Ch 2 · 23 Nuclear Physics · A Level Physics 9702
Topic 2 of 2 in Quantum and Nuclear Physics — A Level Physics 9702 — 72 lessons.
Introduction
A nucleus weighs less than the protons and neutrons it is made from. That missing mass is the energy that holds it together, and releasing a little of it powers the Sun and nuclear reactors. This topic also turns the randomness of radioactive decay into precise exponential laws. (Syllabus topic 23: sections 23.1–23.2.)
Key ideas — 23.1 (a) Mass–energy equivalence
Mass and energy are equivalent:
🧑🏫 Teacher's explanation
Imagine you have a Lego castle. If you take the castle apart into individual bricks, the total weight of the loose bricks is actually slightly less than the weight of the completed castle. Where did that missing mass go? In the nuclear world, that 'missing mass' was converted into the energy that holds the bricks (nucleons) together. This energy is called binding energy.
📖 Definitions
🔍 Worked example — Easy
Calculate the energy released when a nucleus with a mass of 3.00 × 10^-26 kg decays into products with a total mass of 2.98 × 10^-26 kg. Use c = 3.00 × 10^8 m/s.
🔍 Worked example — Standard
A nuclear reaction results in a mass loss of 0.050 u. Calculate the energy released in MeV. (Given 1 u = 931 MeV)
🔍 Worked example — Exam-style
A nucleus consists of 2 protons and 2 neutrons. The mass of a proton is 1.673 × 10^-27 kg and the mass of a neutron is 1.675 × 10^-27 kg. The actual mass of the nucleus is 6.032 × 10^-27 kg. Calculate the binding energy of this nucleus in Joules.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.1 (b) Mass defect and binding energy
The mass defect of a nucleus is the difference between the total mass of its separate nucleons and the mass of the nucleus:
🧑🏫 Teacher's explanation
Imagine you have a collection of LEGO bricks. If you weigh each brick individually and then weigh the completed LEGO castle, you might notice something strange: the castle actually weighs slightly less than the sum of the individual bricks. In the subatomic world, this 'missing mass' isn't a mistake; it has been converted into the energy that holds the nucleus together. This is the essence of mass-energy equivalence.
📖 Definitions
🔍 Worked example — Easy
A nucleus consists of 2 protons and 2 neutrons. The mass of a proton is 1.673 × 10^-27 \text{ kg}, the mass of a neutron is 1.675 × 10^-27 \text{ kg}, and the mass of the Helium-4 nucleus is 6.644 × 10^-27 \text{ kg}. Calculate the mass defect Δ m.
🔍 Worked example — Standard
Calculate the binding energy of the Helium-4 nucleus from the previous example. Use c = 3.00 × 10^8 \text{ m/s}.
🔍 Worked example — Exam-style
A nucleus has a mass defect of 7.0 × 10^-29 \text{ kg}. Calculate the binding energy per nucleon for this nucleus, given it has 16 nucleons. Use c = 3.00 × 10^8 \text{ m/s}.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.1 (c) Binding energy per nucleon
Binding energy per nucleon (binding energy ÷ nucleon number) measures how stable a nucleus is: the higher it is, the more tightly bound.
🧑🏫 Teacher's explanation
Imagine you are building a Lego structure. If you use very strong, interlocking bricks, it takes a huge amount of energy to pull them apart. If you use loose, flimsy bricks, they fall apart easily. In a nucleus, the 'bricks' are nucleons (protons and neutrons). The 'strength' of how they are stuck together is the binding energy. However, it is not enough to just look at the total energy; we need to know how much energy is holding each individual nucleon in place. This is the binding energy per nucleon.
📖 Definitions
🔍 Worked example — Easy
A nucleus has 12 nucleons and a total binding energy of 108 MeV. Calculate the binding energy per nucleon.
🔍 Worked example — Standard
Calculate the energy released (in MeV) when a nucleus undergoes a reaction that results in a mass loss of 0.005 u. Use 1 u = 931 MeV/c^2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.1 (d) Fusion and fission
Relevance of binding energy per nucleon: energy is released in any reaction whose products have a higher binding energy per nucleon (are more tightly bound) than the reactants.
🧑🏫 Teacher's explanation
Think of a nucleus like a collection of magnets. If you have two small magnets that are loosely held together, they have a certain amount of 'potential' to snap together more tightly. When they snap together into one stronger unit, they release a burst of energy. This is the essence of nuclear fusion.
📖 Definitions
🔍 Worked example — Exam-style
A star undergoes fusion. The reaction is ^4_1H + ^4_1H arrow ^7_2He + energy. If the mass of the reactants is 8.016 u and the mass of the products is 7.018 u, calculate the energy released in J (use 1 u = 1.66 × 10^-27 kg and c = 3.00 × 10^8 m/s).
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.2 (a) Random and spontaneous decay
Radioactive decay is:
🧑🏫 Teacher's explanation
Imagine you have a large jar filled with thousands of popcorn kernels, and you are heating them. You know that eventually, they will all pop, but you cannot predict exactly which kernel will pop next or at what precise millisecond. You can't speed them up by shaking the jar or slow them down by cooling it slightly; the process just happens. This is exactly how radioactive decay works.
📖 Definitions
🔍 Worked example — Easy
A sample of a radioactive isotope has an initial activity of 1.2 × 10^4 Bq. After one half-life, what is the activity of the sample?
🔍 Worked example — Standard
A radioactive source has a decay constant λ = 0.045 s^-1. Calculate the half-life t_1/2 of the source.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.2 (b) Activity, decay constant and half-life
A = λ N
🧑🏫 Teacher's explanation
Imagine you are in a large crowd of people, and every person has a tiny, invisible coin that they might drop at any moment. You cannot predict exactly *when* a specific person will drop their coin, but if you watch the whole crowd, you can predict exactly how many coins will hit the floor every minute. This is exactly how radioactive decay works.
📖 Definitions
🔍 Worked example — Easy
A sample contains 8.0 × 10^20 nuclei. The decay constant λ is 1.5 × 10^-4 s^-1. Calculate the activity A of the sample.
🔍 Worked example — Exam-style
A sample of a radioactive substance has an initial activity of 400 Bq. After 12 hours, the activity has dropped to 50 Bq. Calculate the half-life of the substance.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 23.2 (c) Exponential decay
Because the decay rate is proportional to the number of nuclei left, decay is exponential:
🧑🏫 Teacher's explanation
Imagine you are in a room with 100 people, and every minute, exactly half of the people decide to leave. In the first minute, 50 leave. In the second minute, 25 leave. In the third, 12.5 leave. The number of people leaving gets smaller and smaller, but it never quite hits zero. This is the essence of exponential decay.
📖 Definitions
🔍 Worked example — Easy
A sample of a radioactive isotope has an initial activity of 800 Bq. After 24 hours, the activity has dropped to 100 Bq. Calculate the half-life of the isotope.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked example
Example 1: binding energy. Helium-4 has a nuclear mass of 4.00151 u. Proton mass = 1.00728 u, neutron mass = 1.00867 u.
🔍 Worked example — Easy
Calculate the binding energy of a Helium-4 nucleus in Joules. The mass of a proton is 1.673 × 10^-27 kg, the mass of a neutron is 1.675 × 10^-27 kg, and the mass of the Helium-4 nucleus is 6.644 × 10^-27 kg. Use c = 3.00 × 10^8 m/s.
🔍 Worked example — Standard
A radioactive source has an initial activity of 8.0 × 10^4 Bq. After 3 half-lives, what is the activity of the source?
🔍 Worked example — Exam-style
A sample contains 2.0 × 10^15 nuclei of a radioactive isotope. The decay constant λ is 1.1 × 10^-4 s^-1. Calculate the activity of the sample after 500 seconds.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) Mass defect: total mass of separate nucleons minus the mass of the nucleus; binding energy: energy needed to separate the nucleus into its individual nucleons. (2) Peak near iron-56; fusion of light nuclei and fission of heavy nuclei both produce nuclei with higher binding energy per nucleon, so mass decreases and energy is released. (3) 0.20 × 1.66 × 10^-27 × (3.00 × 10^8)^2 = 3.0 × 10^-11 J (about 186 MeV). (4) The probability per unit time that a nucleus will decay; λ = 0.693/t_1/2. (5) The count rate fluctuates randomly about a mean value.