Learn7408Paper 2 · Paper 2Nuclear physicsRadioactive decay

Nuclear physics · Subtopic 3 of 4

Radioactive Decay

Radioactive decay is random (you can't predict which nucleus will decay next) and spontaneous (unaffected by temperature, pressure or chemistry). The activity is proportional to the number of undecayed nuclei: $A=\lambda N$ , where $\lambda$ is the decay constant. So $N$ , $A$ and the count rate all fall exponentially: $x=x_0e^{-\lambda t}$ . The half-life is $t_{1/2}=\dfrac{\ln2}\lambda=\dfrac{0.693}\lambda$ .

Cambridge 9702 Paper 4Edexcel YPH11 WPH15AQA 7408 Paper 2≈ 40 minRevision 7 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain that decay is random and spontaneous, with evidence.
Define activity and decay constant; use $A=\lambda N$ .
Use $x=x_0e^{-\lambda t}$ and $\lambda=\dfrac{0.693}{t_{1/2}}$ .
Correct count rates for background radiation.

Pause and think

A Geiger counter next to a source clicks irregularly: sometimes three in a row, then a gap. Is the counter faulty?

Section 2 of 14

Core idea

Start hereDice that decay

Roll 1000 dice; remove every six; roll again. About a sixth vanish each round, and the pile shrinks exponentially. Radioactive nuclei behave the same: each has a fixed chance of decaying per second.

The big idea

$A=\lambda N$ ; $N=N_0e^{-\lambda t}$ (and $A$ , count rate). $t_{1/2}=\frac{\ln2}\lambda$ .

1. From randomness to exponentials

Decay of nuclei: each half-life halves what remains

time (half-lives) = 0.0

1\to\tfrac12\to\tfrac14\to\tfrac18

. It never quite reaches zero, but after 10 half-lives less than 0.1% remains.

The decay law

$\frac{dN}{dt}=-\lambda N$
Each nucleus has probability $\lambda$ per second.
$N=N_0e^{-\lambda t}$
Solve (as for capacitor discharge).
$\tfrac12=e^{-\lambda t_{1/2}}$
Half-life.
$t_{1/2}=\frac{\ln2}\lambda$
Take logs.

Simulating randomness: 100 dice, remove the sixes each throw

Each die has the same chance each throw, like each nucleus. The numbers scatter, but the trend is exponential with a "half-life" of about 3.8 throws.

Worked example1.0 μg of iodine-131

$t_{1/2}=8.0$ days, molar mass 131 g.

Goal: $N$ , $\lambda$ , $A$ , fraction left after 20 days.
Strategy: Moles → $N$ ; $\lambda$ in s $^{-1}$ .

$N=\frac{10^{-6}}{131}\times6.02\times10^{23}=4.6\times10^{15}$
Nuclei.
$\lambda=\frac{0.693}{8.0\times86400}=1.0\times10^{-6}\text{ s}^{-1}$
Days to seconds.
$A=\lambda N=4.6\times10^9\text{ Bq}$
Activity.
$e^{-\lambda t}=2^{-20/8}=0.18$
2.5 half-lives.

Answer

4.6\times10^{15}

;

1.0\times10^{-6}

^{-1}

;

4.6\times10^9

Bq; 18%.

✓ CheckBetween

\frac14

and

\frac18

✓.

Common misconceptionMixed time units

\lambda

in s

^{-1}

for activity in Bq.

Pause and think

Half-lives until under 1% remains?

Summary

$A=\lambda N$ ; $N=N_0e^{-\lambda t}$ .
$t_{1/2}=\ln2/\lambda$ .

Decay constant: Probability of decay per unit time.

HistoryRutherford and Soddy. In

1902

Ernest Rutherford and Frederick Soddy showed that radioactive atoms transform into different elements, and that the activity decays exponentially. Soddy reportedly said "Rutherford, this is transmutation!" Rutherford snapped back: "Don't call it transmutation, they'll have our heads off as alchemists!"

Fun factCarbon dating. Living things keep a constant ratio of carbon-14 while alive. After death, it decays with a half-life of

5730

years. Measuring how much remains lets archaeologists date wood, bone and cloth up to about

50\,000

years old.

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