6.4NuclearStretch

Radioactive Decay and Half-Life

No physicist can predict when one particular unstable nucleus will decay — yet given a large enough sample, its behaviour becomes almost perfectly predictable. That apparent contradiction is resolved by the exponential mathematics of half-life.

45 min Video by Science Shorts Module 6.4 Nuclear and particle physics

Radioactivity, Half-Life & Inverse Square LawWatch the full walkthrough before the notes below.

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What you'll be able to do

Describe radioactive decay as random and spontaneous
Use $A = λ N$ and the exponential decay equations
Use $T_{1/2} = \frac{ln 2}{λ}$
Determine the decay constant from experimental data

Random, spontaneous decay

Decay of any individual nucleus is unpredictable and unaffected by external conditions — but a large sample decays entirely predictably on average, just like coin-flipping.

Activity and exponential decay

Activity, $A = λ N$ , is the number of decays per second; $λ$ is the probability of decay per unit time for one nucleus.

A = λ N N = N_{0} e^{- λ t}

Activity, and exponential decay.

1Use

A = λ N

A = 4.0 \times 1 0^{- 8} \times 3.0 \times 1 0^{18}

Answer

A = 1.2 \times 1 0^{11}

Half-life

Half-life, $T_{1/2}$ , is the time for the number of nuclei (or activity) to halve — constant for a given isotope, however much sample remains.

T_{1/2} = \frac{ln 2}{λ}

Half-life.

1Use

T_{1/2} = \frac{ln 2}{λ}

T_{1/2} = \frac{0.693}{1.5 \times 1 0 ^{- 4}}

Answer

T_{1/2} \approx 4620

Tip — Plotting ln(activity) against time gives gradient −λ — the standard experimental method for finding decay constant and half-life.

Equation recap

A = λ N

Activity.

N = N_{0} e^{- λ t}

Exponential decay.

T_{1/2} = \frac{ln 2}{λ}

Half-life.

Common mistakes to avoid

Thinking a specific nucleus’s decay time can be predicted.

Decay is fundamentally random for any individual nucleus — only large-sample behaviour is predictable.

Believing half-life changes as a sample decays.

Half-life is constant for a given isotope, independent of remaining sample size.

Key takeaways

Radioactive decay is random and spontaneous, unaffected by external conditions.
Activity A=λN; N=N₀e^(−λt).
Half-life T½=ln2/λ, constant for a given isotope.

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Radioactive Decay and Half-Life

45 min Video by Science Shorts Module 6.4 Nuclear and particle physics