IB Physics SLTopic 5 — The Atomic & Nuclear WorldPaper 1 & 2remaining = (½)n~8 min read
Activity & Half-Life
You can never say when one particular nucleus will decay — but a whole sample of them behaves with beautiful regularity. Wait one half-life and exactly half have gone. Wait another and half of what’s left goes. This steady halving is the heartbeat of every radioactive source, and it lets us date bones, time medical scans, and measure the age of the Earth.
📘 What you need to know
Activity is the number of nuclei that decay per unit time, measured in becquerels (Bq) — 1 Bq = one decay per second
Half-life (t½) is the time for half the undecayed nuclei — or the activity — to fall to half its value
Half-life is constant for a given isotope, and different isotopes have wildly different half-lives
After n half-lives, the fraction remaining is (½)n
Both the number of nuclei and the activity halve together over each half-life
Half-life can be read straight off a decay curve: the time for the activity to drop from any value to half that value
Accuracy improves by measuring over several half-lives and taking repeats
Activity
The activity of a source is simply how many nuclei are decaying every second. It’s measured in becquerels (Bq): an activity of 1 Bq means one nucleus decays per second. A source with more undecayed nuclei has a higher activity, so as the sample decays away, its activity steadily drops too. We can’t predict any single decay, but with billions of nuclei the average rate is smooth and predictable.
What Half-Life Means
Here’s the key idea. Pick any moment. The half-life is the time it takes for half the nuclei still present to decay — equivalently, the time for the activity to fall to half. The clever part: this time is always the same, no matter where you start on the curve. Going from 100% to 50% takes one half-life; going from 50% to 25% takes exactly the same time again.
100%
1 half-life →
50%
1 half-life →
25%
1 half-life →
12.5%
The Fraction Remaining
Because each half-life halves what’s left, after n half-lives the fraction still remaining is one-half multiplied by itself n times:
Fraction remaining after n half-lives
fraction remaining = (½)n
This one formula answers most half-life questions. The activity remaining is A = A0 × (½)n, and the number of nuclei remaining follows exactly the same pattern. The table shows how fast it drops:
After n half-lives → (½)n remains — the general rule
Reading a Decay Curve
Plot activity against time and you get the classic decay curve: a smooth line that falls steeply at first, then flattens, always halving over equal time steps but never quite reaching zero. To find the half-life from a graph, pick a starting activity, find where it has dropped to half, and read off the time between them. Doing it two or three times and averaging gives a more reliable value.
Every drop to half the current activity takes one half-life. The two equal steps t½ and 2t½ are the same width, even though the activity falls by different amounts — that’s what “constant half-life” looks like.
Quick recap: activity is decays per second (Bq); half-life is the constant time for the activity or number of nuclei to halve; after n half-lives the fraction remaining is (½)n; and you read the half-life off a curve as the time for any value to fall to half.
🧭 Solving a half-life problem
Find the number of half-lives: n = total time ÷ half-life (or count how many times the amount has halved)
Fraction remaining = (½)n
For a fraction decayed: subtract the remaining fraction from 1
To find half-life instead: count the halvings, then divide the total time by that number
Sanity-check the size — more half-lives means far less remaining
WE 1
A radioactive sample has a half-life of 4 hours. After 20 hours, what is the ratio of decayed nuclei to original nuclei?
Step 1 — number of half-lives
n = total time ÷ half-life = 20 ÷ 4
n = 5 half-livesStep 2 — fraction remaining
(½)5 = 1 / 32
Step 3 — turn into a ratio
if 1 / 32 remains, then 31 / 32 has decayed
decayed : original = 31 : 32Careful: the question asks decayed, not remaining — a very common place to slip.
WE 2
A sample starts with 3 200 000 undecayed atoms. After 24 days, only 200 000 remain undecayed. Determine the half-life of the material.
Step 1 — how many times has it halved?
3 200 000 → 1 600 000 → 800 000 → 400 000 → 200 000
that’s 4 halvings, so n = 4 half-livesStep 2 — divide total time by the number of half-lives
half-life = 24 days ÷ 4
half-life = 6 daysQuick check: (½)4 = 1/16, and 3 200 000 ÷ 16 = 200 000 ✓
💡 Top tips
Read the question carefully: “remaining” is (½)n; “decayed” is 1 minus that. Ratios trip people up here
Activity and number of nuclei halve together — the same (½)n works for both
To find a half-life from a graph, pick a clear starting activity and read the time to reach half of it; repeat and average
Half-life is constant for an isotope — it never changes with temperature, amount, or anything else
⚠ Common mistakes
Giving the remaining fraction when the question asked for the decayed fraction (or vice versa)
Dividing the total time by the number remaining instead of the number of half-lives
Thinking the activity reaches zero — the curve keeps halving and only approaches zero
Assuming half-life depends on how much sample you have — it’s constant for the isotope
Up next: Applications of Radioactivity. Now that you can handle activity and half-life, we’ll see how the right choice of half-life and penetrating power powers carbon dating, medical tracers, smoke detectors, and more.
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