Half Life Calculator – Radioactive Decay, Remaining Quantity & Decay Constant
Solve for any variable · Step-by-step solutions · Interactive decay chart · Isotope presets
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| n (Half-Lives) | Time | N(t) | % Remaining |
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Click any isotope to load its half-life into the calculator.
Enter any one value. The other two calculate automatically.
A half life calculator solves the one question radioactive decay always raises. How much is left, and when? Whether you are working a chemistry problem or checking how long caffeine stays in your system, the math is the same.
Half life appears in nuclear physics, pharmacology, environmental science, and archaeology. The formula looks the same in all of them. Only the numbers change.
This guide covers every formula version, every reaction order, and the real-world cases competitors skip entirely.
What Is Half Life? A Definition That Actually Sticks
Half life is the time it takes for exactly half of a quantity to decay or be eliminated. Not all of it. Just half.
After one half-life, 50% remains. After two, 25%. After three, 12.5%. The same fraction disappears every cycle, which is what makes it exponential.
Three types of half-life matter in practice:
- Physical half-life: time for radioactive atoms to decay by half
- Biological half-life: time for the body to eliminate half of a substance
- Effective half-life: combination of both, used in nuclear medicine and pharmacology
Most competitors only cover radioactive decay. But biological and effective half-life matter just as much in medicine.
The Half Life Formula, All Versions in One Place
Core Radioactive Decay Formula
The standard exponential decay formula is:
N₀ is the initial quantity. N(t) is what remains after time t. λ is the decay constant. This form is most common in physics and calculus courses.
Half-Life Formula Using t½ Directly
If you know the half-life but not the decay constant, use:
This version skips λ entirely. It works in algebra 2 and pre-calculus without needing e or natural logs.
First-Order Reaction Half-Life
In chemistry, first-order reactions use the rate constant k:
This is the most tested version in AP Chemistry and on the MCAT. The half-life does not depend on initial concentration, which is the defining feature of first-order kinetics.
Second-Order and Zero-Order Half-Life Formulas
These two are where most content stops. They should not:
Zero order: t½ = [A]₀ / (2k)
Both depend on initial concentration. As [A]₀ decreases, second-order half-life increases. Zero-order half-life decreases. That is the opposite behavior from first order.
Effective Half-Life Formula for Drugs and Medicine
When a substance decays both physically and biologically, the effective half-life is shorter than either alone:
Used heavily in nuclear medicine dosimetry and in clinical pharmacokinetics. Most online calculators ignore this formula entirely.
The Half-Life Equation Explained with Derivation
Where the Equation Comes From
Radioactive decay follows a first-order differential equation:
Integrating both sides gives N(t) = N₀ × e^(−λt). Set N(t) equal to N₀/2, then solve for t. The result is t½ = ln(2) / λ.
That derivation is what most calculus students need. The algebra version skips the integration and starts from the solved form.
Decay Constant, Mean Lifetime, and Half-Life
Three parameters describe the same decay process:
- λ (decay constant) = ln(2) / t½
- τ (mean lifetime) = 1 / λ
- t½ = ln(2) × τ ≈ 0.6931 × τ
Mean lifetime is always longer than half-life. Specifically, τ = t½ / 0.6931, about 44% longer. Most students confuse the two until they see this comparison directly.
How to Find k From Half-Life
Rearrange t½ = ln(2)/k to get k = 0.693 / t½. Units of k match the inverse of the time unit used for t½.
Example: Carbon-14 has a half-life of 5,730 years. k = 0.693 / 5730 = 1.21 × 10⁻⁴ yr⁻¹.
How to Calculate Half-Life, Step by Step
When You Know N₀, N(t), and Time
Solve for λ first: λ = ln(N₀ / N(t)) / t. Then compute t½ = 0.693 / λ.
Example: a sample started at 400g and now measures 50g after 21 days. λ = ln(400/50) / 21 = ln(8)/21 ≈ 0.099 day⁻¹. Half-life = 0.693 / 0.099 = 7 days.
When You Know Half-Life and Want Remaining Quantity
Use N(t) = N₀ × (0.5)^(t / t½). No decay constant needed.
Example: 1000g of Carbon-14, t½ = 5730 years, after 11460 years. That is 2 half-lives. 1000 × 0.5² = 250g remaining.
How to Calculate Without Knowing the Initial Amount
Use the percentage remaining directly. If 25% remains, n = log(0.25) / log(0.5) = 2 half-lives. Then t½ = total time / n.
This approach works for any half-life word problem where N₀ is missing. A fossil showing 12.5% carbon-14 is 3 half-lives old, so 3 × 5730 = 17,190 years.
How to Find Half-Life From a Graph
Locate the time where the quantity first drops to 50% of the starting value. That time is one half-life.
If the graph does not show exactly 50%, read any two points (N₁ at t₁, N₂ at t₂) and use t½ = (t₂ − t₁) × ln(2) / ln(N₁/N₂).
How to Find Half-Life From Rate Constant
For first order: t½ = 0.693 / k. The units of k tell you the time units of t½. If k is in min⁻¹, t½ is in minutes.
For second order, you also need [A]₀. For zero order, you need both [A]₀ and k.
How to Calculate Drug Half-Life
What Drug Half-Life Actually Means
In pharmacokinetics, half-life measures how long plasma concentration takes to drop by half. It is called the elimination half-life.
It is not the same as “how long until the drug is gone.” It is a decay rate, not a cutoff.
The Drug Half-Life Formula
Pharmacologists use:
Vd is the volume of distribution in liters. CL is clearance in L/hr. A large Vd or low clearance both extend half-life. This is why the same drug clears faster in young healthy adults than in elderly patients with reduced liver function.
How Many Half-Lives to Clear a Drug?
The clinical rule is five half-lives for practical clearance:
- After 1 half-life: 50% remains
- After 3 half-lives: 12.5% remains
- After 5 half-lives: ~3% remains
- After 7 half-lives: ~1% remains
Most drug interaction guidelines and surgical prep protocols use five half-lives as the threshold. That is not a hard cutoff but a clinical standard.
Drug Accumulation and Multiple Doses
When a drug is dosed more frequently than it clears, it accumulates. Steady state, the point where intake equals elimination, is reached after 4 to 5 half-lives.
This is why loading doses exist. A higher first dose gets blood levels therapeutic faster, before steady state develops normally.
Half-Life of Common Drugs and Substances
Caffeine Half-Life
Average caffeine half-life in healthy adults is 3 to 7 hours. Pregnancy extends it to up to 15 hours. Cigarette smoking shortens it. Liver disease and certain antibiotics can double it.
A 200mg coffee at 2pm still has 100mg active at 9pm at the short end. At the long end, 100mg may still be present at midnight. That is why late afternoon caffeine disrupts sleep even when it does not feel like it.
Alcohol and ETG Half-Life
Ethanol itself is eliminated at roughly 0.015 g/dL per hour. Its half-life is about 4 to 5 hours. But ETG, the metabolite tested in many workplace and court-ordered screens, has a urine half-life of about 2 to 3 hours in blood and up to 13 hours in urine.
ETG tests can detect alcohol consumption up to 80 hours after drinking, long after blood alcohol returns to zero.
THC Half-Life
Blood half-life of THC is 1 to 3 days for single use. For chronic daily users, the half-life in fat tissue can be 10 to 30 days because THC accumulates in fat and slowly re-releases.
This explains why urine tests detect cannabis use for weeks in regular users, even though the subjective effects last only hours.
Common Medications Half-Lives
| Drug | Half-Life | Clearance (~5 t½) |
|---|---|---|
| Adderall IR | 9 to 14 hours | ~70 hours |
| Alprazolam (Xanax) | 6 to 27 hours | ~135 hours |
| Hydrocodone | 3.8 hours | ~19 hours |
| Acetaminophen | 2 to 3 hours | ~15 hours |
| Tirzepatide (Zepbound) | ~5 days | ~25 days |
| Ibuprofen | 1.8 to 2 hours | ~10 hours |
Mitragynine, the main active alkaloid in kratom, has an estimated half-life of 9 to 24 hours. However, peer-reviewed human pharmacokinetic data is limited. Published figures come largely from case reports and animal models, not controlled studies.
Radioactive Isotopes and Their Half-Lives
Isotopes Used in Radiometric Dating
Different isotopes are used depending on the age range being dated:
Medical Isotopes and Their Half-Lives
Short half-lives are a feature, not a flaw, in medical isotopes. They minimize radiation dose to the patient after the scan or treatment is done.
- Technetium-99m: 6 hours. Most common diagnostic isotope worldwide
- Fluorine-18: 110 minutes. Used in PET scans, produced on-site due to short life
- Iodine-131: 8 days. Standard treatment for thyroid cancer
- Cobalt-60: 5.27 years. External beam radiotherapy source
Nuclear and Environmental Isotopes
Long half-lives determine how dangerous nuclear waste remains and for how long:
- Caesium-137: 30 years. Major fallout concern after Chernobyl and Fukushima
- Strontium-90: 28.8 years. Accumulates in bone like calcium
- Plutonium-239: 24,110 years. Drives the need for deep geological repositories
- Americium-241: 432 years. The alpha source inside smoke detectors
Half-Life in Chemistry by Reaction Order
First-Order Reaction Half-Life
t½ = 0.693 / k. Independent of starting concentration. Always the same regardless of how much you start with.
All radioactive decay is first order. Most drug elimination in the body is also first order. That is why these two fields share the same calculation method.
Second-Order Reaction Half-Life
t½ = 1 / (k × [A]₀). As concentration drops, each successive half-life gets longer. The reaction slows down as reactant is consumed.
Common in bimolecular reactions where two molecules must collide. Reaction rate depends on the concentration of two species simultaneously.
Zero-Order Reaction Half-Life
t½ = [A]₀ / (2k). Half-life shortens as concentration decreases. The reaction proceeds at constant rate regardless of how much reactant remains.
Alcohol metabolism in the liver is approximately zero order at high concentrations. The enzyme alcohol dehydrogenase becomes saturated and processes ethanol at a fixed rate per hour, not proportional to how much is present. That is why drinking more does not speed up elimination.
Half-Life in Different Academic Contexts
Algebra 2 and Pre-Calculus
Use the (1/2)^(t/t½) form. No e, no natural log in the base. Solve for t using common logarithms or log base 0.5.
Word problems typically give you the starting amount, the half-life, and ask for either the remaining amount or time elapsed. Plug and solve.
Calculus
Half-life problems in calculus start from dN/dt = −kN and ask you to derive the exponential solution. The half-life falls out when you solve N(t) = N₀/2 for t.
Separation of variables and integration of 1/N are the core techniques. The result is always t½ = ln(2)/k.
APES (AP Environmental Science)
Focus shifts from math to meaning. Half-life determines how long radioactive contamination persists in soil or water. Ten half-lives is the standard threshold used for “practically safe.”
Caesium-137 at 30 years per half-life means 300 years for practical safety after a nuclear accident. That context appears on APES exams repeatedly.
MCAT and AP Chemistry
Know all three integrated rate laws and their corresponding half-life expressions. First order: t½ = 0.693/k. Second order: t½ = 1/(k[A]₀). Zero order: t½ = [A]₀/(2k).
MCAT questions often give you concentration vs time data and ask you to identify reaction order from the shape of the curve, then find the half-life.
Finding Half-Life From Data: Practical Techniques
From a Table
Scan rows for any point where quantity drops to exactly half of a previous value. The time difference is one half-life. If no such pair exists, use:
From Percent Remaining
Count how many halvings the percentage represents. 50% = 1 half-life. 25% = 2. 12.5% = 3. For arbitrary percentages:
Then t½ = total elapsed time / n. This works for any percentage without needing the decay constant first.
From Decay Rate or Rate Constant
If you have λ (decay constant in s⁻¹), then t½ = 0.693 / λ.
If you have k from a first-order reaction, same formula applies. Units of the answer match the inverse of k’s time unit.
Real-World Applications of Half-Life
Radiocarbon Dating
Living organisms absorb carbon-14 at a constant ratio. After death, C-14 decays with a half-life of 5,730 years. Measuring the remaining fraction tells us when the organism died.
The technique is reliable up to about 50,000 years. Beyond that, the remaining C-14 is too small to measure accurately. Uranium-lead and potassium-argon dating cover older timescales.
Nuclear Waste Storage
Plutonium-239 has a half-life of 24,110 years. Ten half-lives for practical safety means 241,100 years of secure storage. No human institution has existed for that long.
This is not an abstract number. It shapes the engineering requirements of every deep geological repository currently being built or planned.
Medical Dosing and Drug Clearance
Pharmacologists set dosing intervals based on half-life. A drug with a 6-hour half-life dosed every 12 hours will accumulate to steady state after about 5 half-lives (30 hours).
Surgeons require patients to stop anticoagulants 5 half-lives before surgery. Anesthesiologists factor in half-lives when combining agents. Half-life is foundational to safe prescribing.
Radiation Safety After Nuclear Accidents
After Chernobyl, Caesium-137 contaminated large areas of Europe. Its 30-year half-life means activity has dropped by half since 1986, but will take until roughly 2286 to reach one-thousandth of original levels.
Radon-222, with a half-life of only 3.8 days, accumulates in buildings from uranium in the ground. Ventilation solves it quickly because of the short decay time.
Common Half-Life Problems Solved
Carbon-14 Dating Example
A fossil contains 25% of its original carbon-14. Two half-lives have elapsed: 2 × 5,730 = 11,460 years old.
Algebraically: n = log(0.25) / log(0.5) = 2. t = 2 × 5730 = 11,460 years. Same answer both ways.
Drug Clearance Example
Adderall IR, 20mg dose, half-life 10 hours.
- After 10h: 10mg active
- After 20h: 5mg active
- After 40h: 1.25mg active
- After 50h: 0.625mg active (clinically cleared)
The drug is not “gone” after one half-life. Five half-lives (50 hours) is the clinical clearance standard.
How Many Half-Lives for a Safe Radiation Level?
Start with 1,000 Bq. Target: below 10 Bq. Use n = log(1000/10) / log(2) = log(100) / log(2) = 6.64 half-lives.
Round up to 7. This approach works for any isotope, any activity unit, any target threshold.
Frequently Asked Questions
The most common formula is t½ = ln(2) / λ, where λ is the decay constant. For first-order reactions in chemistry: t½ = 0.693 / k. For simple problems without calculus: N(t) = N₀ × (0.5)^(t/t½).
Use t½ = (0.693 × Vd) / CL, where Vd is the volume of distribution in liters and CL is clearance in liters per hour. Patient-specific factors like liver function, age, and weight affect both values and therefore the actual half-life in a given individual.
Because the rate of decay is always proportional to the current amount. The ratio decaying per unit time stays constant. Whether you start with 1000 atoms or 10, the same fraction disappears per half-life. This is not true for zero-order or second-order reactions.
Use the percentage remaining and total time. n = log(% remaining / 100) / log(0.5), then t½ = total time / n. You never need the original quantity when working with ratios or percentages.
The decay constant λ measures the probability of decay per unit time. It relates to half-life by λ = ln(2) / t½ = 0.693 / t½. Mean lifetime τ = 1/λ, which is always longer than half-life by a factor of 1/ln(2) ≈ 1.443.
Very accurate up to about 50,000 years, within a few decades for recent samples. Accuracy requires calibration against tree ring records because atmospheric C-14 levels have varied over time. Contamination and sample handling affect precision more than the decay formula itself.
Conclusion
Half-life is one formula that shows up in physics labs, hospital dosing charts, archaeological digs, and nuclear waste policy. The math does not change. Only the context does.
Use the half life calculator above to solve for any variable. Every result includes the step-by-step derivation so you can see exactly how the answer was reached.
For related calculations, the Math calculators cover elapsed time, date differences, and working hours in one place.