Radioactive Half-Life

What half-life means

Take a sample of an unstable isotope with N₀ nuclei at time t = 0. The number remaining at time t obeys

      N(t) = N_0 · e^(−λ·t)

where λ (the decay constant) is the per-nucleus probability of decay per unit time. The half-life t_1/2 is defined by N(t_1/2) = N₀/2. Substituting:

      N_0 / 2 = N_0 · e^(−λ·t_{1/2})
      1/2     = e^(−λ·t_{1/2})
      −ln(2) = −λ · t_{1/2}
      t_{1/2} = ln(2) / λ ≈ 0.693 / λ

Because no concentration appears in the answer, the half-life is the same whether you start with one mole or one nanogram. Each successive half-life cuts the remaining nuclei by another factor of two: after 1 half-life, 50 % remains; after 2, 25 %; after 7, less than 1 %; after 10, less than 0.1 %. A useful rule of thumb: 10 half-lives ≈ 99.9 % decayed.

Decay constant and activity

The decay constant λ measures intrinsic instability. A short-lived isotope (large λ) has a short half-life; a long-lived isotope (small λ) has a long half-life. The activity — decays per unit time, measured in becquerels (Bq = decays/s) or curies (Ci = 3.7 × 10¹⁰ Bq) — is

      A = λ · N

Activity falls off exponentially with the same time constant: A(t) = A₀·e^−λ·t. So an old radium source has the same long half-life as a fresh one but lower activity because fewer nuclei remain.

Worked example: ¹⁴C dating

Carbon-14 has half-life t_1/2 = 5,730 years (the "Cambridge" value used in calibration; older textbooks quote 5,568, the original Libby value). Decay constant:

      λ = ln(2) / t_{1/2} = 0.693 / 5,730 = 1.210 × 10⁻⁴ year⁻¹
        = 1.210 × 10⁻⁴ / (3.156 × 10⁷ s/year) = 3.83 × 10⁻¹² s⁻¹

While alive, an organism maintains atmospheric ratio ¹⁴C/¹²C ≈ 1.3 × 10⁻¹², held steady by exchange with CO₂. After death, intake stops and the ratio decays. Apply the integrated rate law:

      ln(N/N_0) = −λ · t
      t = −ln(N/N_0) / λ

The 1988 ¹⁴C dating of the Shroud of Turin found N/N₀ ≈ 0.928 (averaging Arizona, Oxford, and Zurich labs):

      t = −ln(0.928) / (1.210 × 10⁻⁴) = 0.0747 / (1.210 × 10⁻⁴) = 617 years

Refining for atmospheric ¹⁴C variations and reporting uncertainties, the published age was 1260–1390 CE — medieval. Living-tissue activity is about 0.23 Bq per gram of carbon; samples older than about 50,000 years have so little ¹⁴C left that activity drops below background and dating becomes impractical.

Worked example: ²³⁸U geological dating

Uranium-238 has t_1/2 = 4.468 × 10⁹ years. Decay constant:

      λ = 0.693 / (4.468 × 10⁹) = 1.551 × 10⁻¹⁰ year⁻¹

²³⁸U decays through 14 intermediate steps to stable ²⁰⁶Pb. Once a zircon crystal forms, it locks in U with no Pb (Pb is rejected by the crystal lattice). Over time, ²⁰⁶Pb accumulates as ²³⁸U decays. The age comes from the daughter-to-parent ratio:

      ²⁰⁶Pb / ²³⁸U = e^(λ·t) − 1
      t = (1/λ) · ln(1 + ²⁰⁶Pb/²³⁸U)

For a sample with ²⁰⁶Pb/²³⁸U = 0.30:

      t = (1 / 1.551 × 10⁻¹⁰) · ln(1.30) = 6.45 × 10⁹ × 0.2624 = 1.69 × 10⁹ years

About 1.7 billion years old — typical of Precambrian shield rocks. The same method, refined with ²³⁵U → ²⁰⁷Pb (t_1/2 = 7.04 × 10⁸ years) as a cross-check, dated the oldest known terrestrial mineral (a Western Australian zircon) to 4.404 ± 0.008 billion years.

Half-lives across the chart of nuclides

Isotope	Half-life	Decay constant λ	Decay mode	Use / significance	Activity of 1 g
²¹²Po	0.299 µs	2.32 × 10⁶ s⁻¹	α	Among the shortest α emitters; standard in α spectroscopy	~6.6 × 10²² Bq (vanishes instantly)
¹³¹I	8.02 days	1.00 × 10⁻⁶ s⁻¹	β⁻ + γ	Thyroid imaging and ablation; Chernobyl plume marker	4.6 × 10¹⁵ Bq
⁹⁹ᵐTc	6.01 hours	3.20 × 10⁻⁵ s⁻¹	γ (isomeric transition)	Workhorse of nuclear medicine; ~30 million scans/year	1.95 × 10¹⁷ Bq
⁶⁰Co	5.27 years	4.17 × 10⁻⁹ s⁻¹	β⁻ + γ	External-beam radiotherapy; sterilization of medical equipment	4.18 × 10¹³ Bq
¹³⁷Cs	30.07 years	7.31 × 10⁻¹⁰ s⁻¹	β⁻ + γ (via ¹³⁷ᵐBa)	Long-term Chernobyl/Fukushima fallout marker; food irradiation	3.21 × 10¹² Bq
⁹⁰Sr	28.79 years	7.63 × 10⁻¹⁰ s⁻¹	β⁻	Bone-seeker fallout from atmospheric bomb tests; RTG fuel	5.11 × 10¹² Bq
²³⁹Pu	24,110 years	9.11 × 10⁻¹³ s⁻¹	α	Weapons-grade fissile; reactor by-product; Pu-240 contaminates above 7 %	2.30 × 10⁹ Bq
¹⁴C	5,730 years	3.83 × 10⁻¹² s⁻¹	β⁻	Radiocarbon dating, atmospheric tracer, biomedical labelling	1.65 × 10¹¹ Bq
²³⁵U	7.04 × 10⁸ years	3.12 × 10⁻¹⁷ s⁻¹	α	Fissile in Hiroshima Little Boy; LWR fuel at 3–5 % enrichment	8.00 × 10⁴ Bq
²³⁸U	4.468 × 10⁹ years	4.92 × 10⁻¹⁸ s⁻¹	α	Most abundant U isotope; geochronology of zircons; depleted U penetrators	1.24 × 10⁴ Bq

Decay curve and successive half-lives

  N/N_0
  1.0 ●
       \\
   0.5  \\●            ←  1 half-life: 50 %
         \\ \\
   0.25   \\ ●          ← 2 half-lives: 25 %
           \\  \\
   0.125    \\  ●        ← 3 half-lives: 12.5 %
              \\\\
   0.0625      \\● ← 4 half-lives: 6.25 %
                \\
                 \\●●●●●●●●●●●●● (10 half-lives ≈ 0.1 % left)
              ──────────────────────► t (in units of t_{1/2})
              0   1   2   3   4   5   6   7   8

Effective half-life in biology and medicine

When a radionuclide is administered to a patient or released into the environment, two clocks run in parallel: physical decay and biological clearance. They combine like resistors in parallel:

      1/t_eff = 1/t_phys + 1/t_bio

For ¹³¹I in the thyroid: t_phys = 8.02 days, t_bio ≈ 80 days. Then 1/t_eff = 1/8.02 + 1/80 = 0.1372 days⁻¹, giving t_eff ≈ 7.3 days. For ⁹⁹ᵐTc: t_phys = 6 h, t_bio ≈ 24 h, so t_eff ≈ 4.8 h. The short effective half-life is what makes Tc-99m the most-used isotope in nuclear medicine — image quickly while patient dose stays low.

Branching ratios and parallel decay paths

Some nuclei have multiple decay channels. Potassium-40 decays in three ways: 89.3 % β⁻ to ⁴⁰Ca, 10.7 % electron capture to ⁴⁰Ar, and tiny β⁺. The total decay constant is the sum: λ_total = λ_β + λ_EC + λ_β+. Half-life t_1/2 = ln(2)/λ_total = 1.248 × 10⁹ years. The K–Ar dating method exploits the EC branch alone: knowing the 10.7 % branching fraction, the daughter Ar-40 accumulation tells age. This is how lunar samples and the Tutankhamun-era Egyptian potsherds get dated.

Real-world applications

• Radiocarbon dating. Living tissue ¹⁴C/¹²C ≈ 1.3 × 10⁻¹². Decay clock runs after death; ages 50,000 years and younger are accessible. Calibration with tree-ring data accounts for atmospheric ¹⁴C fluctuations.
• Chernobyl exclusion zone (1986). Initial radiation dominated by ¹³¹I (gone in months) and ¹³⁴Cs (t_1/2 = 2.06 y, gone in decades). Today's residual activity is dominated by ¹³⁷Cs (t_1/2 = 30.07 y), with ⁹⁰Sr also persisting. Computed: 40 years post-disaster, ¹³⁷Cs is at 0.5⁴⁰⁄³⁰·⁰⁷ ≈ 40 % of 1986 levels — still substantial.
• Pacemaker batteries (1960s–80s). Plutonium-238 (t_1/2 = 87.7 years) thermoelectric batteries powered some implanted pacemakers. Multi-decade lifetime; replaced by lithium chemistries for regulatory rather than performance reasons.
• Voyager RTG. ²³⁸Pu also powers Voyager 1 and 2's radioisotope thermoelectric generators. Output dropped from 470 W (1977 launch) to about 230 W in 2026 — half-life predicts steady decline in instrument availability.
• Smoke detectors. Americium-241 (t_1/2 = 432.2 years) emits α particles that ionize a small air gap; smoke disrupts the ionization current and triggers the alarm. Half-life is long enough that detector sensitivity stays constant for decades.
• Banana radiation. Bananas contain about 422 mg potassium per 100 g, of which 0.0117 % is ⁴⁰K (t_1/2 = 1.248 × 10⁹ y). A banana has roughly 15 Bq activity — the famous "banana equivalent dose" used as an intuitive comparison for everyday radiation.

Why temperature and chemistry don't change half-life

Radioactive decay is governed by the strong and weak nuclear forces operating inside the nucleus, where the relevant energy scale is MeV. Chemical bonds, by contrast, involve electron rearrangements at the eV scale — six orders of magnitude smaller. So changing molecular environment, applying pressure, or heating the sample changes the chemistry but not the nuclear physics. The only known exception is electron-capture decay, where the rate depends weakly on electron density at the nucleus: ⁷Be in metallic Be vs ⁷Be in BeF₂ shows about 0.8 % half-life difference. Otherwise, t_1/2 is one of the most invariant numbers in physics.

Common mistakes

• Confusing half-life with mean lifetime. Mean lifetime τ = 1/λ; half-life t_1/2 = ln(2)/λ = τ·ln(2). The mean lifetime is what appears in the exponent (e^−t/τ = e^−λt), but the half-life is what's tabulated. Don't mix them in calculations.
• Assuming half-life changes with temperature. It doesn't, except for the marginal electron-capture cases noted above. Burning a wood sample to ash before ¹⁴C dating doesn't reset its clock.
• Forgetting the units of λ. λ has units of inverse time. When you compute t_1/2 = 0.693/λ, the answer carries units of whatever time unit you used for λ. Mixing seconds with years gives 10-billion-year errors.
• Linear extrapolation past 10 half-lives. "It's mostly gone" is true at 10 half-lives (0.1 %), but storage-of-spent-nuclear-fuel calculations care about 0.001 % differences. Use the exponential form, not a "10 half-lives done" rule of thumb.
• Confusing physical and effective half-life. A ⁹⁹ᵐTc patient is not radioactively safe after 6 hours — they are after about 4 effective half-lives ≈ 19 hours. Wash-out and physical decay both contribute.
• Treating short-lived daughters as static. A ²³⁸U sample contains all 14 decay-chain daughters at secular equilibrium, each with their own activity. Don't compute total activity from the parent alone.