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IB Physics Half-Life: Definition, Equations and Worked Examples

IB Physics Half-Life: Definition, Equations and Worked Examples

10 min readOliver Kidd (Co-founder)6 Nov 2026

In IB Physics, half-life is the time it takes for half of the radioactive nuclei in a sample to decay. It is linked to the decay constant by λ = ln(2)/t½, and it stays the same however much of the sample is left. This guide shows you how to use it in exam questions, from simple halving to graphs and activity.

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TL;DR

  • Half-life (t½) is the time for half the undecayed nuclei, or half the activity, to disappear.
  • Decay is random for one nucleus but predictable for a large sample, so the half-life is constant.
  • Use repeated halving for whole half-lives. Use N = N₀e^(−λt) for any other time.
  • Link the two ideas with λ = ln(2)/t½ and A = λN.
  • Show every step: the equation, the substitution, the answer and the unit.

Table of Contents

What half-life means in IB Physics

Half-life is a time, not an amount. Start with 1000 undecayed nuclei of an isotope with a half-life of 10 minutes. After 10 minutes about 500 remain. After 20 minutes about 250 remain. The sample never reaches zero in a neat finite step, because each interval removes half of whatever is left.

Proportional decay sequence across two half-lives

The same applies to activity. Activity is the number of decays per second, measured in becquerels (Bq). When the number of nuclei halves, the activity halves too, so you can measure half-life from either.

Why decay is described with probability

You cannot say when one particular nucleus will decay. OpenStax College Physics describes nuclear decay as a purely statistical process. Each nucleus has a 50% chance of surviving for one half-life, and a nucleus that has already survived hundreds of half-lives still has a 50% chance of surviving one more. It works like a coin flip, where the chance of heads is 50% whatever happened before.

A large sample behaves differently. A QuarkNet classroom activity using dice shows that single-particle decay is probabilistic, yet a big sample follows a smooth exponential curve. Each time the sample halves, the number of rolls (time intervals) needed is the same. That is the evidence that decay is best modelled with probability.

The decay constant λ is the probability of decay per unit time. HyperPhysics explains that the half-life and the decay constant carry the same information, so either one characterises the decay.

Why decay is described with probability

The equations to know

The IB Physics data booklet gives the decay equations for the current syllabus, so you do not need to memorise them. You do need to know when to use each one.

Quantity Equation Use it when
Exponential decay N = N₀e^(−λt) The time is not a whole number of half-lives
Activity A = λN You need decays per second from the number of nuclei
Half-life and decay constant t½ = ln(2)/λ You are converting between the two
Whole half-lives N = N₀ × (1/2)ⁿ, n = t/t½ The time is an exact multiple of t½

The link between half-life and decay constant is λ = ln(2)/t½ ≈ 0.693/t½. OpenStax shows why: put t = t½ into N = N₀e^(−λt) and you get N = N₀e^(−0.693) = 0.500N₀. OpenStax Chemistry makes the same point from the chemistry side, noting that radioactive decay follows first-order kinetics.

You can also write the decay law as N = N₀ × 2^(−t/t½). This form is often quicker, because you can type the number of half-lives straight into your calculator.

Worked examples

These practice questions are written in the style of IB short-answer and structured questions.

Example 1: whole half-lives

A sample contains 80 g of an isotope with a half-life of 6.0 hours. How much remains after 18 hours?

  1. Number of half-lives: n = 18 ÷ 6.0 = 3.
  2. Remaining mass = 80 × (1/2)³ = 80 ÷ 8 = 10 g.

Chemistry LibreTexts uses the same method, with a 100.0 g sample leaving 12.5 g after three half-lives.

Example 2: a time that is not a whole number of half-lives

A sample has 4.0 × 10⁶ nuclei of an isotope with a half-life of 5.0 days. How many remain after 8.0 days?

  1. Decay constant: λ = 0.693 ÷ 5.0 = 0.139 day⁻¹.
  2. N = N₀e^(−λt) = 4.0 × 10⁶ × e^(−0.139 × 8.0).
  3. The exponent is −1.11, so e^(−1.11) ≈ 0.33.
  4. N ≈ 1.3 × 10⁶ nuclei.

Sanity check: 8.0 days is 1.6 half-lives. That falls between one half-life (50%) and two (25%), and 33% fits. Checking like this is a good habit, because it catches most calculator slips. OpenStax makes the same point: repeated halving works only for whole half-lives, and any other time needs the exponential.

Example 3: activity

A source has a half-life of 20 minutes and contains 2.0 × 10⁹ undecayed nuclei. What is its activity?

  1. Convert to seconds: 20 × 60 = 1200 s.
  2. λ = 0.693 ÷ 1200 = 5.8 × 10⁻⁴ s⁻¹.
  3. A = λN = 5.8 × 10⁻⁴ × 2.0 × 10⁹ ≈ 1.2 × 10⁶ Bq.

Unit conversion is the most common place to lose marks here. If λ is in s⁻¹, the half-life must be in seconds.

Example 4: finding half-life from a graph

You may get a decay curve of N or activity against time. Read off the time for the value to fall by half, then repeat from a different starting point on the curve. If both give the same half-life, that confirms the exponential pattern.

A linear graph is also possible. MiniPhysics notes that plotting ln N (or ln A) against t gives a straight line with gradient −λ. Calculate the gradient, change its sign to get λ, then use t½ = ln(2)/λ.

Common mistakes to avoid

  • Thinking half-life changes as the sample shrinks. It does not. The activity falls because A = λN, but λ belongs to the nuclide and stays constant.
  • Saying “the sample is gone after two half-lives”. It has fallen to a quarter, not zero.
  • Mixing units. Keep t and t½ in the same unit, and use seconds when you want becquerels.
  • Using halving for awkward times. If n is not a whole number, use the exponential form.
  • Forgetting background count. If a question gives a raw count rate and a background rate, subtract the background first.
  • Describing decay as predictable for one nucleus. Say it is random for one nucleus and predictable for a large sample.

How to show your working

Examiners award marks for method, so lay your answer out in a clear sequence:

  1. Write the equation you are using.
  2. Substitute values with units.
  3. Give the answer to a sensible number of significant figures, with its unit.
  4. For “explain” questions, name the idea: random decay, constant decay probability, large sample.

If a question says “estimate”, repeated halving with a clear note is usually enough. If it says “calculate”, show the equation.

Practising decay questions with exam-style material

Half-life questions are easy marks once the method is automatic, and that takes repetition on questions that look like the real paper. Our IB Physics E3 radioactive decay questions cover the basics, and the further radioactive decay set takes you into exponential and graph work. If you are revising the neighbouring ideas, the atomic structure questions are a natural next step.

Our view on half-life revision

We see the same pattern again and again. Students understand half-life in class, then lose marks on units, on non-integer half-lives, or on the wording of “explain” questions. Content that mirrors the real exam fixes this fastest.

That is why Tiber Tutor’s resources are written by actual IB examiners and linked to each other. A question points to the notes, the notes to the animated video, the video to flashcards and cram sheets, and your results feed performance tracking and syllabus monitoring. Our student-led answer review lets you mark yourself against examiner mark schemes, so you learn what earns the mark.

We believe this combination of exam-faithful content and detailed performance insight is what makes Tiber Tutor the leading IB STEM platform. For the price, we think it is the best value available, because it puts the work of real examiners within reach of every student.

Oliver

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FAQ

What is the half-life formula in IB Physics?

The key link is t½ = ln(2)/λ, which is the same as λ = 0.693/t½. For any time, use N = N₀e^(−λt), or N = N₀ × 2^(−t/t½). The data booklet provides these equations.

Does half-life change if I have a bigger sample?

No. Half-life depends on the decay constant, which is a property of the nuclide. A bigger sample has a higher activity at the start, but it still halves in the same time.

How do I find half-life from a graph?

Read the time for N or activity to fall to half its value, then check it from another point on the curve. Alternatively, plot ln N against t, take the gradient to get −λ, and calculate t½ = ln(2)/λ.

Can I use repeated halving in every question?

Only when the time is a whole number of half-lives. For anything else, such as 1.6 half-lives, use the exponential equation. OpenStax gives the same guidance.

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