Half-Life

Because radioactive decay is random, we can never say when a particular nucleus will decay. Yet a large sample behaves in a remarkably predictable way: in a fixed period of time, a fixed fraction of the unstable nuclei always decays. This regularity is captured by the half-life — the single most useful number for describing how quickly a radioactive source decays. Half-life lets us predict how much of a source is left after a given time, choose the right isotope for a job, and date ancient objects. This lesson, part of Topic P6 (Radioactivity) of OCR Gateway Science A, defines half-life, shows how to read and use a decay curve, and works through the calculations you will be expected to do.

By the end of this lesson you should be able to define half-life, find the fraction or amount of a source remaining after a whole number of half-lives, read a half-life from a decay curve, and (Higher tier) work with the ratio of decayed to remaining nuclei.

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What Is Half-Life?

The half-life of a radioactive isotope is the time taken for half the undecayed nuclei in a sample to decay. Equivalently — and just as usefully — it is the time taken for the activity (or count-rate) of the sample to fall to half its value.

These two definitions are really the same statement, because the activity is proportional to the number of undecayed nuclei present: halve the number of unstable nuclei and you halve the rate at which they decay. The half-life is a fixed property of a given isotope: it is always the same, no matter how big the sample is or how old it is. Half-lives vary enormously between isotopes — from tiny fractions of a second to billions of years.

The key feature is that the decay is repeated halving. After one half-life, half the original nuclei remain. After two half-lives, half of that half — a quarter — remains. After three half-lives, an eighth remains, and so on. The amount never reaches exactly zero; it just keeps halving.

Number of half-lives	Fraction of original remaining	As a fraction
0	all	$1$1
1	a half	$\dfrac{1}{2}$21​
2	a quarter	$\dfrac{1}{4}$41​
3	an eighth	$\dfrac{1}{8}$81​
4	a sixteenth	$\dfrac{1}{16}$161​
$n$n	—	$\left(\dfrac{1}{2}\right)^{n}$(21​)n

Exam Tip: Define half-life as the time for half the undecayed nuclei to decay (or the time for the count-rate to halve). The fraction remaining after $n$n half-lives is $\left(\tfrac{1}{2}\right)^{n}$(21​)n: a half, then a quarter, then an eighth, and so on. The half-life is fixed for a given isotope.

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The Decay Curve

If you plot the count-rate (or the number of undecayed nuclei) of a source against time, you get a decay curve — a smooth curve that falls steeply at first and then more gently, getting closer and closer to zero without ever quite reaching it. This characteristic shape is called exponential decay.

To find the half-life from a decay curve: pick a starting value on the count-rate axis, find the time at which the count-rate has fallen to half of it, and read off the time interval. A good check is to do it twice — from the start value to its half, and from that half to a quarter — and confirm you get the same time both times, because the half-life is constant. In the curve above, the count-rate falls from $800$800 to $400$400 in one half-life, from $400$400 to $200$200 in the next, and from $200$200 to $100$100 in the next: each step takes the same time.

Exam Tip: To read a half-life off a graph, find the time for the count-rate to drop from any value to half of that value (e.g. $800 \to 400$800→400, or $600 \to 300$600→300). Reading it at two different places and getting the same answer confirms you've done it correctly.

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Calculating the Amount Remaining

The repeated-halving pattern lets you calculate how much of a source remains after a whole number of half-lives. The method is simple: work out how many half-lives have passed, then halve the starting amount that many times (or multiply by $\left(\tfrac{1}{2}\right)^{n}$(21​)n).

Worked Example — Mass remaining after several half-lives

A sample contains $80 \text{ g}$80 g of a radioactive isotope with a half-life of $6$6 years. What mass of the isotope remains after $18$18 years?

Step 1 — find the number of half-lives: $18 \div 6 = 3$18÷6=3 half-lives.

Step 2 — halve the mass three times: $80 \to 40 \to 20 \to 10 \text{ g}$80→40→20→10 g.

Answer: after $18$18 years, $10 \text{ g}$10 g of the isotope remains. (Check: $80 \times \left(\tfrac{1}{2}\right)^{3} = 80 \times \tfrac{1}{8} = 10 \text{ g}$80×(21​)3=80×81​=10 g.)

Worked Example — Count-rate after several half-lives

A source has an initial count-rate of $2400 \text{ Bq}$2400 Bq. Its half-life is $15$15 minutes. What is the count-rate after $1$1 hour?

Step 1 — find the number of half-lives in $1$1 hour: $60 \div 15 = 4$60÷15=4 half-lives.

Step 2 — halve the count-rate four times: $2400 \to 1200 \to 600 \to 300 \to 150 \text{ Bq}$2400→1200→600→300→150 Bq.

Answer: after $1$1 hour the count-rate is $150 \text{ Bq}$150 Bq. (Check: $2400 \times \left(\tfrac{1}{2}\right)^{4} = 2400 \times \tfrac{1}{16} = 150 \text{ Bq}$2400×(21​)4=2400×161​=150 Bq.)

Worked Example — Finding the number of half-lives

A radioactive source has an activity of $640 \text{ Bq}$640 Bq. After some time, its activity has fallen to $40 \text{ Bq}$40 Bq. How many half-lives have passed?

Step 1 — halve repeatedly until you reach $40$40: $640 \to 320 \to 160 \to 80 \to 40$640→320→160→80→40.

Step 2 — count the halvings: that is 4 halvings.

Answer: $4$4 half-lives have passed (since $640 \times \tfrac{1}{16} = 40 \text{ Bq}$640×161​=40 Bq).

Exam Tip: The reliable method is: (1) find how many half-lives have passed (time ÷ half-life), then (2) halve the starting amount that many times. To find the number of half-lives from before-and-after values, keep halving the start value and count how many halvings reach the end value.

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Using Half-Life in Real Time

Sometimes you are given the half-life and asked for the total time for a source to fall to a certain level — the reverse calculation.

Worked Example — Finding the time taken