Half-Life Calculator Radioactive Decay • Drug Elimination • Carbon-14 Dating — Step-by-Step
The only free half-life calculator that serves both audiences — physics/chemistry and pharmacology — in one clean tool. Pharmacology mode: enter dose, half-life, and time elapsed to see remaining amount, percentage cleared, and a 10-row elimination table. Choose from 60 drug presets (caffeine, warfarin, fluoxetine, diazepam…) for instant calculation. Steady-state timeline shown for every result. Physics mode: solve for any of the four variables — N₀, N, t, or t½ — with the decay constant and mean lifetime. 25 radioisotope presets (C-14, I-131, Tc-99m, U-238…). Carbon-14 dating calculator — enter % C-14 remaining, get sample age. Effective half-life for nuclear medicine (physical + biological). Full step-by-step working for every calculation. All browser-side, no account required.
Radioactive Decay & Drug Elimination — Two Modes, One Tool
Steady state is reached after 4–5 half-lives of regular dosing. At steady state, drug intake equals elimination.
Half-life quick reference
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Half-Life Definition, Formula & Equation — Physics and Pharmacology
Half-life (t½) is the time required for any quantity to reduce to exactly half its initial value. It is a constant characteristic of the substance regardless of how much you start with — whether you have 1 gram or 1 tonne of a radioactive isotope, the fraction that remains after one half-life is always 50%. This property of first-order kinetics is what makes the half-life concept so powerful: it describes exponential decay with a single, simple number.
Half-life applies to two very different but mathematically identical phenomena that often confuse students: radioactive decay (nuclear physics) and drug elimination (pharmacokinetics). The equations are the same; only the physical interpretation differs.
The half-life formula — all four variants
N(t) = N₀ × e^(-λ × t) [exponential decay constant]
Solve for each variable:
Remaining: N = N₀ × (1/2)^(t/t½)
Time: t = t½ × log₂(N₀/N)
Half-life: t½ = t × ln(2) / ln(N₀/N)
Initial: N₀ = N / (1/2)^(t/t½)
Related constants: λ = ln(2)/t½ τ = t½/ln(2) λτ = 1
How to calculate half-life — step by step
Step 1: ratio = N₀/N = 80/10 = 8
Step 2: ln(8) = 2.0794
Step 3: t½ = t × ln(2) / ln(N₀/N) = 24 × 0.6931 / 2.0794
Step 4: t½ = 16.635 / 2.0794 = 8.0 hours
Verify: after 3 half-lives (24 h): 80 × (0.5)^3 = 80 × 0.125 = 10 g ✓
Decay table — what percentage remains after n half-lives?
| Half-lives | Fraction remaining | % remaining | % eliminated | Clinical meaning |
|---|---|---|---|---|
| 1 | 1/2 | 50% | 50% | Half the substance present |
| 2 | 1/4 | 25% | 75% | Three-quarters eliminated |
| 3 | 1/8 | 12.5% | 87.5% | ~87% cleared |
| 4 | 1/16 | 6.25% | 93.75% | Nearly 94% cleared |
| 5 | 1/32 | 3.125% | 96.875% | Clinical clearance rule |
| 6 | 1/64 | 1.563% | 98.44% | ~99% cleared |
| 7 | 1/128 | 0.781% | 99.22% | Near-complete clearance |
| 10 | 1/1024 | 0.098% | 99.90% | Practically none remaining |
Drug Half-Life Calculator — How Long Does a Drug Stay in Your System?
In pharmacokinetics, the elimination half-life (t½) is the time required for the plasma concentration of a drug to decrease by 50%. It determines dosing frequency, time to steady state, and time to clearance. Most drugs follow first-order kinetics — the same exponential decay equation as radioactive isotopes.
The 5 half-lives rule — when is a drug fully cleared?
A drug is considered effectively eliminated after 4–5 half-lives (94–97% cleared). This is a pharmacokinetic convention, not a biological absolute — traces technically remain indefinitely, but concentrations fall below clinical relevance. Time to clearance = 4.32 to 5 × t½.
Drug half-life reference table
| Drug | Half-life | Time to clear (5 t½) | Category |
|---|---|---|---|
| Aspirin | 20 minutes | ~1.7 hours | Analgesic |
| Adrenaline (epinephrine) | 2–3 minutes | 10–15 minutes | Cardiac emergency |
| Penicillin G | 30 minutes | ~2.5 hours | Antibiotic |
| Ibuprofen | 2 hours | ~10 hours | Analgesic/NSAID |
| Paracetamol (acetaminophen) | 2 hours | ~10 hours | Analgesic |
| Morphine | 2–3 hours | 10–15 hours | Opioid analgesic |
| Caffeine | 5 hours | ~25 hours | Stimulant |
| Lorazepam (Ativan) | 12 hours | ~2.5 days | Benzodiazepine |
| Lisinopril (ACE inhibitor) | 12 hours | ~2.5 days | Antihypertensive |
| Warfarin | 40 hours | ~8 days | Anticoagulant |
| Diazepam (Valium) | 50 hours | ~10 days | Benzodiazepine |
| Fluoxetine (Prozac) | 4 days | ~20 days | SSRI antidepressant |
| Levothyroxine (T4) | 7 days | ~35 days | Thyroid hormone |
| Aripiprazole | 75 hours | ~16 days | Antipsychotic |
| Amiodarone | 65 days | ~325 days | Antiarrhythmic |
What is steady state in pharmacokinetics?
When a drug is taken at regular intervals, concentrations accumulate until the amount absorbed per dose equals the amount eliminated per dose — this is steady state. It is reached after approximately 4–5 half-lives. At steady state: the plasma concentration oscillates between a predictable peak (C_max) and trough (C_min). Steady-state average concentration = (dose / dosing interval) / clearance.
How to calculate drug clearance time
Time to 94% elimination (4.32 t½): clinical clearance
Time to 97% elimination (5 t½): conservative clearance
Time to 99% elimination (6.64 t½): thorough clearance
Formula: t_clear = n × t½, where n = desired half-lives
Example: Diazepam (t½ = 50 h) clears in: 5 × 50 = 250 hours = ~10.4 days
Radioactive Decay Calculator — Half-Life in Physics
In nuclear physics, the half-life is an intrinsic property of each radioisotope — it is determined by the forces inside the atomic nucleus and cannot be changed by temperature, pressure, chemical bonding, or any external condition. Each unstable nucleus has a fixed probability of decaying per unit time, resulting in exponential population decline.
Decay constant, mean lifetime, and half-life — the three equivalent descriptions
Decay constant λ: probability of decay per unit time
Mean lifetime τ: average lifetime of one nucleus
Relationships:
λ = ln(2) / t½ = 0.6931 / t½
τ = 1 / λ = t½ / ln(2) = 1.4427 × t½
t½ = ln(2) / λ = τ × ln(2) = 0.6931 × τ
Radioisotope half-life reference table
| Isotope | Half-life | Application |
|---|---|---|
| Fluorine-18 (F-18) | 110 minutes | PET scanning |
| Technetium-99m (Tc-99m) | 6.01 hours | Medical imaging (most common) |
| Iodine-131 (I-131) | 8.02 days | Thyroid cancer treatment |
| Cobalt-60 (Co-60) | 5.27 years | Radiation therapy, industrial |
| Cesium-137 (Cs-137) | 30.17 years | Nuclear fallout, food irradiation |
| Strontium-90 (Sr-90) | 28.8 years | Nuclear fallout, RTGs |
| Tritium (H-3) | 12.32 years | Nuclear weapons, luminescent devices |
| Carbon-14 (C-14) | 5,730 years | Radiocarbon dating |
| Plutonium-239 (Pu-239) | 24,110 years | Nuclear weapons, reactors |
| Uranium-235 (U-235) | 703 million years | Nuclear fuel, geology |
| Uranium-238 (U-238) | 4.468 billion years | Earth age dating |
| Potassium-40 (K-40) | 1.248 billion years | Geological dating, body radioactivity |
Carbon-14 dating — how to calculate the age of a sample
Carbon-14 (t½ = 5,730 years) is continuously produced in the atmosphere and absorbed by living organisms. When an organism dies, C-14 intake stops and existing C-14 decays. By measuring the fraction of C-14 remaining, the age can be determined:
age = 5730 × log₂(100% / % remaining)
Examples:
50% remaining ⟹ 5,730 years
25% remaining ⟹ 11,460 years (2 half-lives)
12.5% remaining ⟹ 17,190 years (3 half-lives)
1% remaining ⟹ 38,069 years
Limit of dating: ~50,000–60,000 years (too little C-14 remains)
Effective half-life in nuclear medicine
When a radioactive substance is used in the body (e.g., Iodine-131 for thyroid therapy), two separate decay processes occur simultaneously: the physical decay of the isotope, and the biological elimination by kidneys, liver, and sweat. The effective half-life is always shorter than both:
t_eff = (t_phys × t_biol) / (t_phys + t_biol)
Example: I-131 in thyroid (t_phys = 8 days, t_biol = 120 days):
t_eff = (8 × 120) / (8 + 120) = 960/128 = 7.5 days
Example: Tc-99m (t_phys = 6.01 h, t_biol = 24 h):
t_eff = (6.01 × 24) / (6.01 + 24) = 4.81 hours
LazyTools vs Other Half-Life Calculators
| Feature | LazyTools | Omni Calculator | Calculator.net | GraphCalc |
|---|---|---|---|---|
| Physics + Pharmacology in one tool | ✅ Both modes | ❌ Separate tools | ⚠ Physics only | ⚠ Basic only |
| Drug presets (60 medications) | ✅ 60 drugs | ⚠ Small table | ❌ None | ❌ None |
| Radioisotope presets (25 isotopes) | ✅ 25 isotopes | ❌ None | ❌ None | ❌ None |
| Carbon-14 dating calculator | ✅ Built-in | ⚠ Separate tool | ✅ Yes | ❌ None |
| Effective half-life (nuclear medicine) | ✅ Built-in | ❌ None | ❌ None | ❌ None |
| Steady-state timeline | ✅ Auto-shown | ⚠ Separate | ❌ None | ❌ None |
| 10-row decay table | ✅ Auto-generated | ✅ Yes | ❌ No | ❌ No |
| Step-by-step working | ✅ Every result | ❌ No | ❌ No | ❌ No |
| No account required | ✅ Yes | ✅ Yes | ✅ Yes | ✅ Yes |
Half-Life Calculator FAQ
N(t) = N₀ × (1/2)^(t/t½). Four variants: Remaining N = N₀(1/2)^(t/t½). Time t = t½ × log₂(N₀/N). Half-life t½ = t × ln2/ln(N₀/N). Decay constant λ = ln2/t½.
Given initial amount N₀, remaining amount N, and time t: t½ = t × ln(2) / ln(N₀/N). Example: 80g decays to 10g in 24 hours. t½ = 24 × 0.6931 / ln(8) = 8 hours.
N = N₀ × (0.5)^n. After 1 half-life: 50%. After 3: 12.5%. After 5: 3.125%. Example: 200 mg, half-life 6h, after 18h (3 half-lives): 200 × 0.125 = 25 mg.
Aspirin: 20 min. Ibuprofen: 2h. Caffeine: 5h. Morphine: 2–3h. Warfarin: 40h. Diazepam: 50h. Fluoxetine: 4 days. Amiodarone: 65 days. Select any from the 60-drug preset above.
C-14 has t½ = 5,730 years. Measure % remaining. Age = 5730 × log₂(1/fraction). 25% remaining = 11,460 years. Use the C-14 Dating Calculator in Physics mode above.
4–5 half-lives clears 94–97% of a drug. Ibuprofen (t½=2h): ~10h. Caffeine (5h): ~25h. Diazepam (50h): ~10 days. Fluoxetine (96h): ~20 days. The calculator above shows exact clearance timelines.
N(t) = N₀ × e^(-λt) where λ = ln2/t½. Also: N(t) = N₀ × (1/2)^(t/t½). Decay constant λ = ln2/t½. Mean lifetime τ = t½/ln2. All three constants describe the same decay process.
Steady state = drug intake equals elimination rate. Reached after ~4.32–5 half-lives of regular dosing. 94% SS at 4.32 t½. 97% SS at 5 t½. A drug with t½=12h reaches SS after ~52–60 hours of regular dosing.
λ (lambda) = ln(2) / t½ = 0.6931 / t½. Probability of decay per unit time. Larger λ = faster decay. Mean lifetime τ = 1/λ = t½/ln2 = 1.4427 × t½.
Switch to Physics mode. Enter any 3 of: N₀, N, t, t½ — the 4th is solved. Shows λ and τ. Decay table auto-generated. 25 radioisotope presets including C-14, I-131, Cs-137, U-238.
Combines physical and biological decay: 1/t_eff = 1/t_phys + 1/t_biol. t_eff = (t_phys × t_biol)/(t_phys + t_biol). Always shorter than both. I-131 example: t_eff = (8×120)/(8+120) = 7.5 days.
t = t½ × log₂(N₀/N). Example: 100g to 12.5g (t½=6h): log₂(8)=3. t=6×3=18h. In Physics mode: enter N₀=100, N=12.5, t½=6, select solve-for=t.
3.125% remaining, 96.875% eliminated. In pharmacology: clinical drug clearance. After 10 half-lives: 0.098%. The decay table above shows each half-life step.
~5 hours in healthy adults (range 3–7h). 25 hours to clear 97%. A morning coffee at 8 AM: ~6% remains at midnight. Extended by pregnancy (15–33h), liver disease, oral contraceptives. Shortened by smoking (~3h). Select Caffeine from the drug preset above.
Two modes — Pharmacology (drug presets, clearance timeline, steady-state) and Physics (4 formula variants, decay constant, C-14 dating, effective t½). Full step-by-step working. 10-row decay table. Free, no account.
Medical imaging: Tc-99m (6h) for SPECT scans. Cancer treatment: I-131 (8 days) for thyroid cancer. Nuclear dating: C-14 (5730y) for archaeology, U-238 (4.5 Gy) for geology. Nuclear safety: Cs-137 (30y) in fallout assessment. Industrial: Co-60 (5.3y) for radiation therapy equipment.