3.8.1NuclearStretch

Radioactive Decay and Half-Life

You can never predict exactly when a single unstable nucleus will decay — but hand a physicist a large enough sample, and they can tell you almost exactly how much of it will be left in an hour, a year, or a millennium. Radioactive decay is completely random at the level of one nucleus, yet remarkably, precisely predictable at the level of a whole sample — the key is the exponential mathematics of $half-life$ .

45 min Video by Science Shorts 3.8.1 Radioactivity

Radioactivity, Half-Life & Inverse Square Law — GCSE & A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Describe radioactive decay as a random and spontaneous process
Define activity and the decay constant, and use $A = λ N$
Use the exponential decay equations $N = N_{0} e^{- λ t}$ and $A = A_{0} e^{- λ t}$
Define half-life and use $T_{1/2} = \frac{ln 2}{λ}$
Determine the decay constant or half-life from experimental data

Random and spontaneous decay

Radioactive decay is $random$ : it is fundamentally impossible to predict which specific nucleus in a sample will decay next, or exactly when. It is also $spontaneous$ : it happens entirely on its own, completely unaffected by external conditions such as temperature, pressure, or which chemical compound the atom is part of.

Although any individual decay is unpredictable, a sample containing a huge number of identical nuclei behaves entirely predictably on average — exactly like flipping millions of coins: you can’t predict one flip, but you can predict very precisely that close to half will land heads.

Tip — If an exam question asks "why can’t we predict when a specific nucleus will decay?", the answer is simply that decay is a genuinely random process — there is no hidden mechanism or trigger to identify.

Activity and the decay constant

The $activity$ , $A$ , of a sample is the number of decays occurring per second, measured in becquerels (Bq); one becquerel is one decay per second. Activity is proportional to the number of undecayed nuclei currently present, $N$ — more undecayed nuclei means more decays happening per second, purely by probability.

The constant of proportionality, $λ$ , is the $decay constant$ : the probability that any one particular nucleus will decay in a given unit of time (units s⁻¹). A larger decay constant means a more unstable isotope, decaying faster.

A = λ N

Activity = decay constant × number of undecayed nuclei.

1Use

A = λ N

A = 3.5 \times 1 0^{- 8} \times 2.0 \times 1 0^{18}

Answer

A = 7.0 \times 1 0^{10}

Exponential decay

Because the activity (rate of decay) is proportional to the number of nuclei remaining, and each decay reduces that number, the number of undecayed nuclei falls $exponentially$ with time — mathematically identical in form to a discharging capacitor. Activity, being directly proportional to $N$ , follows exactly the same exponential shape.

N = N_{0} e^{- λ t} A = A_{0} e^{- λ t}

Exponential decay of the number of nuclei and of activity.

1Use

N = N_{0} e^{- λ t}

N = 8.0 \times 1 0^{15} \times e^{- 2.0 \times 1 0^{- 4} \times 3000} = 8.0 \times 1 0^{15} \times e^{- 0.60}

e^{- 0.60} \approx 0.549

Answer

N \approx 4.4 \times 1 0^{15}

nuclei

Tip — This is exactly the same mathematical shape as capacitor discharge — if you can use Q=Q₀e^(−t/RC), you already know how to use N=N₀e^(−λt); λ simply plays the role that 1/RC played before.

Half-life

The $half-life$ , $T_{1/2}$ , is the time taken for the number of undecayed nuclei (or the activity) to fall to half its original value. Because the decay is exponential, half-life does not depend on how much of the sample is left — it takes exactly as long to halve from a large quantity as from a small one, which is why half-life is a constant, characteristic property of each isotope.

T_{1/2} = \frac{ln 2}{λ}

Half-life from the decay constant.

1Use

T_{1/2} = \frac{ln 2}{λ}

T_{1/2} = \frac{0.693}{1.2 \times 1 0 ^{- 4}}

Answer

T_{1/2} \approx 5780

s (about 96 minutes)

Tip — Taking natural logs of A=A₀e^(−λt) gives ln A = ln A₀ − λt — a straight line of gradient −λ, exactly the method used to find RC for a capacitor. This is the standard experimental way to find λ (and hence half-life) from real decay data.

Equation recap

A = λ N

Activity from decay constant and number of nuclei.

N = N_{0} e^{- λ t}, A = A_{0} e^{- λ t}

Exponential decay of nuclei and activity.

T_{1/2} = \frac{ln 2}{λ}

Half-life from the decay constant.

Common mistakes to avoid

Thinking a specific individual nucleus can be predicted to decay at a known time.

Decay is fundamentally random for any individual nucleus — only the behaviour of a large sample, statistically, is predictable.

Believing half-life changes as a sample decays and gets smaller.

Half-life is a constant property of the isotope, independent of how much sample remains — a direct consequence of the decay being exponential.

Assuming temperature, pressure or chemical state can change the decay constant.

Decay is spontaneous and entirely unaffected by external physical or chemical conditions — λ depends only on the specific isotope.

Key takeaways

Radioactive decay is random (unpredictable for a single nucleus) and spontaneous (unaffected by external conditions).
Activity A = λN; the decay constant λ is the probability of decay per unit time for one nucleus.
Both N and A decay exponentially: N=N₀e^(−λt), A=A₀e^(−λt).
Half-life T½ = ln2/λ is constant for a given isotope, however much sample remains.

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Radioactive Decay and Half-Life

half-life

45 min Video by Science Shorts 3.8.1 Radioactivity