Radioactive Half-Life Calculator

About Radioactive Half-Life Calculator

The Exponential Decay Formula

N(t) = N₀ x (1/2)^(t/t½), where N₀ is the initial amount, t is elapsed time, and t½ is the half-life. An equivalent form uses the decay constant lambda: N(t) = N₀ x e^(-lambda t), where lambda = ln(2)/t½ = 0.6931/t½.

Solve For	Formula	Example
Remaining amount	N = N₀ x (1/2)^(t/t½)	100 g with t½ = 8 days, after 24 days: 100 x (1/2)³ = 12.5 g
Time elapsed	t = t½ x log₂(N₀/N)	From 100 g to 6.25 g with t½ = 5 yr: 5 x log₂(16) = 20 years
Half-life	t½ = t / log₂(N₀/N)	80 g decays to 20 g in 10 hr: 10 / log₂(4) = 5 hours
Initial amount	N₀ = N x 2^(t/t½)	25 g remaining after 12 days, t½ = 4 days: 25 x 2³ = 200 g

Worked example: Iodine-131 has a half-life of 8.02 days. A patient receives 100 mCi for thyroid treatment. After 24 days (3 half-lives), remaining activity = 100 x (1/2)³ = 12.5 mCi. After 40 days (about 5 half-lives), remaining = 100 x (1/2)^5 = 3.125 mCi, about 3% of the original dose.

Decay Over Multiple Half-Lives

Half-Lives Elapsed	Fraction Remaining	Percentage Remaining	Percentage Decayed
0	1	100%	0%
1	1/2	50%	50%
2	1/4	25%	75%
3	1/8	12.5%	87.5%
5	1/32	3.125%	96.875%
7	1/128	0.78%	99.22%
10	1/1,024	0.098%	99.9%
20	~1/1,000,000	0.0001%	~100%

A useful rule of thumb: after 10 half-lives, less than 0.1% of the original material remains. This is often used as the threshold for considering a substance "effectively gone" in waste management and medicine.

Common Radioactive Isotopes

Isotope	Half-Life	Primary Use	Decay Type
Fluorine-18	109.77 min	PET scans (medical imaging)	Positron emission
Technetium-99m	6.01 hours	Most common medical imaging isotope	Gamma emission
Iodine-131	8.02 days	Thyroid treatment and imaging	Beta and gamma
Phosphorus-32	14.3 days	DNA labelling in research	Beta emission
Cobalt-60	5.27 years	Radiation therapy, food irradiation	Beta and gamma
Strontium-90	28.8 years	Nuclear fallout concern, RTG power	Beta emission
Cesium-137	30.17 years	Nuclear fallout, industrial gauges	Beta and gamma
Carbon-14	5,730 years	Archaeological dating	Beta emission
Plutonium-239	24,110 years	Nuclear weapons, reactor fuel	Alpha emission
Uranium-235	703.8 million years	Nuclear fission fuel	Alpha emission
Uranium-238	4.47 billion years	Geological dating, depleted uranium	Alpha emission

Radiocarbon Dating with Half-Life

Carbon-14 dating measures the ratio of C-14 to C-12 in organic material. Living organisms maintain a constant C-14/C-12 ratio by absorbing CO₂ from the atmosphere. After death, C-14 decays with a half-life of 5,730 years while C-12 stays constant. By measuring the remaining C-14, archaeologists can date organic material up to about 50,000 years old (roughly 9 half-lives, when C-14 levels drop below reliable detection).

C-14 Remaining	Age (years)	Archaeological Period
100%	0 (just died)	Modern
50%	5,730	Neolithic, early agriculture
25%	11,460	End of last Ice Age
12.5%	17,190	Late Paleolithic cave paintings
1.5%	~38,000	Early modern humans in Europe

Half-Life Beyond Radioactivity

The exponential decay formula applies to any process where a fixed fraction is lost per time period.

Process	Typical Half-Life	Application
Drug metabolism (caffeine)	~5 hours	Pharmacokinetics - how long drugs stay in your system
Drug metabolism (ibuprofen)	~2 hours	Dosing schedules
Capacitor discharge (RC circuit)	0.693 x R x C	Electronics, timing circuits
Beer foam decay	~80 - 120 seconds	Quality testing in brewing
Population decline (some species)	Varies	Conservation biology

How to Derive the Decay Constant

The decay constant lambda links half-life to the more fundamental first-order rate equation dN/dt = -lambda x N. Integrating gives N(t) = N₀ x e^(-lambda t). Setting N(t) = N₀/2 and solving yields lambda = ln(2)/t½ = 0.6931/t½. So an isotope with a 10-year half-life has lambda = 0.06931 per year, meaning about 6.93% of remaining atoms decay each year on average. The two formulas - the (1/2)^(t/t½) form and the e^(-lambda t) form - are algebraically identical, just expressed differently.

Worked example (decay constant): Radon-222 has a half-life of 3.82 days. Its decay constant is lambda = 0.6931 / 3.82 = 0.1815 per day. Starting with 1,000 atoms, after one day roughly 1,000 x e^(-0.1815) = 834 atoms remain. After a week, 1,000 x e^(-1.271) = 280 atoms. This matches the half-life form: 1,000 x (1/2)^(7/3.82) = 280.

Half-Life	Decay Constant lambda	Fraction Remaining After 1 Unit of Time
1 second	0.6931 /s	50.0%
1 hour	0.6931 /hr	50.0%
1 day	0.6931 /day	50.0%
10 years	0.06931 /yr	93.3% (per year)
1,000 years	6.931 x 10^-4 /yr	99.93% (per year)

What Is Activity and How Is It Measured?

Activity is the rate of decay, measured in becquerels (Bq, one decay per second) or curies (Ci, 3.7 x 10^10 Bq). Activity A = lambda x N, so a sample's activity drops with the same exponential curve as its atom count. A 1-gram sample of Radium-226 (half-life 1,600 years) has an activity of about 1 Ci - which is how the curie was historically defined. Modern radiation safety uses becquerels and sieverts (for biological dose). The NRC limit for unrestricted public exposure is 1 millisievert per year above background, per 10 CFR 20.

Medical doses are often reported in millicuries (mCi) or megabecquerels (MBq): 1 mCi = 37 MBq. A typical Technetium-99m cardiac scan injects 20-30 mCi (740-1,110 MBq), which decays through six half-lives (36 hours) to less than 2% of the injected activity.

Pharmacokinetic Half-Life - How Long Do Drugs Stay in Your Body?

Biological half-life follows the same exponential decay formula, but the mechanism is metabolism and excretion rather than nuclear decay. After five half-lives, roughly 97% of a drug has been cleared - the standard pharmacological benchmark for when a medication is considered effectively eliminated. This is why antidepressants with short half-lives need daily dosing while fluoxetine (half-life 4-6 days for parent drug, up to 16 days for active metabolite) can be taken weekly.

Substance	Biological Half-Life	Notes
Caffeine	3-5 hours (healthy adults)	Doubles in late pregnancy, halved in heavy smokers
Alcohol (ethanol)	Zero-order, not exponential	Removed at fixed ~15 mg/dL per hour regardless of concentration
Ibuprofen	~2 hours	Peak plasma at 1-2 hr after oral dose
Paracetamol (acetaminophen)	1.5-3 hours	Prolonged in overdose and liver disease
Nicotine	~2 hours	Cotinine metabolite: 16-20 hours (drug test marker)
THC (cannabis)	~1.3 days (occasional), 5-13 days (chronic use)	Stored in adipose tissue
Warfarin	~40 hours	Steady state takes 5-7 days
Diazepam (Valium)	20-50 hours (parent), 30-100 hours (metabolite)	Accumulates with repeated dosing
Fluoxetine (Prozac)	1-4 days (parent), 7-15 days (norfluoxetine)	Long half-life reduces discontinuation symptoms

The "five half-lives to clearance" rule has real clinical consequences. Warfarin's 40-hour half-life means a dose change won't reach its new steady state for about a week, which is why INR is re-checked 5-7 days after any adjustment. For a quick sense of elimination timing, pair this tool with the BMR calculator (metabolic rate) or the percentage calculator for quick fraction conversions.

What Is Radioactive Equilibrium?

When a radioactive parent decays into a radioactive daughter, the daughter's activity grows until it reaches a balance with the parent. If the parent's half-life is much longer than the daughter's, the daughter reaches secular equilibrium, where its activity equals the parent's after about seven daughter half-lives. This is why a Technetium-99m generator can be "milked" daily: Molybdenum-99 (half-life 66 hours) continuously produces Tc-99m (half-life 6 hours), and the column maintains a roughly constant Tc-99m supply until the parent Mo-99 is depleted. Hospitals typically use a Mo-99/Tc-99m generator for about a week before replacement.

Uranium-238 decay also reaches secular equilibrium with its many daughters. Every gram of natural uranium ore carries about 0.34 micrograms of Radium-226 and a small amount of Radon-222, which is why some basements in granite-rich regions need radon mitigation per EPA guidance (action level 4 pCi/L or higher).

Common Mistakes When Calculating Half-Life

• Confusing half-life with mean lifetime. Mean lifetime tau = 1/lambda = t½/ln(2) = 1.4427 x t½. A particle's "average" lifetime is about 44% longer than its half-life. Physics papers often quote tau; medicine usually quotes t½.
• Using t/t½ as an integer only. The formula works for any real number of half-lives. At 2.7 half-lives, about 15.3% remains - not the 12.5% you'd get by rounding down to three.
• Mixing time units. If half-life is in days but elapsed time is in hours, the exponent must be converted. The calculator above handles this automatically when units match.
• Treating radiocarbon dating as accurate beyond 50,000 years. After 9 half-lives, C-14 falls below reliable atmospheric-ratio detection. For older samples, potassium-argon or uranium-series dating is used instead.
• Assuming zero-order kinetics follow half-life. Alcohol elimination is zero-order (fixed rate per hour), so "half-life" doesn't apply in the exponential sense.
• Ignoring branching decays. Some isotopes decay through multiple channels with different probabilities. Potassium-40, for example, beta-decays 89% of the time and captures an electron 11% of the time. The total half-life accounts for both.

For the energy released during nuclear decay, the E = mc² calculator converts mass defect to energy. For wave properties of radiation emitted during decay, the wavelength calculator finds photon energy from frequency. All calculations run locally in your browser with no data sent anywhere.

Sources

• National Nuclear Data Center (Brookhaven) - NuDat 3 isotope database
• NIST - Radioactivity and half-life reference values
• IAEA - Nuclear data resources
• eCFR - 10 CFR Part 20 Standards for Radiation Protection
• EPA - Radon information and action levels
• Cancer Research UK - Tests and Scans
• FDA - Office of Clinical Pharmacology