Playground
Exponential decay curve N(t) with adjustable half-life. Shows remaining fraction and marks half-lives on the timeline.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Number of nucleioutput Remaining undecayed nuclei at time t | count | 1 | 0 – 1e+24 | |
| Initial number of nuclei Population at t = 0 | count | 1 | 1 – 1e+24 | |
| Decay constant Probability per unit time of decay | 1/s | T⁻¹ | 1e-12 – 1 | |
| Time Elapsed time since t = 0 | s | T | 0 – 1000000000000 |
Deep dive
Derivation
Each nucleus has constant decay probability λ per unit time. For N nuclei, dN/dt = −λN. Separating variables: dN/N = −λ dt → ln N = −λt + C. Applying N(0) = N_0 gives N(t) = N_0 e^{−λt}. Half-life satisfies ½ = e^{−λt₁ᐟ₂} → t₁ᐟ₂ = (ln 2)/λ.
Experimental verification
Rutherford's α-counting experiments confirmed exponential behavior. Modern liquid scintillation counters and HPGe γ-spectroscopy verify the law over dozens of half-lives. Radiocarbon dating (Libby, 1949 Nobel) cross-checks with tree-ring dendrochronology to <50-yr accuracy.
Common misconceptions
- Half-life is NOT the average lifetime — ⟨τ⟩ = 1/λ = t₁ᐟ₂/ln 2 ≈ 1.44 t₁ᐟ₂.
- Decay is genuinely random for each nucleus; only the bulk follows the smooth curve.
- Decay constants are essentially independent of temperature, pressure, and chemistry (electron-capture isotopes show tiny exceptions).
Real-world applications
- Carbon-14 dating of organic materials up to ~50,000 years.
- U-Pb and K-Ar dating of rocks (deep time geology).
- Medical imaging: ⁹⁹ᵐTc (t₁ᐟ₂ = 6 hr) for SPECT, ¹⁸F (110 min) for PET.
- Radiotherapy dosing and nuclear reactor fuel burn-up calculations.
Worked examples
Carbon-14 dating of a 25% sample
Given:
- N/N_0:
- 0.25
- t_half:
- 5730
Find: age t
Solution
0.25 = (1/2)^(t/5730) → t/5730 = 2 → t = 11460 yr
Activity of 1 g of Co-60 (t₁ᐟ₂ = 5.27 yr)
Given:
- m:
- 0.001
- M:
- 0.06
- t_half:
- 5.27
Find: Activity A = λN
Solution
λ = ln 2 / (5.27×3.156×10⁷) ≈ 4.17×10⁻⁹ s⁻¹; N ≈ 1.0×10²² → A ≈ 4.2×10¹³ Bq
Scenarios
What if…
- scenario:
- What if you doubled the sample mass?
- answer:
- N_0 doubles, so the activity at every time doubles, but the half-life and shape of the curve are unchanged.
- scenario:
- What if you cooled radioactive material to 1 K?
- answer:
- Decay rate is essentially unchanged. Nuclear decay depends on nuclear physics, not thermal energy (~meV vs MeV barriers).
- scenario:
- What if a daughter is also radioactive?
- answer:
- You get the Bateman equations: secular equilibrium when t₁ᐟ₂(parent) ≫ t₁ᐟ₂(daughter), so daughter activity matches parent — basis of radiogenerators (e.g., ⁹⁹Mo → ⁹⁹ᵐTc).
Limiting cases
- condition:
- t = 0
- result:
- N = N_0
- explanation:
- Initial population — no decays yet.
- condition:
- t = t₁ᐟ₂
- result:
- N = N_0/2
- explanation:
- Half-life: defines the timescale of the decay process.
- condition:
- t → ∞
- result:
- N → 0
- explanation:
- All nuclei eventually decay, but never literally zero in finite time (statistical asymptote).
- condition:
- λt ≪ 1
- result:
- N ≈ N_0(1 − λt)
- explanation:
- Linear decay regime for small times — useful for activity measurements.
Context
Ernest Rutherford and Frederick Soddy · 1902
Rutherford and Soddy showed radioactive substances transmute exponentially, founding nuclear physics.
Hook
How can we date a 5000-year-old artifact from the carbon inside it?
A sample contains 25% of its original Carbon-14. Given t½ = 5730 yr, how old is it?
Dimensions: [λt] dimensionless: T⁻¹ · T = 1 ✓; both sides of N = N_0 e^{−λt} are pure counts.
Validity: Valid for any large ensemble of identical, statistically independent unstable nuclei. Breaks down for small N (Poisson fluctuations dominate) and for daughter chains where Bateman equations are needed.