting of a heavy nucleus; the correction is to state that fusion joins light nuclei while fission splits a heavy nucleus.
• Subtracting masses in the wrong order, giving a negative mass difference; the correction is to subtract total product mass from total reactant mass.
• Forgetting to convert atomic mass units to kilograms before using ΔE = Δmc² in SI units; the correction is to multiply by 1.66 × 10⁻²⁷ kg per u.
• Using the mass of a whole atom instead of the nuclear mass when the question specifies nuclear masses; the correction is to use the values given and not add or remove electron masses unless instructed.
• Treating the vertical axis as total binding energy rather than binding energy per nucleon; correct this by dividing total binding energy by A before plotting or reading values.
• Assuming the most stable nucleus has the largest nucleon number; correct this by identifying the peak near iron-56 as the region of greatest stability.
• Confusing binding energy with the energy released in a reaction; correct this by using differences in binding energy per nucleon between initial and final nuclei to find energy release.
• Claiming fission releases energy because heavy nuclei are unstable in a general sense; correct this by comparing binding energy per nucleon of the heavy nucleus with that of its fragments.
• Stating that fusion always releases energy for any two nuclei; correct this by restricting energy-releasing fusion to light nuclei whose products move towards the peak.
• Using total binding energy instead of binding energy per nucleon when comparing reactants and products; correct this by comparing values per nucleon before multiplying by nucleon number.
• Treating mass as being converted into energy in a vague everyday sense; the correction is that a measurable mass difference Δm corresponds to energy ΔE = Δmc².
• Assuming that scientific knowledge automatically determines the best policy; the correction is that decisions also depend on costs, risks, ethics and public values.
• Confusing the energy released per fission with the energy released per kilogram of fuel; the correction is that per-kilogram values require the number of nuclei in the sample, found using Avogadro's constant.
• Claiming Bertozzi measured kinetic energy solely from the accelerating voltage: the error misses the independent verification; the correction is that he used calorimetry (target temperature rise) to measure kinetic energy independently.
• Stating that electrons exceeded the speed of light at the highest energies: the correction is that measured speeds approached c asymptotically and never surpassed it.
• Using Eₖ = ½mv² for the high-speed data: the correction is to use the relativistic expression Eₖ = (γ − 1)mc² when v is comparable to c.
• Students often think mass is 'lost' in nuclear reactions. In fact, mass is converted into energy, and the total mass-energy is conserved. The missing mass appears as energy (often kinetic energy of products).
• Many believe that binding energy is stored in the nucleus. Actually, binding energy is the energy required to break the nucleus apart; it is released when the nucleus forms. A higher binding energy per nucleon means a more stable nucleus.
• Some students confuse the mass defect with the mass of the missing particles. The mass defect is simply the difference in mass before and after the reaction, not a physical missing piece.

1. 1Start by reviewing the definition of mass defect and binding energy. Use the equation E = mc^2 to calculate energy from mass defect, practising unit conversions between u, kg, J, and MeV.
2. 2Draw and annotate the binding energy per nucleon graph. Memorise the key features: peak at iron-56, values for light and heavy nuclei, and the regions where fusion and fission release energy.
3. 3Work through past paper questions on fission and fusion calculations. Focus on setting out calculations clearly, showing the mass defect, conversion to kg, and final energy in J or MeV.
4. 4Create flashcards for key definitions and common misconceptions. Test yourself on explaining why fusion releases energy for light nuclei and why fission releases energy for heavy nuclei.
5. 5Attempt extended-response questions (6 marks) on nuclear stability and energy release. Practise structuring your answers with clear definitions, explanations, and examples.

• 📋Calculation questions: Calculate the energy released in a fission or fusion reaction given masses in u. Advice: Always convert to kg, use E = mc^2, and check units. Show all steps.
• 📋Graph interpretation: Sketch or interpret the binding energy per nucleon curve. Advice: Label axes, mark the peak, and explain the significance of the shape for fusion and fission.
• 📋Explanation questions: Explain what is meant by mass defect and binding energy, and how they relate to nuclear stability. Advice: Use precise definitions and link to E = mc^2.
• 📋Extended writing: Discuss the conditions necessary for fusion and why it is difficult to achieve on Earth. Advice: Include temperature, pressure, and Coulomb barrier, and relate to binding energy.