Everything about half-life, activity and decay constants flows from one simple idea: the more undecayed nuclei you have, the faster the decays happen. That single statement is the radioactive decay law, and it produces the smooth exponential curve that never quite reaches zero. Once you can wield N = N0e−λt and its cousins for activity and count rate, you can solve any decay problem — from dating a rock to powering a space probe.
📚 What you need to know
The number of undecayed nuclei falls exponentially and never quite reaches zero
N = N0e−λt, where N0 is the number at t = 0
The same law applies to activity and count rate: A = A0e−λt and C = C0e−λt
A steeper curve means a larger decay constant λ
All decay curves start on the y-axis at the initial value (N0, A0 or C0)
The symbol e ≈ 2.718; its inverse is the natural log, ln
What the law says
In radioactive decay, the number of undecayed nuclei drops very fast at first, then more and more slowly, tailing off towards — but never reaching — zero. This shape is called exponential decay, and it comes straight from A = λN: when there are many nuclei, there are many decays per second, so the number plummets; as nuclei run low, decays become rare, so the fall slows.
Both curves start at N0 on the y-axis. The steeper red curve has the larger decay constant and decays faster; the blue one is slower. Neither ever reaches zero.
The key features examiners look for: every decay curve starts on the y-axis at the initial value, and a steeper slope means a bigger λ. If a question shows two isotopes on one graph, the one that drops faster is the more radioactive — higher decay constant, shorter half-life.
The decay equations
The law is written most often for the number of nuclei, but because activity and count rate are both proportional to N, they follow the identical exponential form. Same equation, different symbol:
The radioactive decay lawN = N0e−λtA = A0e−λtC = C0e−λtN nuclei • A activity (Bq) • C count rate (cpm) • subscript 0 = initial value
You don’t have to memorise three separate equations — it’s one law wearing three hats. Whatever quantity you’re given (nuclei, activity, or count rate), it decays the same way. Just pop the right symbol into X = X0e−λt and solve. And remember: on your calculator, ex and ln are inverse buttons — to undo e you take ln, and vice versa.
WE 1
Strontium-90 has a decay constant of 0.025 year−1. Determine the activity of a sample after 5.0 years, as a fraction of its initial activity A0.
Step 1 — write the activity form of the lawA = A₀e⁻λᵗStep 2 — rearrange for the ratioA/A₀ = e⁻λᵗ = e⁻⁽⁰․₀₂₅ × ⁵₋Step 3 — evaluateA/A₀ = e⁻⁰․¹²₅ = 0.88A/A₀ = 0.88 (activity falls by ~12%)Because λ and t are both in years, no conversion is needed. Dividing by A₀ cancels the initial value, leaving a clean fraction. 5 years isn’t a whole half-life, so only the exponential law gives the exact answer.
A worked application: a space probe
The decay law shows up in real engineering. A space probe uses a radioactive source for power — as the source decays, its power output falls exponentially, so we can predict exactly how long the probe will keep running.
WE 2
A probe carries 4.0 kg of plutonium-238 (molar mass 238 g mol−1, half-life 87.7 years). Each alpha decay releases 5.5 MeV, converted to electricity at 32% efficiency. The probe needs at least 0.4 kW. Estimate how long the source can power it. (NA = 6.02 × 1023, 1 eV = 1.6 × 10−19 J)
Step 1 — number of nucleiN₀ = (4000 × 6.02×10²³) ÷ 238 = 1.01 × 10²⁵Step 2 — initial activity (λ = ln2/t½)A₀ = λN₀ = 2.54 × 10¹⁵ BqStep 3 — initial power (energy per decay × activity)E = 5.5×10⁶ × 1.6×10⁻¹⁹ = 8.8 × 10⁻¹³ JP₀ = A₀E × 0.32 = ~704 WStep 4 — solve P = P₀e⁻λᵗ for t at P = 400 Wt = −(t½/ln2) × ln(400/704)t ≈ 71.5 yearsPower tracks activity, so it obeys the same exponential law. Build up N₀ → A₀ → P₀, then solve backwards for the time when power drops to the 400 W limit. The ln undoes the e.
⚛ Working a decay law question
Pick the right quantity:N, A or C — all use X = X0e−λt.
Need λ first? Get it from λ = ln 2 / t½.
Finding a fraction? Divide by the initial value to cancel it.
Solving for time? Take ln of both sides to bring t down from the exponent.
Match units of λ and t throughout.
💡 Top tips
It’s one law for nuclei, activity and count rate — same exponential shape.
To solve for time, take ln of both sides.
Divide by the initial value to get a clean fraction.
Curves always start on the y-axis at the initial value.
Steeper = larger λ = faster decay.
⚠ Common mistakes
Dropping the minus sign in e−λt
Forgetting to find λ from the half-life first
Mismatched units for λ and t
Trying to solve for twithout taking ln
Thinking the curve reaches zero — it only approaches it
Reading a steeper curve as a smaller decay constant
Quick recap: The radioactive decay law says the number of undecayed nuclei falls exponentially: N = N0e−λt, with the same form for activity and count rate. Curves start at the initial value on the y-axis, approach zero without reaching it, and a steeper curve means a larger λ. To solve for time, take ln of both sides; to get a fraction, divide by the initial value.
That completes the core of radioactivity: what isotopes are, why they decay, what they emit, how to balance the equations, and how to predict the amounts over time. From here, the topic branches into applications (carbon dating, medical tracers, smoke detectors) and the deeper nuclear ideas of mass defect and binding energy — where the energy released by decay ultimately comes from.
The decay law equations still a blur?
Book a free meeting and we’ll drill N = N₀e⁻λᵗ, solving for time with ln, and reading decay curves confidently.
