Half-Life Calculator
Half-life (t₁/₂) is the time required for half of the radioactive atoms in a sample to undergo decay. It's a fundamental concept in nuclear chemistry and first-order kinetics that describes exponential decay processes. Half-life is constant for each radioactive isotope, regardless of the initial amount.
Half-life determines radioactive hazard duration, enables radiometric dating, guides medical treatments, informs nuclear waste storage, and helps understand chemical reaction rates. It's crucial for safety, medicine, archaeology, and energy production.
Key half-life concepts:
- Exponential decay: Constant fractional decay per time unit
- First-order kinetics: Rate ∝ current amount
- Statistical process: Cannot predict individual atom decay
- Temperature independence: Nuclear decay unaffected by conditions
- Mean lifetime: τ = 1/λ = t₁/₂/ln(2) ≈ 1.443 × t₁/₂
This calculator solves for any variable in the half-life equation when you know the others:
- Find Remaining: Enter initial amount, half-life, and time → Get remaining amount
- Find Time: Enter initial, remaining, and half-life → Get elapsed time
- Find Half-Life: Enter initial, remaining, and time → Get half-life
The calculator provides:
- Complete decay parameters: Remaining amount, fraction, half-lives passed
- Decay constant (λ): Probability of decay per unit time
- Visual decay progression: Graphical representation of decay
- Common isotope presets: Reference half-lives for quick calculations
- Multiple unit support: Years, days, hours, seconds, grams, moles, atoms
- First-order kinetics confirmation: Verifies exponential decay pattern
Half-lives of important radioactive isotopes used in science, medicine, and industry:
| Isotope | Half-Life | Decay Mode | Applications | Notes |
|---|---|---|---|---|
| Carbon-14 | 5,730 years | β⁻ | Radiocarbon dating | Forms in atmosphere, used for organic materials ≤ 50k years |
| Uranium-238 | 4.47×10⁹ years | α | Nuclear fuel, dating | Primordial isotope, decays to Pb-206, Earth age dating |
| Iodine-131 | 8.02 days | β⁻ | Medical therapy | Thyroid treatment, nuclear medicine |
| Cobalt-60 | 5.27 years | β⁻ | Cancer therapy | Gamma source for radiotherapy, sterilization |
| Radium-226 | 1,600 years | α | Historical luminescence | Once used in glow-in-dark paints, dangerous |
| Technetium-99m | 6.01 hours | γ | Medical imaging | Most common medical radioisotope, low radiation |
| Potassium-40 | 1.25×10⁹ years | β⁻/EC | Geological dating | Natural in bananas, human body, decays to Ar-40 |
| Tritium (H-3) | 12.32 years | β⁻ | Luminescence, fusion | Self-powered lighting, hydrogen bomb component |
| Plutonium-239 | 24,100 years | α | Nuclear weapons | Fissile material, long-term waste concern |
| Americium-241 | 432.2 years | α | Smoke detectors | Ionization source in household detectors |
Very short (<1 hour): Medical diagnostics, rapid decay minimizes exposure
Short (hours-days): Medical therapy, industrial tracers
Medium (years-centuries): Industrial sources, long-term studies
Long (millions-billions years): Geological dating, nuclear fuel
Stable (infinite): Non-radioactive, no decay
Below are answers to frequently asked questions about half-life calculations:
Use the exponential decay formula: N = N₀ × (1/2)^(t/t₁/₂)
Initial: 100 g, Half-life: 10 years, Time: 15 years
Half-lives passed: 15/10 = 1.5 half-lives
Remaining = 100 × (1/2)^(1.5) = 100 × (0.5)^(1.5)
(0.5)^(1.5) = √(0.5³) = √(0.125) = 0.3536
Remaining = 100 × 0.3536 = 35.36 g
Alternative: Use natural logarithm: N = N₀ × e^(-λt) where λ = ln(2)/t₁/₂. Our calculator handles all cases automatically.
Decay constant (λ) and half-life (t₁/₂) are inversely related through natural logarithm of 2:
t₁/₂ = ln(2)/λ ≈ 0.693147/λ
λ = ln(2)/t₁/₂ ≈ 0.693147/t₁/₂
Mean lifetime (τ) = 1/λ = t₁/₂/ln(2) ≈ 1.4427 × t₁/₂
Exponential decay: N = N₀ × e^(-λt) = N₀ × (1/2)^(t/t₁/₂)
Physical meaning: λ = probability of decay per unit time (units: time⁻¹). Larger λ = faster decay = shorter half-life.
Carbon-14 dating compares C-14/C-12 ratio in sample to atmospheric ratio:
| Step | Process | Calculation |
|---|---|---|
| 1. Formation | Cosmic rays create C-14 in atmosphere | C-14 + O₂ → ¹⁴CO₂ (enters carbon cycle) |
| 2. Uptake | Plants absorb CO₂, animals eat plants | Living organisms maintain atmospheric C-14/C-12 ratio |
| 3. Death | Organism stops exchanging carbon | C-14 decays, C-12 stable (ratio decreases) |
| 4. Measurement | Measure current C-14/C-12 ratio | Use accelerator mass spectrometry |
| 5. Calculation | Compare to atmospheric ratio | t = [ln(R₀/R)/ln(2)] × 5730 years |
Limitations: Maximum ~50,000 years (after ~10 half-lives, too little C-14 remains). Calibration needed due to historical atmospheric variations.
Half-life concepts apply to drug elimination and radioactive tracers:
- Drug dosing: Biological half-life determines dosing frequency
- Steady state: After ~5 half-lives, drug reaches steady concentration
- Elimination: After 4-5 half-lives, drug essentially eliminated
- Medical imaging: Tc-99m (6h half-life) ideal for scans, minimizes exposure
- Radiotherapy: I-131 (8d) treats thyroid cancer, decays in body
- Diagnostics: Short-lived isotopes for PET/CT scans
Example: Drug with 6-hour half-life: Given 100mg dose → 50mg after 6h → 25mg after 12h → 12.5mg after 18h → ~3mg after 24h (almost eliminated).
Radioactive decay is a quantum mechanical process with constant probability per unit time:
- Quantum randomness: Cannot predict which atom decays when
- Constant probability: Each atom has same decay probability per time
- Statistical law: Large numbers → predictable exponential decay
- Nuclear stability: Half-life depends on nuclear binding energy
- Environmental independence: Temperature, pressure, chemistry don't affect nuclear decay rates
- Parent-daughter: Decay product may also be radioactive (decay chains)
Exception: Electron capture decay rates can be slightly affected by chemical environment (bound electron density). Most decays are completely constant.
Many radioactive isotopes decay through series of daughter products:
| Decay Chain | Example Sequence | Half-Lives | Equilibrium Condition |
|---|---|---|---|
| Uranium-238 | ²³⁸U → ²³⁴Th → ²³⁴Pa → ²³⁴U → ... → ²⁰⁶Pb | 4.47B yr → 24d → 1.2m → 245ky → ... → stable | After ~10× longest daughter half-life |
| Thorium-232 | ²³²Th → ²²⁸Ra → ²²⁸Ac → ... → ²⁰⁸Pb | 14B yr → 5.75 yr → 6.15h → ... → stable | Different equilibrium types |
| Uranium-235 | ²³⁵U → ²³¹Th → ... → ²⁰⁷Pb | 704M yr → 25.5h → ... → stable | Complex multi-step chains |
Secular equilibrium: When parent half-life >> daughter half-life, daughter activity equals parent activity after enough time. Example: ²²⁶Ra (1600yr) → ²²²Rn (3.8d). After ~40 days, Rn production rate = decay rate.