Radioactive Decay Law

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing energy in the form of particles or electromagnetic radiation. The crucial word here is spontaneous: no external trigger causes it, and nothing — not temperature, pressure, nor chemical bonding — can speed it up or slow it down. Each unstable nucleus has a fixed, characteristic probability of decaying per unit time, completely independent of how long it has already existed or what its neighbours are doing. Because every nucleus in a large sample decays independently and at random, the overall behaviour of the sample is beautifully predictable through statistics. When you have billions of atoms, the randomness averages out into a smooth, mathematically precise pattern called exponential decay. Think of it like a large crowd where every person independently flips a coin each second and sits down on heads: even though each individual flip is random, the number of standing people decreases by a predictable fraction every second. The same logic governs radioactive nuclei. The concept of half-life captures this behaviour in an intuitive single number: it is the time required for exactly half of any given sample of a radioactive isotope to decay. Whether you start with a gram or a kilogram of carbon-14, after one half-life (about 5,730 years) exactly half remains. This consistency makes radioactive decay an extraordinarily powerful tool — from dating ancient artefacts and rocks, to designing medical tracers, to engineering nuclear reactors.

Key takeaways
  • The Decay Rate is Proportional to How Many Nuclei Remain
  • Solving the Differential Equation
  • Connecting λ to Half-Life T½
  • Activity: The Measurable Quantity
  • Successive Half-Lives: A Practical Shortcut

The Decay Rate is Proportional to How Many Nuclei Remain

If N(t) is the number of undecayed nuclei at time t, then the rate at which nuclei disappear is proportional to N itself. In calculus notation: dN/dt = −λN, where λ (the decay constant, units s⁻¹) is the probability per unit time that any single nucleus will decay. The minus sign shows the number is decreasing. This simple differential equation is the heartbeat of the entire topic.

Solving the Differential Equation

Separating variables gives dN/N = −λ dt. Integrating both sides: ln(N) − ln(N₀) = −λt, so ln(N/N₀) = −λt. Exponentiating: N(t) = N₀ e^(−λt). This is the Radioactive Decay Law. It tells us that the number of undecayed nuclei decreases exponentially with time, with the decay constant λ controlling how fast the curve drops.

Connecting λ to Half-Life T½

At t = T½ exactly half the nuclei remain, so N(T½) = N₀/2. Substituting: N₀/2 = N₀ e^(−λT½). Dividing by N₀ and taking the natural log of both sides: ln(1/2) = −λT½, giving −ln 2 = −λT½, therefore T½ = ln 2 / λ ≈ 0.693 / λ. A large λ means rapid decay and a short half-life; a small λ means slow decay and a long half-life.

Activity: The Measurable Quantity

We rarely count individual nuclei directly — instead we measure activity A, the number of decay events per second. Since A = λN, and N(t) = N₀ e^(−λt), we get A(t) = A₀ e^(−λt), where A₀ = λN₀ is the initial activity. Activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second. An older unit, the curie (Ci), equals 3.7 × 10¹⁰ Bq. Activity also falls off exponentially with the same half-life as the number of atoms.

Successive Half-Lives: A Practical Shortcut

After n half-lives, the fraction remaining is (1/2)ⁿ. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. After 10 half-lives only about 0.1% remains. This geometric sequence is extremely useful for quick estimates without a calculator. For example, if a medical isotope has a 6-hour half-life, after 24 hours (4 half-lives) only 1/16 ≈ 6.25% of the original activity remains in the patient.

The Key Equation

N(t)=N0 e−λt,T1/2=ln⁡2λN(t) = N_0 \, e^{-\lambda t}, \qquad T_{1/2} = \frac{\ln 2}{\lambda}
  • N(t) — Number of undecayed nuclei (dimensionless (count)): The number of radioactive nuclei that have not yet decayed at time t.
  • N_0 — Initial number of nuclei (dimensionless (count)): The number of radioactive nuclei present at t = 0.
  • \lambda — Decay constant (s⁻¹ (per second)): The probability per unit time that a single nucleus will decay. Each isotope has a unique value.
  • t — Time (s (seconds)): The elapsed time since the start of observation.
  • T_{1/2} — Half-life (s (seconds)): The time for half of the nuclei in a sample to decay. Related to λ by T½ = ln2/λ.

Worked Example

Problem. A freshly prepared sample of iodine-131 (used in thyroid cancer treatment) has an initial activity of 4.0 × 10⁸ Bq. Iodine-131 has a half-life of 8.0 days. (a) Calculate the decay constant λ. (b) Find the activity after 24 days. (c) How many days must pass before the activity falls below 5.0 × 10⁶ Bq?

λ≈0.0866 day−1;A(24)=5.0×107 Bq;t≈50.6 days\lambda \approx 0.0866 \text{ day}^{-1};\quad A(24) = 5.0\times10^7\text{ Bq};\quad t \approx 50.6\text{ days}

Answer. λ ≈ 1.00 × 10⁻⁶ s⁻¹ (or 0.0866 day⁻¹); activity after 24 days = 5.0 × 10⁷ Bq; time to reach 5.0 × 10⁶ Bq ≈ 50.6 days.

Figure
Diagram
Radioactive Decay: N(t) = N₀ e^(−λt)Time (half-lives)N(t) / N₀1.00.50.250.125012345T½: N = N₀/22T½: N = N₀/43T½: N₀/8N(t) = N₀ e^(−λt)T½ = ln2 / λ
Origins · 1 minErnest Rutherford and how this was discovered
ER
Ernest Rutherford
1871–1937 · New Zealand–British

Rutherford, working with Frederick Soddy at McGill University in 1902–1903, first established that radioactive decay involves the transmutation of one element into another and introduced the concept of half-life to describe the rate of decay quantitatively. He identified and named alpha and beta radiation, and later discovered the atomic nucleus. His systematic measurements of decay rates laid the empirical foundation for the exponential decay law. Rutherford won the Nobel Prize in Chemistry in 1908 — not in Physics, which amused him greatly given that he once said all science is either physics or stamp collecting. He is buried in Westminster Abbey near Isaac Newton.

  • Types of Radioactive Decay (α, β, γ) — Understanding what actually happens during a decay event — which particle or radiation is emitted and why — gives physical meaning to the abstract decay constant λ.
  • Nuclear Binding Energy and Stability — The binding energy per nucleon determines whether a nucleus is stable or unstable, explaining why certain isotopes have very large λ (decay quickly) and others have λ ≈ 0 (essentially stable).
  • Radiometric Dating — Carbon-14 dating and uranium-lead dating are direct applications of the decay law, using the known half-life to work backwards from the measured ratio of parent to daughter nuclei to find the age of a sample.
  • Exponential Functions and Natural Logarithms — The decay law N(t) = N₀ e^(−λt) is a pure exponential function; fluency with e^x and ln(x) is essential to solve decay problems algebraically.