Half-Life Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 19, 2026Decay That Never Quite Reaches Zero
Radioactive decay, drug elimination, and even the fading of a chemical reagent all share the same mathematical signature: whatever amount remains, half of it disappears in a fixed interval, no matter how much or how little was there to start. That interval is the half-life, and it lets you predict remaining quantity at any point in time — or work backward from measurements to find the half-life itself.
The Formula
N0 is the starting amount, t is the elapsed time, and t_half is the half-life in the same time units as t. Given a starting and ending amount instead, the calculator rearranges this to solve for the half-life directly: t_half = t × ln(2) ÷ ln(N0 / N).
Where This Calculation Matters
- Radiometric dating — measuring the remaining fraction of a radioactive isotope reveals the age of a sample.
- Nuclear medicine dosing — radioactive tracers and treatments are timed around their half-life to balance diagnostic signal against radiation exposure.
- Pharmacokinetics — many drugs clear the body following the same exponential decay pattern, which determines dosing intervals.
- Radioactive waste management — storage timelines are planned around how many half-lives are needed to reach a safe activity level.
Fraction Remaining After Successive Half-Lives
| Half-Lives Elapsed | Fraction Remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
| 7 | 0.7812% |
By 5 half-lives, less than 5% of the original quantity remains — a common rule of thumb for when a decaying substance is considered practically gone.
How to Use This Calculator
- Choose whether you're solving for the amount remaining or for the half-life itself.
- To find remaining amount, enter Initial Amount (N0), Elapsed Time, and Half-Life in matching time units.
- To find the half-life, enter Initial Amount (N0), Final Amount (N), and Elapsed Time instead.
- Select Calculate to get the result.
Related Calculations
Working with population growth instead of decay? The biology section's Population Growth Calculator uses the same exponential math in the opposite direction. For simple mole-to-mass conversions of a decaying isotope, see the Moles Calculator.
Mathematical Kinetics of First-Order Decay and Half-Life
The half-life (designated as t1/2) of a quantity is the time required for a decaying entity to decrease to exactly one-half of its initial value. In nuclear physics, radiochemistry, pharmacokinetics, and environmental toxicology, half-life describes processes governed by first-order exponential decay kinetics, where the instantaneous rate of disappearance is strictly proportional to the remaining quantity present at that moment.
Where:
- N(t): Remaining quantity or activity of the decaying substance at elapsed time t.
- N0: Initial baseline quantity or activity at time t = 0.
- t1/2: Half-life duration (seconds, hours, days, or years).
- λ (Lambda, Decay Constant): First-order fractional rate constant: λ = ln(2) / t1/2 ≈ 0.693147 / t1/2.
Nuclear Radiometric Dating and Carbon-14 Chronology
Radiometric geochronology exploits invariant nuclear half-lives as absolute geological clocks:
- Radiocarbon Dating (Carbon-14, t1/2 = 5,730 years): Cosmic rays in the upper atmosphere continuously convert nitrogen-14 into radioactive carbon-14, which enters living biological carbon cycles. Upon organism death, metabolic intake ceases, allowing archaeologists to date organic wood, bones, and textiles up to 50,000 years old.
- Potassium-Argon Dating (K-40 → Ar-40, t1/2 = 1.25 billion years): Used by geologists to date ancient volcanic rock formations and early hominid fossil strata.
- Uranium-Lead Chronology (U-238 → Pb-206, t1/2 = 4.468 billion years): Establishes the absolute crystallization age of zircon minerals and the 4.54-billion-year age of Earth.
Pharmacokinetics and Drug Elimination Clearance
In clinical pharmacology, drug biological half-life determines dosage frequency and steady-state blood plasma concentrations:
- Five Half-Life Elimination Rule: Clinicians consider a therapeutic drug to be effectively cleared (>96.8% eliminated) from the human body after five consecutive elimination half-lives (1/25 = 3.125% remaining).
- Steady-State Accumulation: Continuous maintenance IV infusion achieves 96.9% of its final steady-state therapeutic plasma concentration after approximately 5 elimination half-lives.
Step-by-Step Worked Calculation Example
Example: Radiopharmaceutical Iodine-131 Thyroid Treatment Decay
Problem: A nuclear medicine patient receives an oral therapeutic dose of Iodine-131 (radioactive half-life t1/2 = 8.02 days) with an initial administered radioactivity of 500 Megabecquerels (MBq). Calculate: (1) The radioactive decay constant λ in days-1; and (2) The remaining radioactivity after exactly 24.0 days.
Step 1: Calculate the first-order decay constant λ:
λ = ln(2) / t1/2 = 0.693147 / 8.02 days = 0.086427 day-1
Step 2: Determine elapsed half-life cycles (n):
n = t / t1/2 = 24.0 days / 8.02 days = 2.9925 cycles
Step 3: Calculate remaining radioactivity N(t):
N(t) = N0 × (0.5)2.9925 = 500 MBq × 0.12565 = 62.83 MBq
Alternatively using exponential form: N(t) = 500 × e(-0.086427 × 24.0) = 500 × e-2.0742 = 62.83 MBq
Conclusion: After 24.0 days (approx. 3 half-lives), 62.83 MBq of active Iodine-131 remains (12.57% of original dose).
Common Pitfalls in Half-Life Calculations
- Mixing Linear Decay with Exponential Kinetics: First-order decay never decreases by a fixed absolute amount per unit time; the decay rate drops continuously as the remaining parent isotope diminishes.
- Time Unit Discordance: Ensure that the elapsed duration t and half-life t1/2 share identical time dimensions (e.g., hours, days, or years) before exponentiation.
Environmental Ecotoxicology and Pesticide Soil Persistence
In environmental chemistry and agricultural science, the environmental persistence of pesticides, herbicides, and persistent organic pollutants (POPs) is quantified by field dissipation half-life (DT50). Chemical compounds like glyphosate, atrazine, and PFAS "forever chemicals" undergo microbial biodegradation, photolysis, and soil sorption governed by pseudo-first-order kinetics:
Hydrological modelers use DT50 values to simulate agricultural runoff into municipal groundwater aquifers and predict long-term ecological bioaccumulation across food webs.
Nuclear Reactor Poisoning: Xenon-135 Pit and Equilibrium
In commercial nuclear power reactors, Xenon-135 is a potent fission product with an enormous thermal neutron absorption cross-section (2.6 × 106 barns) and a radioactive half-life of 9.14 hours. Following an emergency reactor shutdown (scram), decaying Iodine-135 (t1/2 = 6.57 hours) continues to feed Xenon-135 production, causing xenon concentrations to peak approximately 10 to 12 hours post-shutdown (the "xenon pit"). Nuclear reactor operators calculate exact half-life decay kinetics to determine when sufficient excess reactivity exists to safely restart the reactor.
Tritium Radioluminescent Emergency Exit Sign Lifespans
Commercial aviation and institutional buildings employ self-luminous emergency exit signs powered by gaseous Tritium (Hydrogen-3, ³H, half-life t1/2 = 12.32 years). Low-energy beta particles continuously excite zinc sulfide phosphor coatings, producing greenish luminescence that requires zero external electrical wiring or battery backup. Safety regulations require sign replacement after 10 to 20 years as radioactive decay diminishes luminous flux below mandatory fire safety egress thresholds.
Positron Emission Tomography (PET) Radiotracer Synthesis
In diagnostic oncology, Fluorine-18 labeled fluorodeoxyglucose (¹&sup8;F-FDG) has a short radioactive half-life of 109.8 minutes. Medical cyclotrons synthesize ¹&sup8;F-FDG on-demand daily, requiring rapid hospital logistics before decay limits patient imaging efficacy.