Half-Life Calculations

Half-life is the time for half the nuclei in a sample to decay, and every 0625 half-life calculation follows one pattern: repeated halving. It earns 3-5 marks on nearly every Paper 4, and the Extended version adds a single twist of subtracting background radiation first. Master the halving table and the graph read-off and the marks are yours.

What is half-life and how do you calculate with it?

The half-life of an isotope is the time taken for half the nuclei in a sample to decay. Equivalently, it is the time for the count rate (activity) from a source to fall to half its original value. Each isotope’s half-life is fixed: cobalt-60’s is about 5.3 years, carbon-14’s about 5,700 years, regardless of sample size or conditions.

Calculations use repeated halving. A halving table keeps it tidy:

Number of half-lives	Fraction remaining	Count rate (start 800/s)
0	1	800
1	$\dfrac{1}{2}$	400
2	$\dfrac{1}{4}$	200
3	$\dfrac{1}{8}$	100

Three question types cover everything. Type 1: given half-life and time, find what remains (count the halvings). Type 2: given start and end values, find the half-life (count halvings, divide the time). Type 3: read the half-life from a decay graph (find the time for the curve to drop from any value to half that value). In words for type 2: half-life equals total time divided by number of halvings.

How does background radiation change the calculation?

Extended questions often give a measured count rate that includes background. Subtract the background from every reading before you halve. A measured 410/min with background 10/min is a true source rate of 400/min. After one half-life the source gives 200/min, but the detector reads 210/min. Reverse the correction at the end if the question asks for the detector reading. Forgetting this subtraction is the single biggest mark-loser in 0625 half-life questions.

Worked Exam Question

A detector near a sealed source reads 1,610 counts/min. The background count rate is 10 counts/min. The isotope has a half-life of 8.0 days. (a) Calculate the count rate the detector reads after 32 days. (4 marks) (b) State why the source remains hazardous even after this time. (1 mark)

Solution. (a) Correct for background: $1610 - 10 = 1600$ counts/min from the source. Number of half-lives: $32 \div 8.0 = 4$. Halve four times: $1600 \rightarrow 800 \rightarrow 400 \rightarrow 200 \rightarrow 100$ counts/min. Add background back: detector reads $100 + 10 = 110$ counts/min. (b) The source still emits ionising radiation: activity has fallen to $\dfrac{1}{16}$, not to zero.

Mark scheme:

• M1: subtracts background ($1610 - 10 = 1600$).
• M1: identifies 4 half-lives ($32 / 8.0$).
• M1: halves correctly four times to 100 counts/min.
• A1: 110 counts/min (background re-added) with unit.
• B1: radiation is still emitted / activity never reaches zero.

Common Mistakes

• Dividing by the number of half-lives. Fix: 4 half-lives means halve 4 times ($\div 16$), not $\div 4$. Build the halving chain explicitly.
• Skipping the background correction. Fix: if a background value appears anywhere in the question, subtract before halving, then re-add if asked for the detector reading.
• Graph read-offs from the wrong starting point. Fix: half-life is the time from any count rate to half that rate. Draw the horizontal and vertical construction lines on the graph; they often carry a mark.
• Saying the sample is “safe after two half-lives”. Fix: activity falls to a quarter; it never reaches zero. Use “falls to 1/4”, not “gone”.
• Mixing units of time. Fix: keep half-life and elapsed time in the same unit before dividing.

Exam Technique Tip

Write the halving chain as an arrow line, $1600 \rightarrow 800 \rightarrow 400 \rightarrow 200 \rightarrow 100$, and label the time above each arrow (8 d, 16 d, 24 d, 32 d). This single line shows the examiner every method step, makes self-checking instant, and converts a fiddly mental calculation into visible M-mark working. For graph questions, leave your construction lines on the grid.

How This Is Examined

A CS subtopic. Core candidates (Papers 1 and 3) handle straightforward halving and graph read-offs with whole numbers of half-lives. Extended candidates (Papers 2 and 4) add background correction and two-step problems, such as finding half-life from a table then projecting forward. Paper 6 links here through decay-curve plotting: expect to plot count rate against time, draw a smooth curve and extract the half-life with construction lines, a direct application of our graph-skills guide. Set a 5-minute target per half-life question; with the arrow-chain method most students finish in three.