Half-life is the time required for the number of undecayed radioactive nuclei in a sample to fall to half its initial value. Equivalently, it is the time required for the sample's activity to decrease to half its initial activity, provided the sample contains a single radionuclide and no new radioactive nuclei are being produced.
The central idea is statistical. Half-life is not a countdown assigned to each nucleus, and it does not tell you when a particular nucleus will decay. It describes the predictable behaviour of a large population whose individual decays are random.
This half-life explained guide focuses on that single concept: its probabilistic meaning, exponential mathematics, relationship with activity and count rate, and the methods IB Physics students need for exam questions. For wider syllabus coverage, including decay equations, nuclear binding energy, and radiation types, use the IB Physics Atomic and Nuclear Explained exam guide.
What does half-life mean in radioactive decay?
A radioactive nucleus is unstable and can transform into another nuclear state or nuclide by emitting radiation. The process is spontaneous, meaning it occurs without needing an external trigger, and random, meaning the exact decay time of an individual nucleus cannot be predicted.
Suppose a sample initially contains 8,000 undecayed nuclei and has a half-life of 10 minutes. The expected pattern is:
Time elapsedHalf-lives elapsedExpected nuclei remainingFraction remaining0 min08,000110 min14,0001/220 min22,0001/430 min31,0001/840 min45001/16
Each half-life removes half of the nuclei that remain, not half of the original sample every time. The number lost therefore becomes smaller during successive intervals: approximately 4,000 in the first interval, 2,000 in the second, and 1,000 in the third.
This is why radioactive decay is exponential rather than linear. A linear model would subtract the same number in every equal time interval and would eventually predict a negative number of nuclei, which is physically impossible.
Why is half-life probabilistic rather than a fixed countdown?
An individual radioactive nucleus does not have a known decay appointment. Two apparently identical nuclei of the same nuclide can decay at very different times, even though they have the same probability of decaying during any given interval.
For a nucleus with half-life T₁/₂:
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The probability that it survives for one half-life is 1/2.
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The probability that it survives for two half-lives is (1/2)² = 1/4.
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The probability that it survives for three half-lives is (1/2)³ = 1/8.
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The probability that it has decayed after one half-life is 1 - 1/2 = 1/2.
A nucleus that survives one half-life is not now overdue for decay. It still has a 50% probability of surviving the next half-life. Radioactive decay is therefore described as memoryless: the probability of future decay does not depend on how long the nucleus has already existed.
The familiar statement that “half the nuclei decay in one half-life” is an expected large-sample result. It should not be interpreted as a guarantee that exactly 50 out of 100 nuclei will decay. With a small number of nuclei, random fluctuations can be significant; with the enormous numbers found in ordinary samples, the measured fraction is usually very close to the statistical prediction.
This distinction is a frequent source of errors in IB physics radioactivity questions. Half-life characterizes a radionuclide or decay process, not the predetermined lifetime of one selected nucleus. The statistical interpretation is developed further in RevisionDojo's explanation of how half-life expresses the statistical nature of decay.
How is half-life calculated?
Using repeated halving
For an integer number of half-lives, the most efficient equation is:
Here:
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N₀ is the initial number of undecayed nuclei.
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N is the number remaining.
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n = t/T₁/₂ is the number of half-lives elapsed.
The same fractional relationship applies to activity and corrected count rate:
For example, a source has an initial activity of 960 Bq and a half-life of 6 hours. After 18 hours, three half-lives have elapsed, so:
Notice that 120 Bq remains. The activity that has been lost is 960 - 120 = 840 Bq. Always check whether a question asks for the amount remaining or the amount that has decayed.
Using the exponential decay law
For arbitrary times, exponential notation is more general:
The activity follows the same form:
The decay constant λ represents the decay probability per unit time in the appropriate small-time limit. It is related to half-life by:
A large decay constant means a high probability of decay per unit time and therefore a short half-life. A small decay constant means slower decay and a longer half-life.
The units must be consistent. If λ is measured in s⁻¹, time must be in seconds; if λ is in day⁻¹, time must be in days. The product λt must have no units because it appears as an exponent.
In the current IB Physics course, students at both levels work with half-life, activity, count rate, integer numbers of half-lives, and background radiation. The exponential decay law, decay constant, and the relationship T₁/₂ = ln 2/λ are additional Higher Level content. Official specimen questions demonstrate the expected use of this relationship, while RevisionDojo's HL quantitative analysis of decay notes provide focused practice.
What is the difference between nuclei, activity, and count rate?
These quantities all decrease with the same half-life under suitable conditions, but they are not interchangeable definitions.
QuantityMeaningTypical unitNNumber of undecayed radioactive nucleiNo unitActivity, ANumber of nuclear decays per unit time in the sourcebecquerel, BqCount rateNumber of events recorded by a detector per unit timecounts s⁻¹ or counts min⁻¹Background count rateDetector reading from environmental and other sourcescounts s⁻¹ or counts min⁻¹
One becquerel means one decay per second. Activity is connected to the number of undecayed nuclei by:
A detector generally records fewer events than the source actually produces because not all emitted radiation travels towards the detector or is detected. The measured count rate may also include background radiation, so it is not automatically equal to activity.
If the background rate remains constant, calculate:
The corrected count rate is proportional to source activity and can therefore be halved to determine half-life. RevisionDojo's E.3 Radioactive Decay resources place these quantities within the current syllabus without turning this single-concept explanation into a complete nuclear physics topic page.
How do you determine half-life from a graph?
A decay graph normally places time on the horizontal axis and nuclei remaining, activity, or count rate on the vertical axis. The curve falls steeply at first and then becomes progressively less steep because fewer undecayed nuclei remain.
Use this method:
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If the graph shows measured count rate, identify and subtract the background count rate.
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Choose a convenient corrected vertical value on the curve.
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Divide that value by two.
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Read the time at the original value and at the halved value.
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Subtract the two times to obtain the half-life.
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Repeat with another pair of values and average if the graph is noisy.
For example, if corrected activity falls from 800 Bq at 2 minutes to 400 Bq at 9 minutes, the half-life is 9 - 2 = 7 minutes. It should also take approximately 7 minutes to fall from 400 Bq to 200 Bq.
A graph of uncorrected count rate may approach the background level rather than zero. Halving the raw measured value in that situation gives the wrong result because the background component is not decaying with the source. This is one of the most important graph-reading checks in IB questions.
Why does a radioactive sample never reach zero in the model?
After every half-life, the model multiplies the number remaining by one-half. No finite number of halvings makes a positive continuous quantity exactly zero, so an ideal exponential curve approaches zero asymptotically.
A real sample contains a whole number of nuclei, however, not a continuous fraction of a nucleus. Eventually the final radioactive nucleus will decay at an unpredictable time. The smooth exponential equation gives an expected population and becomes less representative of an individual sample when only a very small number of nuclei remain.
This does not mean a radioactive source remains equally dangerous forever. Its activity can become negligibly small compared with background or below a relevant measurement threshold. “Not mathematically zero” and “practically insignificant” are different statements.
Common IB Physics half-life mistakes
Treating decay as linear
Do not subtract 50% of the original amount during every half-life. Multiply the current amount by one-half, producing 100%, 50%, 25%, 12.5%, and so on.
Saying every nucleus lives for one half-life
Half-life is not the lifetime of an individual nucleus. State that it is the time for the expected number of undecayed nuclei, or the activity of a large sample, to fall to half its initial value.
Confusing decayed and remaining fractions
After three half-lives, 1/8 remains, but 7/8 has decayed. Write the remaining fraction first, then subtract it from one if the question asks how much has decayed.
Halving an uncorrected count rate
Subtract background before halving detector readings. Otherwise, the calculated half-life is systematically distorted.
Mixing time units
Convert all times into a common unit before using λ, T₁/₂, or an exponential equation. A decay constant in s⁻¹ cannot be combined directly with a time in hours.
Confusing half-life with mean lifetime
The mean lifetime is τ = 1/λ, whereas the half-life is T₁/₂ = ln 2/λ. They describe related but different statistical times, so they are not numerically equal.
A useful revision method is to classify errors rather than simply reread solutions. RevisionDojo's guide to common atomic and nuclear physics mistakes can help you identify whether an error came from wording, graph interpretation, equation selection, or units.
How to prepare for half-life exam questions
Learn the definition precisely, but practise switching among verbal descriptions, tables, graphs, and equations. An IB question may test the same concept through a multiple-choice calculation, a data-based decay curve, or an explanation of randomness.
A reliable sequence is:
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Identify the quantity: nuclei, activity, raw count rate, or corrected count rate.
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Check for background radiation.
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Calculate the number of half-lives if the elapsed time is an integer multiple.
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Use repeated halving when possible.
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Use the exponential law for arbitrary intervals when it is part of your assessed course content.
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Distinguish the fraction remaining from the fraction decayed.
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Include units and sensible significant figures.
After reviewing the concept, complete targeted questions in the IB Physics E.3 Radioactive Decay Questionbank. The Topic E nuclear and quantum physics flashcards are useful for active recall of definitions and equations, while Jojo AI can help diagnose a specific step without replacing your own calculation.
Conclusion
Half-life is the time required for the expected number of undecayed nuclei, and therefore the activity of a pure radioactive sample, to fall to half its initial value. It describes exponential population behaviour produced by random, independent nuclear decays, not a fixed countdown for each nucleus.
For exams, remember repeated halving, distinguish activity from detector count rate, correct for background radiation, and use consistent time units. RevisionDojo's E.3 notes, Questionbank, flashcards, and Jojo AI are most useful when you combine precise definitions with repeated graph and calculation practice.