Ch 7 · 5.2.5 Half-life · O Level Physics 5054
Topic 7 of 8 in Nuclear Physics — O Level Physics 5054 — 48 lessons.
Introduction
You cannot predict when one radioactive nucleus will decay, but with billions of them the pattern is completely regular: in a fixed time, half of them decay. That fixed time, the half-life, lets us date ancient bones and choose the right isotope for a hospital or a factory. (Syllabus section 5.2.5.)
Key ideas — Half-life
The half-life of a particular isotope is the time taken for half the nuclei of that isotope in any sample to decay.
🧑🏫 Teacher's explanation
Imagine you have a large bag of 100 candies, and every hour, you eat exactly half of whatever is left in the bag. In the first hour, you eat 50 candies (leaving 50). In the second hour, you eat 25 (leaving 25). In the third hour, you eat 12.5 (leaving 12.5). Even though the amount you eat changes, the 'time taken to halve' remains a constant 1 hour. This is exactly how radioactive decay works at the atomic level.
📖 Definitions
🔍 Worked example — Easy
A sample of a radioactive isotope has an initial activity of 80 Bq. After 24 days, the activity has dropped to 10 Bq. Calculate the half-life of the isotope.
🔍 Worked example — Standard
A sample contains 400 nuclei of an isotope. After 15 minutes, only 50 nuclei remain. Calculate the half-life.
🔍 Worked example — Exam-style
An archaeological artifact contains a certain amount of Carbon-14. The half-life of Carbon-14 is 5700 years. If the current activity of the Carbon-14 in the artifact is 25\% of its original activity, how old is the artifact?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Finding the half-life from a decay curve
A graph of count rate (or number of undecayed nuclei) against time is a decay curve: it falls steeply at first, then more and more slowly, never quite reaching zero.
🧑🏫 Teacher's explanation
Imagine you have a large bag of candies, and every hour, exactly half of the candies in the bag are eaten by your friends. In the first hour, you lose many candies. In the second hour, you lose fewer, because there were fewer to start with. Even though the number of candies decreases, the *time* it takes to lose half of whatever is currently in the bag remains constant. This constant time interval is the half-life.
📖 Definitions
🔍 Worked example — Easy
A sample of a radioactive isotope has an initial activity of 800 counts/min. After 24 minutes, the activity is 200 counts/min. Calculate the half-life of the isotope.
🔍 Worked example — Standard
A radioactive source has a background radiation of 20 counts/s. The measured activity is 102 counts/s. After one half-life, what is the new measured activity?
🔍 Worked example — Exam-style
An isotope is used in a medical scan. The initial activity is 1200 Bq. After 5 hours, the activity is 150 Bq. Calculate the half-life.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Carbon-14 dating
It works for objects up to roughly 50 000 years old; after that there is too little carbon-14 left to measure.
🧑🏫 Teacher's explanation
Imagine you have a jar full of 1000 magical candies that disappear at a constant rate. Every hour, exactly half of the candies remaining in the jar vanish. After one hour, you have 500; after two hours, you have 250. The time it takes for that 'halving' to happen is the half-life.
📖 Definitions
🔍 Worked example — Easy
A sample of Carbon-14 has an initial activity of 80 Bq. After one half-life, what is the activity of the sample?
🔍 Worked example — Exam-style
A sample of an isotope has a half-life of 20 minutes. If the initial activity is 400 Bq, calculate the activity remaining after 1 hour.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Choosing an isotope for a job
Two things matter: the type of radiation (how penetrating and ionising it is) and the half-life (long enough to be useful, short enough not to stay dangerous).
🧑🏫 Teacher's explanation
Imagine you are choosing a flashlight for different jobs. If you need to light up a dark cave for a week, you want a battery that lasts a long time. If you want to use a laser to perform surgery, you want a beam that is very precise and doesn't linger in the patient's eye. In nuclear physics, we choose isotopes based on two 'features': the type of radiation they emit and their half-life.
📖 Definitions
🔍 Worked example — Easy
A sample of an isotope has a half-life of 10 years. If the initial activity is 800 Bq, what is the activity after 30 years?
🔍 Worked example — Standard
A medical tracer used for diagnosis has a half-life of 6 hours. A patient is injected with 400 MBq of the tracer. How much activity remains in the patient after 24 hours?
🔍 Worked example — Exam-style
A sample of Carbon-14 has an activity of 12 Bq. The initial activity was 96 Bq. Calculate the age of the object using the half-life of Carbon-14, which is 5730 years.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked example
Example 1. A sample has an activity of 1200 counts/min. Its half-life is 6 hours. What is its activity after 24 hours?
🔍 Worked example — Easy
A radioactive source has an initial activity of 1200 counts/min. If the half-life of the isotope is 6 hours, calculate its activity after 24 hours.
🔍 Worked example — Standard
A sample contains 8.0 mg of a radioactive isotope. After 30 days, only 1.0 mg remains. Determine the half-life of the isotope.
🔍 Worked example — Exam-style
An archaeologist finds a wooden tool. The activity of Carbon-14 in the tool is 25\% of the activity found in living trees. Given the half-life of Carbon-14 is 5700 years, calculate the age of the tool.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) The time taken for half the nuclei of that isotope in any sample to decay. (2) 3 half-lives: 1/8. (3) 2400 → 1200 → 600 → 300 is 3 half-lives in 15 minutes, so the half-life is 5 minutes. (4) Alpha ionises the air well, so the current is easily reduced by smoke, but it is stopped by a few cm of air and the case, so it is safe; a long half-life means the source does not need replacing. (5) Gamma can pass out of the body to be detected, and it is the least ionising; a short half-life means it stops emitting soon after the scan, limiting the dose to the patient.