🧪 Half Life Calculator: Decay, Time and Half-Life

What is the Half Life Calculator?

Half-life describes any process where a quantity shrinks by a fixed fraction in equal time steps. After one half-life, half of the original amount remains. After two half-lives, a quarter remains. After three, an eighth. The defining feature is that the same proportion disappears in every equal interval, which is what makes the decay exponential rather than linear. This pattern shows up in radioactive isotopes, drug concentrations in the bloodstream, the discharge of a capacitor, and the cooling of an object toward room temperature.

The working formula is N = N0 times 0.5^(t/T), where N0 is the initial amount, N is the amount remaining after time t, and T is the half-life. The exponent t/T is simply the number of half-lives that have elapsed. If t/T equals 1 you have one half-life and N is half of N0; if t/T equals 3.5 you have three and a half half-lives and N is 0.5^3.5, or about 8.8 percent, of N0. Because the base is one half, every whole step in the exponent halves the amount again. The same equation rearranges to answer the reverse questions. To find the elapsed time from a known fraction remaining, use t = T times log base 2 of (N0/N). To find the half-life from a measured decay, use T = t divided by log base 2 of (N0/N). These forms come straight from taking a base-2 logarithm of the original equation, and they are exactly what this calculator applies when you switch the solve-for selector. The amount units cancel out, so only the ratio N0/N matters, not whether you measure in grams, milligrams, counts per minute, or percent.

Scientists and textbooks often write the same decay using a different base: N = N0 times e^(-k times t), where e is Euler's number and k is the decay constant. This is the same curve, just expressed with a natural exponential instead of a base-2 one, and it is the form used in most physics derivations because it falls straight out of the differential equation dN/dt = -kN. The two forms convert into each other through k = ln(2) / T, so a half-life of 10 years corresponds to a decay constant of about 0.0693 per year. The related mean lifetime, tau, equals 1/k and is always a bit longer than the half-life (tau = T / ln 2, about 1.44 times T). Mixing up half-life, decay constant and mean lifetime is one of the most common sources of wrong answers in homework and lab reports, because a source that reports the time constant or lambda is not reporting the half-life directly.

In radioactivity specifically, half-life connects to activity, the rate at which a sample emits decay events. Activity A equals the decay constant k times the number of remaining atoms N, measured in becquerels (one decay per second) or the older unit curies. Because both k and N change over time in opposite ways inside that product, activity itself also falls off with the same half-life T as the quantity of material, just measured in decays per second instead of grams or atoms. This is why a radiation source becomes measurably safer well before the material is gone: after 5 half-lives about 96.9 percent of the original activity has decayed away, and after 10 half-lives less than 0.1 percent remains, even though in principle a tiny amount of the isotope is always technically still present.

The same exponential decay model underlies pharmacokinetics, where it is called the elimination half-life of a drug: the time for its concentration in the blood to fall by half through metabolism and excretion. Clinicians use a well-known rule of thumb that a drug is essentially cleared from the body after about 5 half-lives, since 0.5^5 is roughly 3.1 percent of the starting dose (50%, 25%, 12.5%, 6.25%, 3.125% remaining after each successive half-life). That is also roughly how long it takes a repeated fixed dose to build up to its steady-state concentration. Not every substance follows this model, though: ethanol (alcohol) is metabolized at a roughly constant rate regardless of concentration, called zero-order kinetics, so it does not have a true half-life in the exponential sense, unlike caffeine or most prescription medications, which follow first-order kinetics and do decay half-life style.

Carbon-14 dating is one of the best-known real-world applications of this math outside a physics classroom. Living organisms maintain a roughly constant ratio of radioactive carbon-14 to stable carbon-12 while they are alive, because they keep exchanging carbon with the atmosphere. Once an organism dies, that exchange stops and the carbon-14 it contains decays away with a half-life of 5,730 years. By measuring how much carbon-14 remains relative to the expected starting ratio, scientists can solve the rearranged formula for elapsed time, t = T times log base 2 of (N0/N), to estimate how long ago the organism died. This method is reliable for dates up to roughly 50,000 years, after which the remaining carbon-14 becomes too little to measure accurately.

When to use it

• Working out how much of a radioactive sample is left after a given number of years for a physics or chemistry problem.
• Estimating how long a medication stays meaningfully active in the body based on its published elimination half-life.
• Finding an unknown half-life from a measured before-and-after amount and the time between the two readings.
• Checking carbon-14 dating style calculations, where the fraction of the original isotope reveals the age of a sample.
• Figuring out how many half-lives, and roughly how long, it takes a substance to reach a "negligible" or "safe" leftover level.
• Converting between a stated half-life and a decay constant or mean lifetime when a source uses a different convention.

How to use the Half Life Calculator

1. Choose what to solve for: the remaining amount, the elapsed time, or the half-life.
2. Enter the initial amount (N0) and the other two known values in the same time unit.
3. Read the headline answer, which updates instantly as you type.
4. Check the half-lives elapsed and percent remaining panels to sanity-check the result.

Formula & method

Worked examples

Fraction and percent remaining after each half-life

Half-life of some well-known isotopes

Commonly cited elimination half-lives (approximate, varies by individual and dose)

Common mistakes to avoid

• Mixing time units between t and T. The elapsed time and the half-life must use the same unit, because only the ratio t/T matters. If the half-life is in days, the elapsed time must also be in days. Plugging in 16 hours against an 8-day half-life gives a wildly wrong answer.
• Confusing half-life with the decay constant or mean lifetime. The half-life T, the decay constant k, and the mean lifetime tau are three different numbers describing the same decay, linked by k = ln(2)/T and tau = 1/k. If a problem gives you k or tau instead of T, convert it first, or your answer will be off by a factor of about 0.693 or 1.44.
• Treating decay as linear. Decay does not subtract a fixed amount each step; it multiplies by one half each half-life. After two half-lives you have a quarter left, not zero. Assuming the quantity reaches exactly zero after a fixed small number of half-lives is a frequent error; technically it only approaches zero.
• Entering a remaining amount larger than the initial. In a decay process the remaining amount N can never exceed the initial amount N0. If N is bigger than N0 the formula has no valid solution, so double-check that you have not swapped the two values.
• Assuming every substance decays exponentially. Not all elimination follows a half-life pattern. Alcohol (ethanol) is broken down by the liver at a roughly fixed amount per hour (zero-order kinetics) rather than a fixed percentage, so it does not have a true half-life the way caffeine or a radioactive isotope does. Applying this calculator to a zero-order process will give a misleading answer.
• Using a single half-life as a hard safety cutoff. A half-life tells you when half the material or drug is gone, not when it is gone entirely. Radiation and drug safety guidance is generally based on the accepted rule that about 5 half-lives (roughly 97 percent decayed) are needed before a quantity is considered negligible, not just one.

Glossary

Half-life (T)

The time required for a quantity to decay to half of its current value. It is constant for a given decay process regardless of the starting amount.

Initial amount (N0)

The starting quantity before any decay has taken place, measured at time t = 0.

Remaining amount (N)

The quantity left after a time t has elapsed, given by N = N0 times 0.5^(t/T).

Exponential decay

A process where a quantity decreases by the same proportion in each equal time interval, producing a curve that flattens toward zero but mathematically never reaches it.

Decay constant (k)

The rate parameter of decay in the natural-exponential form N = N0 e^(-kt), equal to ln(2) divided by the half-life. It sets how fast the quantity falls.

Mean lifetime (tau)

The average time a single particle or molecule survives before decaying, equal to 1/k, which is about 1.44 times longer than the half-life.

Activity (A)

In radioactivity, the rate of decay events per second, measured in becquerels or curies. Activity falls off with the same half-life as the quantity of material.

Number of half-lives

The value t/T, telling you how many halving steps have occurred. A value of 3 means the amount has halved three times.

Frequently asked questions

Sources

• Half-life and radiometric dating , OpenStax Physics (2024)
• Elimination Half-Life of Drugs , StatPearls, National Center for Biotechnology Information (NCBI) (2023)