Half-life is the time required for a quantity to reduce to half of its initial value. It is most commonly used in the context of radioactive decay, where it describes how long it takes for half of the atoms in a radioactive sample to decay into a different element or isotope. The concept also applies to drug elimination in pharmacology and any process following exponential decay.

What is the half-life formula?

The half-life formula is N(t) = N₀ × (1/2)^(t/t½), where N(t) is the remaining quantity after time t, N₀ is the initial quantity, t is the elapsed time, and t½ is the half-life. This formula can be rearranged to solve for half-life as t½ = −t × ln(2) / ln(N/N₀), or for elapsed time as t = −t½ × ln(N/N₀) / ln(2).

How many half-lives until a substance is gone?

Mathematically, a substance never completely disappears through half-life decay — it just gets smaller and smaller. However, after about 7 half-lives, less than 1% of the original amount remains (0.78%), and after 10 half-lives, less than 0.1% remains. In practice, most scientists consider a substance effectively gone after 5 to 7 half-lives.

What is the half-life of carbon-14?

Carbon-14 has a half-life of 5,730 years. This property makes it ideal for radiocarbon dating of organic materials up to about 50,000 years old. After this time, the remaining carbon-14 is too small to measure reliably. Living organisms continuously absorb carbon-14 from the atmosphere, but once they die, the carbon-14 decays at this predictable rate.

What does half-life mean for medications?

In pharmacology, half-life refers to the time it takes for the concentration of a drug in your blood to decrease by half. A drug with a short half-life (like ibuprofen at 2 hours) needs to be taken more frequently, while a drug with a long half-life (like fluoxetine at 1-3 days) can be taken less often. It takes approximately 5 half-lives for a drug to be almost completely eliminated from the body.

Can anything change the half-life of a radioactive isotope?

Under normal conditions, no. Radioactive half-life is an intrinsic nuclear property that is unaffected by temperature, pressure, chemical bonding, or electromagnetic fields. However, extreme conditions such as those inside stellar cores or in highly ionized atoms can slightly alter some decay rates. For all practical Earth-based applications, half-life is considered a fixed constant.

Half-Life Calculator — Calculate Radioactive Decay

What Is Half-Life?

Half-life is one of the most fundamental concepts in nuclear physics, chemistry, and pharmacology. It refers to the time required for a quantity to reduce to exactly one-half of its initial value. The term is most commonly associated with radioactive decay, where unstable atomic nuclei lose energy by emitting radiation, but it applies equally to any process that follows exponential decay kinetics.

The concept was first introduced in 1907 by Ernest Rutherford, who discovered that radioactive elements transform into other elements at predictable rates. He observed that regardless of the starting amount of a radioactive substance, it always took the same amount of time for half of the material to decay. This remarkable consistency is what makes half-life such a powerful and reliable measurement in science.

What makes half-life particularly interesting is that it remains constant throughout the decay process. After one half-life, 50% of the original substance remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains. This predictable pattern continues indefinitely, following a smooth exponential curve. In theory, the substance never fully disappears — it simply becomes vanishingly small.

Key insight: Half-life is an intrinsic property of a substance. It cannot be altered by changing external conditions such as temperature, pressure, or chemical environment. A radioactive atom will decay at the same rate whether it is in a laboratory, deep underground, or floating in outer space.

Half-life values in nature span an extraordinary range. Some subatomic particles have half-lives measured in fractions of a second, while certain isotopes like tellurium-128 have half-lives exceeding 10²⁴ years — trillions of times longer than the current age of the universe. This incredible diversity makes half-life calculations essential across many scientific and medical disciplines.

The Half-Life Formula

The mathematical relationship governing half-life is derived from the general exponential decay equation. The core formula that our half-life calculator uses is:

N(t) = N₀ × (1/2)^t/t½

Exponential Decay Formula — Half-Life Form

In this equation, N(t) represents the quantity remaining after time t, N₀ is the initial quantity at time zero, t is the elapsed time, and t½ is the half-life period. The ratio t/t½ tells us how many half-lives have elapsed, which directly determines how much of the original substance remains.

This formula can be rearranged to solve for any of the three unknowns. To find the half-life when you know the initial quantity, remaining quantity, and elapsed time:

t½ = −t × ln(2) / ln(N/N₀)

Solving for half-life

To find the elapsed time when you know the initial quantity, remaining quantity, and half-life:

t = −t½ × ln(N/N₀) / ln(2)

Solving for elapsed time

The natural logarithm of 2 (approximately 0.6931) appears frequently in these equations because it represents the mathematical constant that connects the exponential decay rate to the halving time. The decay constant, often denoted by the Greek letter lambda (λ), is related to the half-life by the equation λ = ln(2) / t½. This decay constant represents the probability per unit time that a given atom will decay.

How to Calculate Half-Life

Calculating half-life problems becomes straightforward once you understand the formula and its rearrangements. Here we walk through detailed examples for each type of calculation that our half-life calculator supports.

Example 1: Finding Remaining Quantity

Suppose you have 800 grams of a radioactive isotope with a half-life of 4 years. How much remains after 12 years?

Step 1: Identify the values: N₀ = 800 g, t½ = 4 years, t = 12 years
Step 2: Calculate number of half-lives: n = 12 / 4 = 3 half-lives
Step 3: Apply the formula: N(t) = 800 × (0.5)³ = 800 × 0.125 = 100 grams

After 12 years (3 half-lives), only 100 grams of the original 800 grams remain. The other 700 grams have decayed into other elements or isotopes.

Example 2: Finding Half-Life

A laboratory starts with 500 mg of a substance. After 20 hours, only 62.5 mg remain. What is the half-life?

Step 1: Identify the values: N₀ = 500 mg, N = 62.5 mg, t = 20 hours
Step 2: Apply the formula: t½ = −20 × ln(2) / ln(62.5/500)
Step 3: Compute: t½ = −20 × 0.6931 / ln(0.125) = −13.863 / −2.0794 = 6.667 hours

Example 3: Finding Elapsed Time

A sample of iodine-131 has a half-life of 8.02 days. If you start with 200 grams, how long until only 25 grams remain?

Step 1: Identify the values: N₀ = 200 g, N = 25 g, t½ = 8.02 days
Step 2: Apply the formula: t = −8.02 × ln(25/200) / ln(2)
Step 3: Compute: t = −8.02 × (−2.0794) / 0.6931 = 16.677 / 0.6931 = 24.06 days

Quick check: Since 25 is one-eighth of 200, exactly 3 half-lives are needed (200 → 100 → 50 → 25). Three half-lives of 8.02 days gives 24.06 days, confirming our answer.

Half-Life in Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting particles or electromagnetic radiation. This is the original and most widely known application of the half-life concept. Every radioactive isotope, or radioisotope, has a characteristic half-life that is unique to that isotope and remains constant under all physical and chemical conditions.

There are several types of radioactive decay, each involving different particles and energy levels:

Alpha decay: The nucleus emits an alpha particle (2 protons and 2 neutrons), reducing the atomic number by 2 and mass number by 4. Common in heavy elements like uranium and thorium.
Beta decay: A neutron converts into a proton (or vice versa), emitting a beta particle (electron or positron). This changes the element while keeping the mass number the same.
Gamma decay: The nucleus releases excess energy as a gamma ray photon without changing the number of protons or neutrons. Often follows alpha or beta decay.

The statistical nature of radioactive decay is a key feature. We cannot predict when any individual atom will decay — it is a fundamentally random quantum mechanical process. However, when dealing with extremely large numbers of atoms (as we always do in practice), the law of large numbers ensures that the overall decay rate is highly predictable. This statistical reliability is what allows scientists to use radioactive isotopes as precise clocks for dating materials and as consistent sources for medical treatments and industrial applications.

Safety note: Radioactive materials require proper handling and shielding. The biological hazard of a radioactive substance depends on its half-life, the type of radiation emitted, and the amount present. Short-lived isotopes can be intensely radioactive but become safe relatively quickly, while long-lived isotopes pose chronic exposure risks.

In nuclear power generation, understanding half-lives is critical for managing nuclear waste. Spent fuel contains a mixture of isotopes with vastly different half-lives. Some fission products like cesium-137 (half-life of 30.17 years) and strontium-90 (28.8 years) remain dangerously radioactive for centuries, while others like plutonium-239 (24,110 years) pose risks for tens of thousands of years. These long time scales make nuclear waste storage one of the greatest engineering challenges of our era.

Half-Life in Pharmacology

In pharmacology and medicine, half-life refers to the time it takes for the concentration of a drug in the blood plasma to decrease to half of its peak level. This pharmacological half-life, also called the elimination half-life or biological half-life, is one of the most important parameters used to determine dosing schedules for medications.

When you take a medication, your body immediately begins processing it through metabolism (primarily in the liver) and excretion (primarily through the kidneys). The rate at which a drug is eliminated follows first-order kinetics in most cases, meaning it obeys the same exponential decay mathematics as radioactive decay. This is why our half-life calculator works equally well for pharmacological calculations.

Pharmacists and physicians use drug half-lives to determine several critical treatment parameters:

Dosing frequency: Drugs with short half-lives need to be taken more frequently to maintain therapeutic levels. For example, ibuprofen has a half-life of about 2 hours, which is why it is typically taken every 4-6 hours.
Time to steady state: It generally takes about 4-5 half-lives for a drug to reach a steady-state concentration in the blood when taken at regular intervals. This is why some medications take several days or weeks to reach full effectiveness.
Washout period: After discontinuing a drug, it takes approximately 4-5 half-lives for the drug to be effectively eliminated from the body. This is important when switching medications or planning surgical procedures.
Drug interactions: Understanding half-lives helps predict how drugs will interact when taken together, as the timing of peak concentrations relative to each other can affect both efficacy and safety.

Clinical rule of thumb: After approximately 5 half-lives, about 97% of a drug has been eliminated from the body (100% × (1/2)⁵ = 3.125% remaining). This is generally considered complete elimination for practical medical purposes.

Some medications are deliberately designed with specific half-life characteristics. Extended-release formulations slow down absorption to effectively lengthen the half-life, allowing once-daily dosing instead of multiple doses. Prodrugs are inactive compounds that the body converts into active drugs, sometimes with different half-life profiles than the active drug itself. Understanding these pharmacokinetic properties is essential for modern drug design and patient care.

Carbon Dating and Half-Life

Radiocarbon dating, commonly known as carbon-14 dating, is one of the most famous practical applications of half-life. Developed by Willard Libby in 1949 (for which he received the Nobel Prize in Chemistry in 1960), this technique revolutionized archaeology, geology, and paleontology by providing a reliable method for determining the age of organic materials up to about 50,000 years old.

The method relies on carbon-14, a naturally occurring radioactive isotope of carbon with a half-life of 5,730 years. Carbon-14 is continuously produced in the upper atmosphere when cosmic rays interact with nitrogen atoms. This radioactive carbon is incorporated into carbon dioxide, which is then absorbed by plants during photosynthesis and subsequently enters the food chain. As a result, all living organisms contain a small but measurable proportion of carbon-14 relative to the stable isotopes carbon-12 and carbon-13.

When an organism dies, it stops absorbing new carbon-14. The existing carbon-14 begins to decay at its characteristic rate, while the stable carbon-12 remains unchanged. By measuring the ratio of carbon-14 to carbon-12 in a sample and comparing it to the known ratio in living organisms, scientists can calculate how many half-lives have elapsed since the organism died, and therefore determine its age.

Carbon Dating Calculation Example

Suppose an archaeological wood sample has 25% of the carbon-14 expected in a living sample. Since 25% represents two half-lives of decay (100% → 50% → 25%), the sample is approximately 2 × 5,730 = 11,460 years old. For samples where the remaining percentage is not a neat power of two, the half-life formula provides exact results.

Carbon dating has limitations. It is only effective for materials younger than about 50,000 years (approximately 8-9 half-lives), after which the remaining carbon-14 is too small to measure reliably. For older materials, scientists use other radiometric dating methods with longer-lived isotopes, such as potassium-argon dating (potassium-40 half-life: 1.25 billion years) or uranium-lead dating (uranium-238 half-life: 4.47 billion years).

Common Half-Lives in Science

Half-life values span an enormous range across different isotopes and substances. Below is a reference of some commonly encountered half-lives organized by field of application.

Radioactive Isotopes

Polonium-214: 164 microseconds — one of the shortest known half-lives among naturally occurring isotopes
Radon-222: 3.82 days — a naturally occurring gas that can accumulate in basements and cause health risks
Iodine-131: 8.02 days — widely used in thyroid cancer treatment and diagnostic imaging
Phosphorus-32: 14.3 days — used in molecular biology research and some cancer treatments
Cobalt-60: 5.27 years — used in radiation therapy and industrial radiography
Tritium (Hydrogen-3): 12.32 years — used in self-illuminating exit signs and as a tracer in environmental studies
Strontium-90: 28.8 years — a dangerous fission product found in nuclear fallout
Cesium-137: 30.17 years — another major fission product and environmental contaminant
Carbon-14: 5,730 years — the basis for archaeological radiocarbon dating
Plutonium-239: 24,110 years — used in nuclear weapons and some reactor designs
Uranium-235: 703.8 million years — the fissile isotope used in nuclear reactors and weapons
Uranium-238: 4.468 billion years — the most abundant uranium isotope, used in geological dating

Common Drug Half-Lives

Aspirin: 15-20 minutes (rapidly converted to salicylic acid, which has a half-life of 2-3 hours)
Ibuprofen: 1.8-2 hours
Acetaminophen (Tylenol): 2-3 hours
Amoxicillin: 1-1.5 hours
Caffeine: 3-5 hours (varies significantly between individuals)
Diazepam (Valium): 20-100 hours (including active metabolites)
Fluoxetine (Prozac): 1-3 days (active metabolite: 4-16 days)
Amiodarone: 40-55 days — one of the longest drug half-lives in clinical use

Did you know? The element bismuth-209 was long considered the heaviest stable isotope, but in 2003, scientists discovered it is actually radioactive with a half-life of approximately 1.9 × 10¹⁹ years — over a billion times the age of the universe. For all practical purposes, it is stable, but technically it is the longest-lived unstable isotope directly measured.

Understanding these values is essential for professionals working in nuclear medicine, radiation safety, environmental science, and pharmacology. Our half-life calculator allows you to quickly compute decay quantities using any of these known half-life values, making it a valuable tool for students, educators, and scientists alike.

Half-Life Calculator